title
Algorithms Course - Graph Theory Tutorial from a Google Engineer

description
This full course provides a complete introduction to Graph Theory algorithms in computer science. Knowledge of how to create and design excellent algorithms is an essential skill required in becoming a great programmer. You will learn how many important algorithms work. The algorithms are accompanied by working source code in Java to solidify your understanding. 💻 Code: https://github.com/williamfiset/algorithms 🔗 Slides: https://github.com/williamfiset/Algorithms/tree/master/slides/graphtheory 🎥 Course created by William Fiset. Check out his YouTube channel: https://www.youtube.com/channel/UCD8yeTczadqdARzQUp29PJw ⭐️ Course Contents ⭐️ ⌨️ (0:00:00) Graph Theory Introduction ⌨️ (0:13:53) Problems in Graph Theory ⌨️ (0:23:15) Depth First Search Algorithm ⌨️ (0:33:18) Breadth First Search Algorithm ⌨️ (0:40:27) Breadth First Search grid shortest path ⌨️ (0:56:23) Topological Sort Algorithm ⌨️ (1:09:52) Shortest/Longest path on a Directed Acyclic Graph (DAG) ⌨️ (1:19:34) Dijkstra's Shortest Path Algorithm ⌨️ (1:43:17) Dijkstra's Shortest Path Algorithm | Source Code ⌨️ (1:50:47) Bellman Ford Algorithm ⌨️ (2:05:34) Floyd Warshall All Pairs Shortest Path Algorithm ⌨️ (2:20:54) Floyd Warshall All Pairs Shortest Path Algorithm | Source Code ⌨️ (2:29:19) Bridges and Articulation points Algorithm ⌨️ (2:49:01) Bridges and Articulation points source code ⌨️ (2:57:32) Tarjans Strongly Connected Components algorithm ⌨️ (3:13:56) Tarjans Strongly Connected Components algorithm source code ⌨️ (3:20:12) Travelling Salesman Problem | Dynamic Programming ⌨️ (3:39:59) Travelling Salesman Problem source code | Dynamic Programming ⌨️ (3:52:27) Existence of Eulerian Paths and Circuits ⌨️ (4:01:19) Eulerian Path Algorithm ⌨️ (4:15:47) Eulerian Path Algorithm | Source Code ⌨️ (4:23:00) Prim's Minimum Spanning Tree Algorithm ⌨️ (4:37:05) Eager Prim's Minimum Spanning Tree Algorithm ⌨️ (4:50:38) Eager Prim's Minimum Spanning Tree Algorithm | Source Code ⌨️ (4:58:30) Max Flow Ford Fulkerson | Network Flow ⌨️ (5:11:01) Max Flow Ford Fulkerson | Source Code ⌨️ (5:27:25) Unweighted Bipartite Matching | Network Flow ⌨️ (5:38:11) Mice and Owls problem | Network Flow ⌨️ (5:46:11) Elementary Math problem | Network Flow ⌨️ (5:56:19) Edmonds Karp Algorithm | Network Flow ⌨️ (6:05:18) Edmonds Karp Algorithm | Source Code ⌨️ (6:10:08) Capacity Scaling | Network Flow ⌨️ (6:19:34) Capacity Scaling | Network Flow | Source Code ⌨️ (6:25:04) Dinic's Algorithm | Network Flow ⌨️ (6:36:09) Dinic's Algorithm | Network Flow | Source Code -- Learn to code for free and get a developer job: https://www.freecodecamp.org Read hundreds of articles on programming: https://www.freecodecamp.org/news

detail
{'title': 'Algorithms Course - Graph Theory Tutorial from a Google Engineer', 'heatmap': [{'end': 11170.893, 'start': 10921.607, 'weight': 1}], 'summary': "Tutorial by a google engineer covers graph theory fundamentals, algorithms, shortest path on grid, topological sorting, shortest & longest paths in directed acyclic graphs, shortest path algorithms, floyd warshall algorithm, bridge finding, graph articulation points and bridges, dynamic programming for the traveling salesman problem, solving tsp and eulerian paths, minimum spanning trees, implementing prim's algorithm and ford fulkerson method, compiling and executing java code, network flow and matching in graphs, and various network flow algorithms with detailed explanations, examples, and complexities.", 'chapters': [{'end': 1041.948, 'segs': [{'end': 52.605, 'src': 'embed', 'start': 24.222, 'weight': 0, 'content': [{'end': 28.163, 'text': 'I think everybody should be able to learn love and enjoy graph theory.', 'start': 24.222, 'duration': 3.941}, {'end': 38.974, 'text': 'These first few videos are going to be a ramp up videos to introduce the topics of how we store represent and traverse graphs on a computer.', 'start': 28.863, 'duration': 10.111}, {'end': 48.784, 'text': 'By the way, this whole video series will be taking on a computer science point of view of graph theory rather than a mathematical one.', 'start': 39.714, 'duration': 9.07}, {'end': 52.605, 'text': "So we won't be covering proofs, and so on per se.", 'start': 49.284, 'duration': 3.321}], 'summary': 'Introducing graph theory for computer science, aiming to make it accessible to all.', 'duration': 28.383, 'max_score': 24.222, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY24222.jpg'}, {'end': 143.978, 'src': 'embed', 'start': 111.937, 'weight': 3, 'content': [{'end': 123.865, 'text': 'But the advantage to graph theory is that it allows us to visualize the problem using nodes to represent an article of clothing and edges to represent relationships between them.', 'start': 111.937, 'duration': 11.928}, {'end': 129.967, 'text': 'Another canonical example of a graph theory problem is a social network of friends.', 'start': 124.445, 'duration': 5.522}, {'end': 143.978, 'text': 'a graph representation enables us to answer interesting questions such as how many friends this person x have or how many degrees of separation are there between person x and person y?', 'start': 129.967, 'duration': 14.011}], 'summary': 'Graph theory visualizes relationships, answers friend network questions.', 'duration': 32.041, 'max_score': 111.937, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY111937.jpg'}, {'end': 280.73, 'src': 'embed', 'start': 226.77, 'weight': 6, 'content': [{'end': 233.574, 'text': 'So an incoming edge represents receiving a gift, and an outgoing edge represents giving a gift.', 'start': 226.77, 'duration': 6.804}, {'end': 238.676, 'text': 'Therefore, person E in this graph bought person D gift,', 'start': 234.194, 'duration': 4.482}, {'end': 247.661, 'text': 'person A bought themselves and person B gift and person F bought nobody any gifts and received none.', 'start': 238.676, 'duration': 8.985}, {'end': 261.277, 'text': "So far we've only seen unweighted graphs, but edges on graphs can contain weights to represent arbitrary values such as cost, distance, quantity,", 'start': 249.033, 'duration': 12.244}, {'end': 261.777, 'text': 'you name it.', 'start': 261.277, 'duration': 0.5}, {'end': 267.079, 'text': 'weighted graphs come in both directed and undirected flavors.', 'start': 262.437, 'duration': 4.642}, {'end': 277.827, 'text': 'As a side note, I will usually denote an edge of a graph as a triplet u vw to indicate where an edge is coming from,', 'start': 267.919, 'duration': 9.908}, {'end': 280.73, 'text': "where it's going to and what its weight is.", 'start': 277.827, 'duration': 2.903}], 'summary': 'Graph theory: edges represent gifts, weighted graphs can capture arbitrary values and come in directed or undirected flavors.', 'duration': 53.96, 'max_score': 226.77, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY226770.jpg'}, {'end': 843.529, 'src': 'embed', 'start': 810.671, 'weight': 2, 'content': [{'end': 815.293, 'text': "However, it lacks structure, and that's why it is seldomly used.", 'start': 810.671, 'duration': 4.622}, {'end': 819.618, 'text': 'Advantage to the edge list is as great for sparse graphs.', 'start': 816.434, 'duration': 3.184}, {'end': 823.743, 'text': 'iterating over all the edges is super easy and the structure is simple.', 'start': 820.179, 'duration': 3.564}, {'end': 831.472, 'text': 'The downside is that edge lookup can be slow and you can run into memory issues on large graphs.', 'start': 824.263, 'duration': 7.209}, {'end': 835.702, 'text': "Today I'm going to talk about common problems in graph theory.", 'start': 832.159, 'duration': 3.543}, {'end': 843.529, 'text': 'A lot of problems you will encounter can often be reduced to a famous or well known problem or some variant thereof.', 'start': 836.223, 'duration': 7.306}], 'summary': 'Graph theory problems can often be reduced to well-known problems with advantages and downsides to edge lists.', 'duration': 32.858, 'max_score': 810.671, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY810671.jpg'}, {'end': 940.269, 'src': 'embed', 'start': 860.962, 'weight': 1, 'content': [{'end': 868.868, 'text': "I want you to think about how you would store and represent the graphs and the upcoming problems I'm going to describe.", 'start': 860.962, 'duration': 7.906}, {'end': 874.813, 'text': "In particular, is the graph and the problem I'm describing directed or undirected??", 'start': 869.388, 'duration': 5.425}, {'end': 878.314, 'text': 'are the edges of the graph weighted or unweighted??', 'start': 875.473, 'duration': 2.841}, {'end': 885.676, 'text': 'Is the common use case a graph that is likely to be sparse or dense with edges?', 'start': 879.114, 'duration': 6.562}, {'end': 895.858, 'text': 'And lastly, should I use an adjacency matrix and adjacency list and edge lists or some other structure to represent my graph efficiently?', 'start': 886.376, 'duration': 9.482}, {'end': 904.227, 'text': 'So one of the most, if not the most common problem in graph theory is the shortest path problem.', 'start': 897.222, 'duration': 7.005}, {'end': 911.552, 'text': 'Given a weighted graph, find the shortest path of edges from node A to node B.', 'start': 904.768, 'duration': 6.784}, {'end': 918.738, 'text': "So if we pretend this graph represents a road system and we're at node A and want to get to node H,", 'start': 911.552, 'duration': 7.186}, {'end': 927.762, 'text': 'our shortest path algorithm should be able to find us a list of edges to follow that will lead us from A to H with minimal cost.', 'start': 919.418, 'duration': 8.344}, {'end': 929.743, 'text': 'Lucky for us.', 'start': 928.683, 'duration': 1.06}, {'end': 940.269, 'text': "many algorithms exist to solve the shortest path problem, including a breadth first search for unweighted graphs, Dijkstra's algorithm, Bellman, Ford,", 'start': 929.743, 'duration': 10.526}], 'summary': 'Consider graph representation, edge types, and solve shortest path problem efficiently using various algorithms.', 'duration': 79.307, 'max_score': 860.962, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY860962.jpg'}], 'start': 0.996, 'title': 'Graph theory fundamentals', 'summary': 'Covers an introduction to graph theory, practical applications in computer science, diverse algorithms, and real-world problem-solving with a focus on types of graphs, weighted graphs, special graph types, graph representation, and common graph theory problems, addressing their space efficiency, time complexity, and real-life applications.', 'chapters': [{'end': 247.661, 'start': 0.996, 'title': 'Introduction to graph theory', 'summary': 'Introduces the concept of graph theory and its practical applications, focusing on computer science implementation, diverse algorithms, and real-world problem-solving, with a goal to make graph theory accessible and applicable. it covers the types of graphs, including undirected and directed graphs, and their significance in problem-solving and programming.', 'duration': 246.665, 'highlights': ['Graph theory is a diverse field hugely applicable to real-world applications. Graph theory is described as hugely applicable to real-world applications.', 'The video series focuses on computer science point of view of graph theory rather than a mathematical one. The series emphasizes a computer science perspective on graph theory.', 'Graphs can be used to represent almost any problem, making them interesting and widely applicable. Graphs are versatile and widely applicable for representing various problems.', 'Introduction to types of graphs, including undirected and directed graphs, and their significance in problem-solving and programming. Introduction to the types of graphs, emphasizing their significance in problem-solving and programming.']}, {'end': 439.242, 'start': 249.033, 'title': 'Weighted graphs and special graph types', 'summary': 'Introduces weighted graphs, special types of graphs such as trees, rooted trees, and directed acyclic graphs, which are important in computer science, and discusses their properties and applications in various scenarios.', 'duration': 190.209, 'highlights': ['Introduction to weighted graphs, which can contain values such as cost, distance, quantity, and come in both directed and undirected flavors. weighted graphs can contain arbitrary values such as cost, distance, quantity', 'Explanation of the special type of graph, a tree, as an undirected graph with no cycles, and its equivalent definition as a graph with n nodes and n minus one edges. a tree is an undirected graph with no cycles, and has n nodes and n minus one edges', 'Introduction to rooted trees, which have a designated root node and can be either an arborescence (out tree) or an anti-arborescence (entry). rooted trees have a designated root node and can be an arborescence or an anti-arborescence', 'Explanation of directed acyclic graphs (DAGs), which are graphs with directed edges and no cycles, and their importance in computer science for representing structures with dependencies. DAGs are important in computer science for representing structures with dependencies', 'Discussion of efficient algorithms for dealing with directed acyclic graphs, such as finding the shortest path and producing a topological ordering of nodes. efficient algorithms for dealing with DAGs, such as finding the shortest path and producing a topological ordering of nodes']}, {'end': 835.702, 'start': 440.403, 'title': 'Types of graphs and graph representation', 'summary': 'Discusses types of graphs including bipartite, complete, and their relevance in network flow. it also explains different graph representations such as adjacency matrix, adjacency list, and edge list, highlighting their space efficiency and time complexity. the chapter also addresses common problems in graph theory.', 'duration': 395.299, 'highlights': ['The chapter discusses types of graphs including bipartite, complete, and their relevance in network flow. It explains the concept of bipartite graphs and their application in matching jobs to people, illustrating that the maximum matching on a bipartite graph can be determined. It also highlights the significance of complete graphs as the worst case possible graph due to the number of edges.', 'It also explains different graph representations such as adjacency matrix, adjacency list, and edge list, highlighting their space efficiency and time complexity. The chapter gives an overview of three graph representations: adjacency matrix, adjacency list, and edge list, outlining their advantages and disadvantages based on space efficiency, time complexity, and suitability for dense or sparse graphs.', 'The chapter also addresses common problems in graph theory. It briefly touches upon common problems in graph theory without providing specific examples or details.']}, {'end': 1041.948, 'start': 836.223, 'title': 'Graph theory problems', 'summary': 'Discusses common graph theory problems including shortest path, connectivity, and detecting negative cycles, with examples of algorithms to solve them and their real-life applications.', 'duration': 205.725, 'highlights': ["The shortest path problem is a common problem in graph theory, where algorithms like breadth first search, Dijkstra's algorithm, and Bellman-Ford can be used to find the shortest path of edges from one node to another with minimal cost.", 'Connectivity in graph theory involves determining if there exists a path from one node to another, and solutions include using a union find data structure or basic search algorithms like depth first search or breadth first search.', 'Detecting negative cycles in a directed graph is important in scenarios with negative edge weights, as it can disrupt calculations, and it can be beneficial in contexts like currency trading where inconsistencies in currency exchange prices may arise.']}], 'duration': 1040.952, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY996.jpg', 'highlights': ['Graph theory is hugely applicable to real-world applications.', 'The video series emphasizes a computer science perspective on graph theory.', 'Graphs are versatile and widely applicable for representing various problems.', 'Introduction to types of graphs, emphasizing their significance in problem-solving and programming.', 'Introduction to weighted graphs, which can contain arbitrary values such as cost, distance, quantity.', 'A tree is an undirected graph with no cycles, and has n nodes and n minus one edges.', 'Rooted trees have a designated root node and can be an arborescence or an anti-arborescence.', 'DAGs are important in computer science for representing structures with dependencies.', 'Efficient algorithms for dealing with DAGs, such as finding the shortest path and producing a topological ordering of nodes.', 'The chapter discusses types of graphs including bipartite, complete, and their relevance in network flow.', 'It explains different graph representations such as adjacency matrix, adjacency list, and edge list, highlighting their space efficiency and time complexity.', "The shortest path problem is a common problem in graph theory, where algorithms like breadth first search, Dijkstra's algorithm, and Bellman-Ford can be used to find the shortest path of edges from one node to another with minimal cost.", 'Connectivity in graph theory involves determining if there exists a path from one node to another, and solutions include using a union find data structure or basic search algorithms like depth first search or breadth first search.', 'Detecting negative cycles in a directed graph is important in scenarios with negative edge weights, as it can disrupt calculations.']}, {'end': 2421.787, 'segs': [{'end': 1108.858, 'src': 'embed', 'start': 1064.224, 'weight': 0, 'content': [{'end': 1070.91, 'text': 'There are two well known algorithms that can detect negative cycles and those are Bellman Ford and Floyd to Warshall.', 'start': 1064.224, 'duration': 6.686}, {'end': 1078.55, 'text': 'something that comes up now and again is finding strongly connected components within a graph.', 'start': 1071.866, 'duration': 6.684}, {'end': 1083.814, 'text': 'This is analogous to finding connected components of an undirected graph.', 'start': 1079.331, 'duration': 4.483}, {'end': 1087.096, 'text': 'But for directed graphs.', 'start': 1084.414, 'duration': 2.682}, {'end': 1092.98, 'text': "when looking at strongly connected components, we're looking for self contained cycles within the graph,", 'start': 1087.096, 'duration': 5.884}, {'end': 1099.004, 'text': 'where every vertex in a given cycle should be able to reach every other vertex in that same cycle.', 'start': 1092.98, 'duration': 6.024}, {'end': 1104.392, 'text': 'This is very useful in many algorithms and is usually an intermediate step.', 'start': 1099.707, 'duration': 4.685}, {'end': 1108.858, 'text': "So it's important to know how to find these strongly connected components.", 'start': 1105.013, 'duration': 3.845}], 'summary': 'Bellman ford and floyd to warshall detect negative cycles. strongly connected components are important in algorithms.', 'duration': 44.634, 'max_score': 1064.224, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY1064224.jpg'}, {'end': 1159.289, 'src': 'embed', 'start': 1137.628, 'weight': 7, 'content': [{'end': 1146.318, 'text': 'For example, if your graph is the one on the left, a possible TSP solution is the graph on the right, which has a cost of nine.', 'start': 1137.628, 'duration': 8.69}, {'end': 1151.744, 'text': 'The TSP problem is NP hard, meaning it is computationally challenging problem.', 'start': 1146.838, 'duration': 4.906}, {'end': 1159.289, 'text': 'This is a unfortunate because the TSP problem has several very important applications.', 'start': 1152.285, 'duration': 7.004}], 'summary': 'Tsp problem is np hard with a cost of nine.', 'duration': 21.661, 'max_score': 1137.628, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY1137628.jpg'}, {'end': 1281.84, 'src': 'embed', 'start': 1255.467, 'weight': 2, 'content': [{'end': 1265.033, 'text': 'For example, in the graph below, one of the possible minimum spanning trees is this graph with a least cost of 12.', 'start': 1255.467, 'duration': 9.566}, {'end': 1272.177, 'text': 'Note that all minimum spanning trees of a graph have the same minimal cost, but are not necessarily identical.', 'start': 1265.033, 'duration': 7.144}, {'end': 1281.84, 'text': 'Minimum spanning trees are seen in lots of different applications in computer science, including designing a least cost network circuit design,', 'start': 1273.037, 'duration': 8.803}], 'summary': 'Minimum spanning trees have minimal cost, e.g., 12, and are used in various computer science applications.', 'duration': 26.373, 'max_score': 1255.467, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY1255467.jpg'}, {'end': 1326.071, 'src': 'embed', 'start': 1300.491, 'weight': 6, 'content': [{'end': 1309.9, 'text': 'This last problem I think is the most fascinating and it is about finding the maximum flow through a special type of graph called a flow network.', 'start': 1300.491, 'duration': 9.409}, {'end': 1315.621, 'text': 'flow networks are networks where edge weights represent capacities.', 'start': 1310.356, 'duration': 5.265}, {'end': 1317.182, 'text': 'in some sense,', 'start': 1315.621, 'duration': 1.561}, {'end': 1326.071, 'text': 'capacities might be things like the maximum amount of cars that fit on a road or the maximum amount of volume that can flow through a pipe,', 'start': 1317.182, 'duration': 8.889}], 'summary': 'Finding maximum flow in flow networks with capacity constraints.', 'duration': 25.58, 'max_score': 1300.491, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY1300491.jpg'}, {'end': 1821.786, 'src': 'embed', 'start': 1789.078, 'weight': 9, 'content': [{'end': 1796.879, 'text': 'Also label it with a zero, then backtrack, like you do, a depth first search, then explore node four,', 'start': 1789.078, 'duration': 7.801}, {'end': 1804.581, 'text': 'give it an ID of zero and then finish exploring that component and then move on to the next node in order.', 'start': 1796.879, 'duration': 7.702}, {'end': 1810.483, 'text': 'So go to node one next, then node one, started up for search there.', 'start': 1804.721, 'duration': 5.762}, {'end': 1819.986, 'text': 'So go to node five, label it with a 15 goes to 17, label it with a one, backtrack, go to 16.', 'start': 1810.843, 'duration': 9.143}, {'end': 1821.786, 'text': 'Also label it with a one.', 'start': 1819.986, 'duration': 1.8}], 'summary': 'Using depth-first search, nodes are labeled and explored sequentially.', 'duration': 32.708, 'max_score': 1789.078, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY1789078.jpg'}, {'end': 2058.292, 'src': 'embed', 'start': 2030.265, 'weight': 5, 'content': [{'end': 2038.467, 'text': 'The breadth first search algorithm is particularly useful for one thing finding the shortest path on an unweighted graph.', 'start': 2030.265, 'duration': 8.202}, {'end': 2047.51, 'text': 'a breadth first search starts at a node in the graph and explores its neighbor nodes first, before moving on to explore the next level of neighbors.', 'start': 2038.467, 'duration': 9.043}, {'end': 2051.731, 'text': 'In a sense, a breadth first search explores nodes in layers.', 'start': 2047.93, 'duration': 3.801}, {'end': 2058.292, 'text': 'So if we start a breadth first search at zero, we would visit zero first, then visit all of zeros.', 'start': 2052.251, 'duration': 6.041}], 'summary': 'Breadth first search finds shortest path on unweighted graph by exploring nodes in layers.', 'duration': 28.027, 'max_score': 2030.265, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY2030265.jpg'}, {'end': 2315.103, 'src': 'embed', 'start': 2284.165, 'weight': 8, 'content': [{'end': 2287.406, 'text': 'This array tracks whether or not node i has been visited.', 'start': 2284.165, 'duration': 3.241}, {'end': 2296.529, 'text': 'If the value at index i is true, then the node has either been visited or is being visited and is on the queue.', 'start': 2288.086, 'duration': 8.443}, {'end': 2300.831, 'text': 'And the animation this corresponds to the gray and yellow nodes.', 'start': 2297.19, 'duration': 3.641}, {'end': 2310.44, 'text': "The last thing we'll need is an array called Prev, which will help us reconstruct the shortest path from the start to the end node.", 'start': 2302.076, 'duration': 8.364}, {'end': 2315.103, 'text': 'Initially, this array should be initialized with all null values.', 'start': 2311.181, 'duration': 3.922}], 'summary': 'Array tracks node visitation status and reconstructs shortest path.', 'duration': 30.938, 'max_score': 2284.165, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY2284165.jpg'}], 'start': 1042.589, 'title': 'Graph theory and algorithms', 'summary': 'Discusses using graph theory to detect negative cycles, finding strongly connected components, the traveling salesperson problem, finding bridges and articulation points, minimum spanning tree problem, maximum flow through a flow network, depth first search algorithm, np hard tsp problem, applications of minimum spanning trees, the time complexity of depth first search, using depth first search to identify and count connected components, pseudocode for the algorithm, labeling nodes to identify components, depth first search variant for finding connected components, and the breadth-first search algorithm running in o(v+e) time complexity.', 'chapters': [{'end': 1108.858, 'start': 1042.589, 'title': 'Graph theory and strongly connected components', 'summary': 'Discusses using graph theory to detect negative cycles and emphasizes the importance of finding strongly connected components within a graph, which is crucial in many algorithms as an intermediate step.', 'duration': 66.269, 'highlights': ['Detecting negative cycles through arbitrage using graph theory can lead to risk-free gains by exchanging currencies. (exchanging one currency for another and coming back to the original currency with more money than originally started)', 'The Bellman Ford and Floyd Warshall algorithms are well-known for detecting negative cycles in graphs.', 'Finding strongly connected components in a graph is crucial as it helps in identifying self-contained cycles where every vertex can reach every other vertex in the same cycle, which is an important intermediate step in many algorithms.']}, {'end': 1689.77, 'start': 1109.398, 'title': 'Graph theory and algorithms', 'summary': 'Discusses the traveling salesperson problem, finding bridges and articulation points in a graph, the minimum spanning tree problem, and the maximum flow through a flow network, and the depth first search algorithm, highlighting key points such as np hard tsp problem, applications of minimum spanning trees, and the time complexity of depth first search.', 'duration': 580.372, 'highlights': ['The TSP problem is NP hard, with several important applications, and can be solved using algorithms such as the heeled carb algorithm and approximation algorithms like ant colony optimization.', "Minimum spanning trees have various applications in computer science, including designing least-cost network circuit design and transportation networks, and can be found using algorithms like Kruskal's and Prim's.", 'Flow networks, used to find maximum flow, are crucial in identifying bottlenecks and vulnerabilities in a network, such as volume of water allowed to flow through pipes or the number of cars roads can sustain in traffic.', 'Depth first search algorithm is a core graph theory algorithm with a time complexity of O(V + E), useful for tasks like counting connected components, determining connectivity between nodes, and finding bridges and articulation points.']}, {'end': 1959.143, 'start': 1689.77, 'title': 'Depth first search for connected components', 'summary': 'Explains how to use depth first search to identify and count connected components in a graph, and presents pseudocode for the algorithm. it demonstrates the process of labeling nodes to identify components and describes the depth first search variant for finding connected components.', 'duration': 269.373, 'highlights': ['The chapter explains how to use depth first search to identify and count connected components in a graph It discusses the process of using depth first search to label nodes and count connected components within a graph.', 'It demonstrates the process of labeling nodes to identify components The chapter describes the process of labeling nodes with ID values to identify different components within the graph.', 'It presents pseudocode for the algorithm and describes the depth first search variant for finding connected components The chapter provides pseudocode for the algorithm and outlines a depth first search variant for identifying connected components in a graph.']}, {'end': 2421.787, 'start': 1959.143, 'title': 'Breadth-first search algorithm', 'summary': 'Highlights the breadth-first search algorithm as a fundamental graph traversal algorithm, running in a time complexity of o(v+e) and being particularly useful for finding the shortest path on an unweighted graph, utilizing a queue data structure to explore nodes in layers, and providing pseudocode and methods for solving the problem and reconstructing the path.', 'duration': 462.644, 'highlights': ['The breadth-first search algorithm explores nodes and edges of a graph, running in a time complexity of big O of v plus e, and is particularly useful for finding the shortest path on an unweighted graph. The breadth-first search algorithm explores nodes and edges of a graph, running in a time complexity of big O of v plus e, and is particularly useful for finding the shortest path on an unweighted graph.', 'It utilizes a queue data structure to explore nodes in layers, maintaining a queue to track which node to visit next and adding new nodes to the queue to visit them later. It utilizes a queue data structure to explore nodes in layers, maintaining a queue to track which node to visit next and adding new nodes to the queue to visit them later.', 'The chapter provides pseudocode and methods for solving the problem by executing the breadth-first search and reconstructing the path from the start to the end node. The chapter provides pseudocode and methods for solving the problem by executing the breadth-first search and reconstructing the path from the start to the end node.']}], 'duration': 1379.198, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY1042589.jpg', 'highlights': ['Detecting negative cycles through arbitrage using graph theory can lead to risk-free gains by exchanging currencies.', 'The Bellman Ford and Floyd Warshall algorithms are well-known for detecting negative cycles in graphs.', 'Finding strongly connected components in a graph is crucial as it helps in identifying self-contained cycles where every vertex can reach every other vertex in the same cycle, which is an important intermediate step in many algorithms.', 'The TSP problem is NP hard, with several important applications, and can be solved using algorithms such as the heeled carb algorithm and approximation algorithms like ant colony optimization.', "Minimum spanning trees have various applications in computer science, including designing least-cost network circuit design and transportation networks, and can be found using algorithms like Kruskal's and Prim's.", 'Flow networks, used to find maximum flow, are crucial in identifying bottlenecks and vulnerabilities in a network, such as volume of water allowed to flow through pipes or the number of cars roads can sustain in traffic.', 'Depth first search algorithm is a core graph theory algorithm with a time complexity of O(V + E), useful for tasks like counting connected components, determining connectivity between nodes, and finding bridges and articulation points.', 'The chapter explains how to use depth first search to identify and count connected components in a graph.', 'It demonstrates the process of labeling nodes to identify components.', 'It presents pseudocode for the algorithm and describes the depth first search variant for finding connected components.', 'The breadth-first search algorithm explores nodes and edges of a graph, running in a time complexity of big O of v plus e, and is particularly useful for finding the shortest path on an unweighted graph.', 'It utilizes a queue data structure to explore nodes in layers, maintaining a queue to track which node to visit next and adding new nodes to the queue to visit them later.', 'The chapter provides pseudocode and methods for solving the problem by executing the breadth-first search and reconstructing the path from the start to the end node.']}, {'end': 3339.269, 'segs': [{'end': 2567.205, 'src': 'embed', 'start': 2534.466, 'weight': 4, 'content': [{'end': 2540.028, 'text': 'So I labeled each cell with the numbers zero through six, non inclusive.', 'start': 2534.466, 'duration': 5.562}, {'end': 2546.391, 'text': 'then we actually want to construct an adjacency list and an adjacency matrix based off this grid.', 'start': 2540.028, 'duration': 6.363}, {'end': 2551.393, 'text': "the adjacency list doesn't require any setup, because it's simply a map that we initialize.", 'start': 2546.391, 'duration': 5.002}, {'end': 2558.838, 'text': 'But the adjacency matrix requires us to initialize a matrix of size six by six to represent our graph.', 'start': 2551.653, 'duration': 7.185}, {'end': 2567.205, 'text': "there are six rows and six columns in the new adjacency matrix, because it's how many nodes there are in the grid we're trying to represent.", 'start': 2558.838, 'duration': 8.367}], 'summary': 'Construct adjacency list and matrix from 6x6 grid.', 'duration': 32.739, 'max_score': 2534.466, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY2534466.jpg'}, {'end': 2837.138, 'src': 'embed', 'start': 2806.215, 'weight': 5, 'content': [{'end': 2815.481, 'text': 'This problem statement is an easier version of the problem dungeon master on the caddis online judge, see the problem link in the description.', 'start': 2806.215, 'duration': 9.266}, {'end': 2825.187, 'text': 'The dungeon is a grid of size r by C, and you start at the node with an S character.', 'start': 2816.421, 'duration': 8.766}, {'end': 2828.689, 'text': "And there's an exit at the cell with an E.", 'start': 2825.747, 'duration': 2.942}, {'end': 2833.474, 'text': 'A cell full of rock is indicated by a pound sign or a hashtag.', 'start': 2829.329, 'duration': 4.145}, {'end': 2837.138, 'text': 'And empty cells are represented using a dot.', 'start': 2833.934, 'duration': 3.204}], 'summary': 'Problem: dungeon master on caddis online judge, grid size r by c, start at s, exit at e, rock=#, empty=.', 'duration': 30.923, 'max_score': 2806.215, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY2806215.jpg'}, {'end': 2968.147, 'src': 'embed', 'start': 2937.35, 'weight': 0, 'content': [{'end': 2941.991, 'text': "then you'll need to keep track of the previously visited node for each node.", 'start': 2937.35, 'duration': 4.641}, {'end': 2946.712, 'text': 'Go and rewatch the last video if you need a refresher on how to do that.', 'start': 2942.491, 'duration': 4.221}, {'end': 2953.494, 'text': "I want to talk a little bit about the way we're representing states in our breadth first search.", 'start': 2947.532, 'duration': 5.962}, {'end': 2961.621, 'text': 'So far, we have been storing the next x y position in the queue as an x y pair.', 'start': 2953.974, 'duration': 7.647}, {'end': 2968.147, 'text': 'This works well, but requires an array or an object or wrapper to store the coordinate values.', 'start': 2961.861, 'duration': 6.286}], 'summary': 'Discusses representing states in breadth first search, utilizes x y pairs for queue, suggests revisiting last video for help', 'duration': 30.797, 'max_score': 2937.35, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY2937350.jpg'}, {'end': 3014.835, 'src': 'embed', 'start': 2984.819, 'weight': 6, 'content': [{'end': 2991.202, 'text': "So the alternative approach I'm suggesting is to use one cue for each dimension.", 'start': 2984.819, 'duration': 6.383}, {'end': 2998.746, 'text': 'So in a three dimensional grid, you would have one cue for each of the x, y and z dimensions.', 'start': 2991.542, 'duration': 7.204}, {'end': 3008.231, 'text': "Suppose we're in queuing the coordinate x one y one z one, then we would simply place each coordinate in their respective cues.", 'start': 2999.446, 'duration': 8.785}, {'end': 3014.835, 'text': 'So the x coordinate goes in the x q, the y goes in its own y q, and so on.', 'start': 3008.732, 'duration': 6.103}], 'summary': 'Use one cue for each dimension in a three-dimensional grid.', 'duration': 30.016, 'max_score': 2984.819, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY2984819.jpg'}, {'end': 3204.676, 'src': 'embed', 'start': 3174.7, 'weight': 3, 'content': [{'end': 3181.968, 'text': 'The first thing I do is add the start cells row and column values to the row q and column q.', 'start': 3174.7, 'duration': 7.268}, {'end': 3187.114, 'text': "Then don't forget to mark the start cell as visited because we don't want to go there again.", 'start': 3181.968, 'duration': 5.146}, {'end': 3192.38, 'text': "we're not done our breadth first search until both of our queues are empty.", 'start': 3187.674, 'duration': 4.706}, {'end': 3196.185, 'text': 'I checked the size of the row queue is greater than zero.', 'start': 3192.4, 'duration': 3.785}, {'end': 3204.676, 'text': 'But you can also check that the size of the column queue is greater than zero since their sizes should always be in sync.', 'start': 3196.486, 'duration': 8.19}], 'summary': 'Breadth-first search algorithm adds start cells and marks as visited. it continues until both queues are empty.', 'duration': 29.976, 'max_score': 3174.7, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY3174700.jpg'}], 'start': 2422.267, 'title': 'Shortest path on grid', 'summary': 'Discusses using breadth-first search to find the shortest path on a grid, highlighting the applications of graph theory in solving grid problems and providing techniques for converting grids to adjacency lists or matrices. it also covers solving a shortest path problem on a grid using the direction vector technique, with a focus on a 2d dungeon escape problem, and introduces an alternative approach for representing states in breadth-first search by using separate queues for each dimension.', 'chapters': [{'end': 2755.37, 'start': 2422.267, 'title': 'Shortest path on grid', 'summary': 'Discusses using breadth-first search to find the shortest path on a grid, highlighting the applications of graph theory in solving grid problems and providing techniques for converting grids to adjacency lists or matrices.', 'duration': 333.103, 'highlights': ['The motivation behind learning about grids is that many problems can be represented using a grid, turning into a graph theory problem, and grids are a form of implicit graph, allowing determination of node neighbors based on grid location.', 'Converting a grid to an adjacency list or matrix involves labeling cells with numbers, initializing a map for the adjacency list, and initializing a matrix to represent the graph for the adjacency matrix.', 'The process of converting a grid to an adjacency list or matrix involves reflecting the connections between nodes in the grid, enabling the application of specialized graph algorithms for problem-solving.', 'Using row vectors simplifies the access to neighboring cells from the current position on the grid, and includes checks to ensure that the new position is within the grid boundaries before considering it as a neighboring cell.']}, {'end': 2937.35, 'start': 2755.37, 'title': 'Shortest path problem on a grid', 'summary': 'Discusses the technique of solving a shortest path problem on a grid using the direction vector technique, with a focus on a 2d dungeon escape problem, demonstrating the application of breadth-first search to find the shortest path.', 'duration': 181.98, 'highlights': ['The technique of solving a shortest path problem on a grid using the direction vector technique is easy to code and extends to higher dimensions.', 'The problem involves finding the shortest path to escape a 2D dungeon, composed of unit cubes, with a surrounding solid rock, where it takes one minute to move one unit north, south, east, or west.', 'The approach involves using breadth-first search from the start node to the end node, counting the number of cells traversed during the process, and being mindful of the possibility of not being able to exit the dungeon if the exit is unreachable.', 'The process includes visiting the start node, adding it to the queue, visiting adjacent unvisited neighbors, and continuing the process while avoiding adding rock cells to the queue, terminating early if the exit is reached without visiting all cells in the grid.']}, {'end': 3339.269, 'start': 2937.35, 'title': 'Breadth first search representation', 'summary': 'Introduces an alternative approach for representing states in breadth first search by using separate queues for each dimension, which can lead to less packing and unpacking of values and more efficient handling of multi-dimensional coordinates and grid exploration.', 'duration': 401.919, 'highlights': ['The alternative approach for representing states in breadth first search is to use separate queues for each dimension, which can reduce the need for packing and unpacking values and result in more efficient handling of multi-dimensional coordinates and grid exploration.', 'The alternative approach can scale well in higher dimensions and requires less setup and effort compared to storing the next x y position in the queue as an x y pair.', 'The concept of using separate queues for each dimension is particularly beneficial for working with multi-dimensional coordinates and can lead to more efficient handling of multi-dimensional grid exploration.']}], 'duration': 917.002, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY2422267.jpg', 'highlights': ['Converting a grid to an adjacency list or matrix involves labeling cells with numbers, initializing a map for the adjacency list, and initializing a matrix to represent the graph for the adjacency matrix.', 'The process of converting a grid to an adjacency list or matrix involves reflecting the connections between nodes in the grid, enabling the application of specialized graph algorithms for problem-solving.', 'The motivation behind learning about grids is that many problems can be represented using a grid, turning into a graph theory problem, and grids are a form of implicit graph, allowing determination of node neighbors based on grid location.', 'Using row vectors simplifies the access to neighboring cells from the current position on the grid, and includes checks to ensure that the new position is within the grid boundaries before considering it as a neighboring cell.', 'The technique of solving a shortest path problem on a grid using the direction vector technique is easy to code and extends to higher dimensions.', 'The alternative approach for representing states in breadth first search is to use separate queues for each dimension, which can reduce the need for packing and unpacking values and result in more efficient handling of multi-dimensional coordinates and grid exploration.', 'The approach involves using breadth-first search from the start node to the end node, counting the number of cells traversed during the process, and being mindful of the possibility of not being able to exit the dungeon if the exit is unreachable.', 'The problem involves finding the shortest path to escape a 2D dungeon, composed of unit cubes, with a surrounding solid rock, where it takes one minute to move one unit north, south, east, or west.']}, {'end': 4172.398, 'segs': [{'end': 3661.881, 'src': 'embed', 'start': 3626.854, 'weight': 4, 'content': [{'end': 3631.536, 'text': 'So any graph with a directed cycle is therefore forbidden.', 'start': 3626.854, 'duration': 4.682}, {'end': 3645.562, 'text': 'The only graphs that have valid topological orderings are called directed acyclic graphs, that is graphs with directed edges and no cycles.', 'start': 3633.157, 'duration': 12.405}, {'end': 3654.039, 'text': 'So a natural question to ask is how do I verify that my graph does not contain a directed cycle.', 'start': 3647.136, 'duration': 6.903}, {'end': 3661.881, 'text': 'One method is to use tarjans strongly connected component algorithm, which can detect these cycles.', 'start': 3654.959, 'duration': 6.922}], 'summary': "Graphs with directed cycles are forbidden, valid topological orderings exist in directed acyclic graphs. tarjan's algorithm detects cycles.", 'duration': 35.027, 'max_score': 3626.854, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY3626854.jpg'}, {'end': 3810.551, 'src': 'embed', 'start': 3774.384, 'weight': 0, 'content': [{'end': 3779.827, 'text': 'Now we do a depth first search outwards from h and all possible directions exploring where we can.', 'start': 3774.384, 'duration': 5.443}, {'end': 3783.109, 'text': "Let's go to node j.", 'start': 3781.168, 'duration': 1.941}, {'end': 3788.079, 'text': "Now that I might know J, I'm going to keep exploring.", 'start': 3784.517, 'duration': 3.562}, {'end': 3793.102, 'text': "And so let's go to M.", 'start': 3790, 'duration': 3.102}, {'end': 3795.643, 'text': "Now that we're at M, there's nowhere left to go.", 'start': 3793.102, 'duration': 2.541}, {'end': 3802.767, 'text': 'So we backtrack and add M as last element to the top logical ordering.', 'start': 3795.763, 'duration': 7.004}, {'end': 3808.77, 'text': 'Still at J, we still need to explore L.', 'start': 3804.068, 'duration': 4.702}, {'end': 3810.551, 'text': "Now we're at L.", 'start': 3808.77, 'duration': 1.781}], 'summary': 'Depth-first search explores nodes, reaching m with nowhere left, and backtracking to add m to the topological ordering.', 'duration': 36.167, 'max_score': 3774.384, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY3774384.jpg'}], 'start': 3339.83, 'title': 'Topological sorting and orderings', 'summary': 'Covers topological sorting algorithm, its applications, and the algorithm to find a topological ordering on a directed acyclic graph. it also explains the motivation for top sort and provides examples related to university class enrollment and program build dependencies. additionally, it discusses the concept of topological orderings, exploring the algorithm, pseudocode for top sort, and an optimization to improve its performance.', 'chapters': [{'end': 3867.73, 'start': 3339.83, 'title': 'Topological sorting algorithm', 'summary': 'Covers the concept of topological sort, its applications, and the algorithm to find a topological ordering on a directed acyclic graph. it also explains the motivation for top sort and provides examples related to university class enrollment and program build dependencies.', 'duration': 527.9, 'highlights': ['Top sort algorithm helps in finding a valid ordering for events or tasks in a directed graph, such as university class enrollment or program build dependencies. The top sort algorithm provides a valid ordering for events or tasks in a directed graph, enabling scenarios like university class enrollment and program build dependencies.', 'Topological ordering is used to ensure that no event or task occurs before its dependencies are met, and it is applicable to directed acyclic graphs. Topological ordering ensures that no task occurs before its dependencies are met and is applicable to directed acyclic graphs, providing a structured approach to task scheduling.', 'The algorithm for finding a topological ordering involves a simple process of depth-first search and reverse order addition to the ordering. The algorithm for finding a topological ordering involves a simple process of depth-first search and reverse order addition to the ordering, which is effective for both trees and general directed acyclic graphs.']}, {'end': 4172.398, 'start': 3867.73, 'title': 'Topological orderings and depth first search', 'summary': 'Discusses the concept of topological orderings, exploring the algorithm, pseudocode for top sort, and an optimization to improve its performance.', 'duration': 304.668, 'highlights': ['The chapter explains the concept of topological orderings and points out that they are not unique, leading to the understanding that different predictions are still valid.', 'The pseudocode for top sort is presented, illustrating the use of visited arrays, orderings arrays, and a for loop to iterate over all nodes in the graph.', 'An optimization to improve the performance of the algorithm is discussed, suggesting the direct insertion of found nodes into the orderings array instead of allocating memory for a separate array.']}], 'duration': 832.568, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY3339830.jpg', 'highlights': ['Top sort algorithm provides a valid ordering for events or tasks in a directed graph, enabling scenarios like university class enrollment and program build dependencies.', 'Topological ordering ensures that no task occurs before its dependencies are met and is applicable to directed acyclic graphs, providing a structured approach to task scheduling.', 'The algorithm for finding a topological ordering involves a simple process of depth-first search and reverse order addition to the ordering, which is effective for both trees and general directed acyclic graphs.', 'The chapter explains the concept of topological orderings and points out that they are not unique, leading to the understanding that different predictions are still valid.', 'The pseudocode for top sort is presented, illustrating the use of visited arrays, orderings arrays, and a for loop to iterate over all nodes in the graph.', 'An optimization to improve the performance of the algorithm is discussed, suggesting the direct insertion of found nodes into the orderings array instead of allocating memory for a separate array.']}, {'end': 6385.108, 'segs': [{'end': 4317.915, 'src': 'embed', 'start': 4286.281, 'weight': 2, 'content': [{'end': 4304.509, 'text': 'The essence of the algorithm is that it finds a topological ordering on the graph using the top sort algorithm we saw in the last video and processes each node sequentially to get the shortest path by relaxing each edge as it is seen.', 'start': 4286.281, 'duration': 18.228}, {'end': 4313.173, 'text': 'Relaxing an edge simply means updating to a better value if a shorter path can be obtained using the current edge.', 'start': 4305.169, 'duration': 8.004}, {'end': 4317.915, 'text': "Suppose this is the graph we're working with.", 'start': 4314.133, 'duration': 3.782}], 'summary': 'Algorithm finds shortest path by topological ordering and edge relaxation.', 'duration': 31.634, 'max_score': 4286.281, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY4286281.jpg'}, {'end': 4393.809, 'src': 'embed', 'start': 4366.431, 'weight': 0, 'content': [{'end': 4372.851, 'text': "In this case, since a is the starting node, its initial distance is zero because we're already there.", 'start': 4366.431, 'duration': 6.42}, {'end': 4385.126, 'text': 'From a we want to visit all reachable nodes starting with node B and update the value to be if it is better than what was already there.', 'start': 4373.982, 'duration': 11.144}, {'end': 4393.809, 'text': 'This is the edge relaxation step of the algorithm, we noticed that a value of three is better than infinity.', 'start': 4386.166, 'duration': 7.643}], 'summary': 'Algorithm starts at node a with initial distance zero and updates distance for reachable nodes.', 'duration': 27.378, 'max_score': 4366.431, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY4366431.jpg'}, {'end': 4478.513, 'src': 'embed', 'start': 4456.391, 'weight': 8, 'content': [{'end': 4465.32, 'text': 'Now we move on to the next node in our topological ordering and keep repeating the same steps, where we explore all the nodes,', 'start': 4456.391, 'duration': 8.929}, {'end': 4470.025, 'text': 'try and relax each edge and then move on to the next node in the topological ordering.', 'start': 4465.32, 'duration': 4.705}, {'end': 4478.513, 'text': 'If we repeatedly do this, the array at the bottom of the screen will contain the shortest path from node A to each node.', 'start': 4470.585, 'duration': 7.928}], 'summary': 'Iteratively explore nodes to find shortest paths.', 'duration': 22.122, 'max_score': 4456.391, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY4456391.jpg'}, {'end': 4740.318, 'src': 'embed', 'start': 4714.834, 'weight': 5, 'content': [{'end': 4725.062, 'text': 'then we check, okay, has there ever been a distance set to where we want to go? This is basically the equivalent of infinity.', 'start': 4714.834, 'duration': 10.228}, {'end': 4728.585, 'text': 'And if so, then we just want to give the new distance.', 'start': 4726.103, 'duration': 2.482}, {'end': 4740.318, 'text': "Otherwise we're going to take the minimum of the distance that's already there and our new competing distance, which is this, over here.", 'start': 4729.989, 'duration': 10.329}], 'summary': 'Algorithm checks for existing distance and computes new distance.', 'duration': 25.484, 'max_score': 4714.834, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY4714834.jpg'}, {'end': 4897.818, 'src': 'embed', 'start': 4874.025, 'weight': 9, 'content': [{'end': 4883.269, 'text': "For this slide deck, my goal is to help you understand how implement Dijkstra's algorithm and also how to implement it very efficiently.", 'start': 4874.025, 'duration': 9.244}, {'end': 4889.073, 'text': "We're going to start by looking at the lazy implementation, because it's by far the most common.", 'start': 4884.07, 'duration': 5.003}, {'end': 4897.818, 'text': "And then we'll look at the eager implementation of Dijkstra's algorithm, which uses an index priority queue alongside the decrease key operation.", 'start': 4889.333, 'duration': 8.485}], 'summary': "Help understand and implement dijkstra's algorithm efficiently.", 'duration': 23.793, 'max_score': 4874.025, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY4874025.jpg'}, {'end': 5083.469, 'src': 'embed', 'start': 5052.839, 'weight': 1, 'content': [{'end': 5058.962, 'text': "Internally, priority queues are usually implemented as heaps, but I'm not going to show that visualization here.", 'start': 5052.839, 'duration': 6.123}, {'end': 5066.134, 'text': 'To start with assign a distance of zero to the start nodes index, which is index zero in the distance array.', 'start': 5059.582, 'duration': 6.552}, {'end': 5076.325, 'text': 'Also insert the key value pair zero comma zero into the priority queue to indicate that we intend on visiting node zero with a best distance of zero.', 'start': 5066.539, 'duration': 9.786}, {'end': 5083.469, 'text': 'Then the algorithm actually starts and we look inside the priority queue for the first time and we discover that we should visit node zero.', 'start': 5076.785, 'duration': 6.684}], 'summary': "Dijkstra's algorithm assigns distance 0 to start node index, and indicates visiting node 0 with a best distance of 0.", 'duration': 30.63, 'max_score': 5052.839, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY5052839.jpg'}, {'end': 5310.362, 'src': 'embed', 'start': 5288.076, 'weight': 4, 'content': [{'end': 5297.158, 'text': 'First is G, the adjacency list of the weighted graph, and the number of nodes in the graph, and s the index of the start node.', 'start': 5288.076, 'duration': 9.082}, {'end': 5298.318, 'text': 'Inside the function.', 'start': 5297.418, 'duration': 0.9}, {'end': 5302.179, 'text': "I begin by initializing two arrays to keep track of the information we'll need.", 'start': 5298.318, 'duration': 3.861}, {'end': 5310.362, 'text': 'first is a Boolean array, I called vi s short for visited, which tracks whether node i has been visited or not.', 'start': 5302.179, 'duration': 8.183}], 'summary': 'Initializing arrays to track visited nodes in a weighted graph.', 'duration': 22.286, 'max_score': 5288.076, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY5288076.jpg'}, {'end': 5543.836, 'src': 'embed', 'start': 5514.069, 'weight': 7, 'content': [{'end': 5518.974, 'text': 'this array tracks the index of the node you took to get to node I.', 'start': 5514.069, 'duration': 4.905}, {'end': 5525.981, 'text': 'initially, the previous array should be filled with a sentinel value such as null or minus one.', 'start': 5518.974, 'duration': 7.007}, {'end': 5529.604, 'text': 'But as you perform edge relaxation operations,', 'start': 5526.241, 'duration': 3.363}, {'end': 5536.53, 'text': "you want to update the previous array to say that the node you're going to came from the node you're currently at.", 'start': 5529.604, 'duration': 6.926}, {'end': 5543.836, 'text': 'then, at the end, instead of returning the distance array, also return the previous array, which we will use soon.', 'start': 5536.53, 'duration': 7.306}], 'summary': 'Array tracks node indexes for edge relaxation operations.', 'duration': 29.767, 'max_score': 5514.069, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY5514069.jpg'}, {'end': 6129.301, 'src': 'embed', 'start': 6106.599, 'weight': 6, 'content': [{'end': 6115.389, 'text': "So the question then becomes what is the optimal dairy heap degree I should use to maximize the performance of Dijkstra's algorithm.", 'start': 6106.599, 'duration': 8.79}, {'end': 6122.837, 'text': 'And the answer in general is that the value of D should be equal to the number of edges divided by the number of nodes.', 'start': 6116.49, 'duration': 6.347}, {'end': 6129.301, 'text': 'This is the best degree to use to balance removals against decrease key operations.', 'start': 6123.698, 'duration': 5.603}], 'summary': "Optimal dairy heap degree for dijkstra's algorithm is d = e/n.", 'duration': 22.702, 'max_score': 6106.599, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY6106599.jpg'}], 'start': 4172.398, 'title': 'Shortest & longest paths in directed acyclic graphs', 'summary': "Discusses efficient solutions for finding single source shortest path in directed acyclic graphs with linear time complexity, and also explains the approach for finding the longest path in a graph using negated edge values, providing a github reference for source code. it also covers dijkstra's algorithm, its overview, and optimization techniques including indexed priority queue, indexed binary heap, and fibonacci heap.", 'chapters': [{'end': 4686.808, 'start': 4172.398, 'title': 'Shortest & longest paths in directed acyclic graphs', 'summary': 'Discusses the efficient solution for finding single source shortest path in directed acyclic graphs, with a linear time complexity, and also explains the approach for finding the longest path in a graph using negated edge values, providing a github reference for source code.', 'duration': 514.41, 'highlights': ['Efficient solution for single source shortest path The algorithm for finding the single source shortest path in a directed acyclic graph is efficient with a linear time complexity.', 'Approach for finding the longest path in a graph The approach for finding the longest path in a graph involves negating the edge values, finding the shortest path, and then multiplying the answer by -1.', 'GitHub reference for source code The source code for the shortest path on a directed acyclic graph can be found on GitHub at github.com/Williamfuse/algorithms.']}, {'end': 5287.776, 'start': 4687.928, 'title': "Dijkstra's shortest path algorithm", 'summary': "Explains dijkstra's algorithm as a single source shortest path algorithm for graphs, with a time complexity of typically big o of e log v, and a main constraint of non-negative edge weights for all graph edges. it also outlines the steps required in executing dijkstra's algorithm and emphasizes its efficiency.", 'duration': 599.848, 'highlights': ["Dijkstra's algorithm is a single source shortest path algorithm for graphs, with a time complexity of typically big O of E log V. This algorithm is efficient and has a time complexity of typically big O of E log V, which is competitive against other shortest path algorithms.", "The main constraint for Dijkstra's algorithm is that all edges of the graph need to have a non-negative edge weight. The algorithm requires all edges of the graph to have a non-negative edge weight, ensuring that the optimal distance from the starting node cannot be improved any further by finding a shorter path with a negative weight.", "The steps required in executing Dijkstra's algorithm involve maintaining an array called 'dist' for tracking the shortest distance to every node from the start node and a priority queue of key-value pairs for determining the next node to visit based on minimum sorted value. The algorithm involves maintaining an array called 'dist' for tracking the shortest distance and a priority queue of key-value pairs for determining the next node to visit based on minimum sorted value, which are essential steps in executing Dijkstra's algorithm."]}, {'end': 5901.488, 'start': 5288.076, 'title': "Dijkstra's algorithm overview", 'summary': "Provides an overview of dijkstra's algorithm for finding the shortest path in a weighted graph, covering the initialization, priority queue implementation, edge relaxation, path reconstruction, optimizations, and eager version of the algorithm.", 'duration': 613.412, 'highlights': ['The algorithm initializes a priority queue to store node index best distance pairs sorted by minimum distance, using a built-in priority queue in the programming language. The algorithm initializes a priority queue to store node index best distance pairs sorted by minimum distance, using a built-in priority queue in the programming language.', "The algorithm utilizes edge relaxation to compute the distance to new nodes by adding the best distance from the start node to the current node and the edge cost of getting to the next node, followed by updating the value if it's better. The algorithm utilizes edge relaxation to compute the distance to new nodes by adding the best distance from the start node to the current node and the edge cost of getting to the next node, followed by updating the value if it's better.", "The process includes maintaining a previous array to track the index of the previous node taken to reach the current node, facilitating the reconstruction of the shortest path in another method called 'find shortest path'. The process includes maintaining a previous array to track the index of the previous node taken to reach the current node, facilitating the reconstruction of the shortest path in another method called 'find shortest path'.", 'An optimization involves stopping early by checking if the current node index is the end node, providing a substantial speed up depending on the early encounter of the end node while processing the graph. An optimization involves stopping early by checking if the current node index is the end node, providing a substantial speed up depending on the early encounter of the end node while processing the graph.', "The eager version of Dijkstra's algorithm aims to avoid duplicate key value pairs and support efficient value updates in logarithmic time using an indexed priority queue, which is particularly useful for dense graphs. The eager version of Dijkstra's algorithm aims to avoid duplicate key value pairs and support efficient value updates in logarithmic time using an indexed priority queue, which is particularly useful for dense graphs."]}, {'end': 6385.108, 'start': 5901.928, 'title': "Optimizing dijkstra's algorithm", 'summary': "Discusses optimizing dijkstra's algorithm by using an indexed priority queue, indexed binary heap, and fibonacci heap, and explains the impact of dairy heap degree on the algorithm's performance, recommending the degree value equal to the number of edges divided by the number of nodes.", 'duration': 483.18, 'highlights': ['Indexed priority queue and its advantages over a regular priority queue, including first-class support for key value pairs Using an indexed priority queue instead of a regular priority queue eliminates the need to wrap key value pairs as tuples and provides first-class support for key value pairs.', "Impact of dairy heap degree on Dijkstra's algorithm performance, recommending a degree value equal to the number of edges divided by the number of nodes The optimal dairy heap degree for maximizing the performance of Dijkstra's algorithm is recommended to be equal to the number of edges divided by the number of nodes.", "Fibonacci heap as the current state-of-the-art for Dijkstra's algorithm, with time complexity of O(E + V log V) The Fibonacci heap is considered the best heap for Dijkstra's algorithm, providing a time complexity of O(E + V log V), which is extremely fast."]}], 'duration': 2212.71, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY4172398.jpg', 'highlights': ['Efficient solution for single source shortest path The algorithm for finding the single source shortest path in a directed acyclic graph is efficient with a linear time complexity.', 'Approach for finding the longest path in a graph The approach for finding the longest path in a graph involves negating the edge values, finding the shortest path, and then multiplying the answer by -1.', "Dijkstra's algorithm is a single source shortest path algorithm for graphs, with a time complexity of typically big O of E log V. This algorithm is efficient and has a time complexity of typically big O of E log V, which is competitive against other shortest path algorithms.", "The main constraint for Dijkstra's algorithm is that all edges of the graph need to have a non-negative edge weight. The algorithm requires all edges of the graph to have a non-negative edge weight, ensuring that the optimal distance from the starting node cannot be improved any further by finding a shorter path with a negative weight.", 'The algorithm initializes a priority queue to store node index best distance pairs sorted by minimum distance, using a built-in priority queue in the programming language.', "The algorithm utilizes edge relaxation to compute the distance to new nodes by adding the best distance from the start node to the current node and the edge cost of getting to the next node, followed by updating the value if it's better.", "The process includes maintaining a previous array to track the index of the previous node taken to reach the current node, facilitating the reconstruction of the shortest path in another method called 'find shortest path'.", 'An optimization involves stopping early by checking if the current node index is the end node, providing a substantial speed up depending on the early encounter of the end node while processing the graph.', 'Indexed priority queue and its advantages over a regular priority queue, including first-class support for key value pairs Using an indexed priority queue instead of a regular priority queue eliminates the need to wrap key value pairs as tuples and provides first-class support for key value pairs.', "Impact of dairy heap degree on Dijkstra's algorithm performance, recommending a degree value equal to the number of edges divided by the number of nodes The optimal dairy heap degree for maximizing the performance of Dijkstra's algorithm is recommended to be equal to the number of edges divided by the number of nodes.", "Fibonacci heap as the current state-of-the-art for Dijkstra's algorithm, with time complexity of O(E + V log V) The Fibonacci heap is considered the best heap for Dijkstra's algorithm, providing a time complexity of O(E + V log V), which is extremely fast."]}, {'end': 7724.171, 'segs': [{'end': 6825.567, 'src': 'embed', 'start': 6794.185, 'weight': 7, 'content': [{'end': 6798.688, 'text': 'But right now we are only interested in detecting negative cycles.', 'start': 6794.185, 'duration': 4.503}, {'end': 6802.531, 'text': 'I will label blue nodes as regular nodes,', 'start': 6799.589, 'duration': 2.942}, {'end': 6811.117, 'text': 'red nodes as nodes directly involved in a negative cycle and yellow nodes as those reachable by a negative cycle.', 'start': 6802.531, 'duration': 8.586}, {'end': 6816.805, 'text': 'One way negative cycles can emerge is through negative self loops.', 'start': 6812.484, 'duration': 4.321}, {'end': 6825.567, 'text': 'What happens is that once we reach a self loop, we can stay in that loop for a near infinite amount of time before exiting.', 'start': 6817.505, 'duration': 8.062}], 'summary': 'Detect negative cycles using color-coded nodes and self loops.', 'duration': 31.382, 'max_score': 6794.185, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY6794185.jpg'}, {'end': 7195.311, 'src': 'embed', 'start': 7157.029, 'weight': 5, 'content': [{'end': 7163.096, 'text': 'So this is all the algorithm does processes each edge performing relaxation operations.', 'start': 7157.029, 'duration': 6.067}, {'end': 7167.261, 'text': "I'll let the animation play for the rest of this iteration.", 'start': 7163.617, 'duration': 3.644}, {'end': 7188.105, 'text': 'So iteration one is over and there are eight more iterations to go.', 'start': 7184.221, 'duration': 3.884}, {'end': 7195.311, 'text': "But for simplicity, I'll just play one more iteration to give you an idea of how the algorithm works.", 'start': 7188.665, 'duration': 6.646}], 'summary': 'Algorithm processes each edge, 9 iterations total.', 'duration': 38.282, 'max_score': 7157.029, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY7157029.jpg'}, {'end': 7253.795, 'src': 'embed', 'start': 7224.321, 'weight': 13, 'content': [{'end': 7228.064, 'text': "However, we're not done, we still need to find the negative cycles.", 'start': 7224.321, 'duration': 3.743}, {'end': 7231.206, 'text': "Let's execute the algorithm a second time.", 'start': 7228.724, 'duration': 2.482}, {'end': 7234.688, 'text': 'Same procedure as usual, just relax each edge.', 'start': 7231.866, 'duration': 2.822}, {'end': 7240.292, 'text': 'But when we are able to relax an edge, update the nodes value to negative infinity instead.', 'start': 7235.028, 'duration': 5.264}, {'end': 7245.255, 'text': "Let's process some edges until something interesting happens.", 'start': 7241.753, 'duration': 3.502}, {'end': 7253.795, 'text': 'So it appears that when we went from node two to node three,', 'start': 7249.614, 'duration': 4.181}], 'summary': 'Algorithm executed a second time to find negative cycles.', 'duration': 29.474, 'max_score': 7224.321, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY7224321.jpg'}, {'end': 7295.208, 'src': 'embed', 'start': 7274.059, 'weight': 11, 'content': [{'end': 7282.983, 'text': 'In the distance table, I do not distinguish between nodes which are reachable by a negative cycle, and those which are primarily involved in one.', 'start': 7274.059, 'duration': 8.924}, {'end': 7285.564, 'text': "So there's no way to tell them apart.", 'start': 7283.743, 'duration': 1.821}, {'end': 7290.886, 'text': 'Feel free to add some logic in the Bellman Ford algorithm if you need to make this distinction.', 'start': 7286.084, 'duration': 4.802}, {'end': 7295.208, 'text': 'Continuing on node two is also trapped in the cycle.', 'start': 7291.586, 'duration': 3.622}], 'summary': 'The bellman ford algorithm does not distinguish between nodes reachable by a negative cycle, offering the opportunity to add logic for this distinction.', 'duration': 21.149, 'max_score': 7274.059, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY7274059.jpg'}, {'end': 7384.433, 'src': 'embed', 'start': 7354.135, 'weight': 0, 'content': [{'end': 7359.678, 'text': 'Or you can go to github.com slash William physio slash algorithms.', 'start': 7354.135, 'duration': 5.543}, {'end': 7364.542, 'text': 'here we are on GitHub in my algorithms repository.', 'start': 7361.42, 'duration': 3.122}, {'end': 7372.226, 'text': 'Now, if you scroll down and look for Bellman Ford under the graph theory section,', 'start': 7364.902, 'duration': 7.324}, {'end': 7380.151, 'text': 'you can see that currently there are two different implementations one for graph represented as an edge list,', 'start': 7372.226, 'duration': 7.925}, {'end': 7384.433, 'text': 'another one for a graph represented as an adjacency list.', 'start': 7380.151, 'duration': 4.282}], 'summary': 'Github repository contains two bellman ford implementations for different graph representations.', 'duration': 30.298, 'max_score': 7354.135, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY7354135.jpg'}, {'end': 7481.887, 'src': 'embed', 'start': 7454.361, 'weight': 10, 'content': [{'end': 7462.565, 'text': 'And then just as the pseudocode said, just loop v minus one time, then for each edge, just relax the edge.', 'start': 7454.361, 'duration': 8.204}, {'end': 7464.886, 'text': "So that's what we're doing.", 'start': 7463.865, 'duration': 1.021}, {'end': 7474.382, 'text': 'And here Now this second pass of the algorithm is to detect negative cycles.', 'start': 7464.906, 'duration': 9.476}, {'end': 7477.204, 'text': 'So run the algorithm a second time.', 'start': 7475.263, 'duration': 1.941}, {'end': 7481.887, 'text': 'So loop v minus one times for each edge, relax the edge.', 'start': 7477.424, 'duration': 4.463}], 'summary': 'Algorithm loops v-1 times, relaxes each edge to detect negative cycles.', 'duration': 27.526, 'max_score': 7454.361, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY7454361.jpg'}, {'end': 7609.818, 'src': 'embed', 'start': 7580.566, 'weight': 1, 'content': [{'end': 7588.75, 'text': 'Before we dive too deeply into the Floyd to Warshall algorithm, I want to address when you should and should not use this algorithm.', 'start': 7580.566, 'duration': 8.184}, {'end': 7600.295, 'text': 'This table gives information about various types of graphs and or constraints in the leftmost column and the performance or outcome of common shortest path algorithms.', 'start': 7589.25, 'duration': 11.045}, {'end': 7609.818, 'text': "For example, you can see in the second row that a breadth first search and Dijkstra's can handle large graphs with lots of nodes,", 'start': 7601.035, 'duration': 8.783}], 'summary': 'The floyd-warshall algorithm is discussed in relation to graph types and constraints, highlighting the performance of common shortest path algorithms.', 'duration': 29.252, 'max_score': 7580.566, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY7580566.jpg'}], 'start': 6385.108, 'title': 'Shortest path algorithms', 'summary': "Covers dijkstra's algorithm for finding the shortest path in a graph, bellman ford algorithm for single source shortest path with emphasis on detecting negative cycles and its application, and the floyd warshall algorithm for all pairs shortest path with a time complexity of o(v^3).", 'chapters': [{'end': 6611.959, 'start': 6385.108, 'title': "Dijkstra's algorithm explanation", 'summary': 'Explains the dijkstra algorithm for finding the shortest path in a graph, where it computes the optimal distance to the start node, initializes arrays, and implements the main while loop for priority queue operations.', 'duration': 226.851, 'highlights': ['The optimal distance to the start node at the beginning of the algorithm has a distance of zero. The start node is initialized with a distance value of zero, ensuring the algorithm begins with the correct distance.', 'The while loop contains the bulk of the Dijkstra algorithm implementation, where the priority queue is not empty, and it retrieves the ID of the node with the shortest distance. The while loop forms the core of the Dijkstra algorithm, efficiently processing nodes in the priority queue based on their distance values.', 'The algorithm employs an optimization of returning early if the end node is reached, thereby ensuring efficiency in finding the shortest distance. The algorithm is optimized to return the optimal distance if the end node is reached, preventing unnecessary iterations and enhancing performance.']}, {'end': 6895.704, 'start': 6612.319, 'title': 'Bellman ford algorithm', 'summary': "Explains the bellman ford algorithm, a single source shortest path algorithm used to find the shortest path from a starting node to all other nodes on the graph, particularly useful in detecting negative cycles and its application in finance and economics, while highlighting its time complexity compared to dijkstra's algorithm.", 'duration': 283.385, 'highlights': ['Bellman Ford algorithm: a single source shortest path algorithm used to find the shortest path from a starting node to all other nodes on the graph, particularly useful in detecting negative cycles and its application in finance and economics. The Bellman Ford algorithm is a single source shortest path algorithm that can find the shortest path from a starting node to all other nodes on the graph, particularly useful in detecting negative cycles. It also has an application in finance and economics for performing arbitrage between markets.', "Comparison with Dijkstra's algorithm: Bellman Ford has a much worse time complexity than Dijkstra's algorithm, running in a time complexity proportional to the product of the number of edges and the number of vertices, while Dijkstra's algorithm can do much better at around big O of E plus v log v with a binary heap. Bellman Ford has a much worse time complexity than Dijkstra's algorithm, running in a time complexity proportional to the product of the number of edges and the number of vertices, while Dijkstra's algorithm can do much better at around big O of E plus v log v with a binary heap.", "Application in finance and economics: Bellman Ford's application in finance and economics is particularly evident when performing arbitrage between two or more markets, where negative cycles can manifest and lead to risk-free gains. Bellman Ford's application in finance and economics is particularly evident when performing arbitrage between two or more markets, where negative cycles can manifest and lead to risk-free gains.", "Detection of negative cycles: Bellman Ford algorithm is used when Dijkstra's algorithm fails, particularly in graphs with negative edge weights, where the detection of negative cycles is of critical importance to avoid getting stuck in an infinite loop. Bellman Ford algorithm is used when Dijkstra's algorithm fails, particularly in graphs with negative edge weights, where the detection of negative cycles is of critical importance to avoid getting stuck in an infinite loop."]}, {'end': 7481.887, 'start': 6896.384, 'title': 'Bellman ford algorithm for shortest path', 'summary': 'Explains the bellman ford algorithm for finding the shortest path in a graph, with a detailed example and code implementation, and the process of detecting negative cycles in the graph.', 'duration': 585.503, 'highlights': ['The algorithm starts by setting every entry in D to positive infinity and then sets the distance to the starting node to be zero. The algorithm initializes the distance array by setting every entry to positive infinity and then sets the distance of the starting node to zero.', 'The algorithm relaxes each edge v minus one times, updating the value from where the edge starts to where it ends. The algorithm iterates v minus one times and relaxes each edge, updating the value from the starting node to the ending node.', 'To detect negative cycles, the algorithm runs a second time, checking for any nodes that update to a better value than the known best value. To detect negative cycles, the algorithm runs a second time, checking for nodes that update to a better value and marking them as part of a negative cycle.', 'The algorithm processes each edge by performing relaxation operations, updating the distance array accordingly. The algorithm processes each edge by performing relaxation operations, updating the distance array based on the edge costs.', 'The algorithm propagates negative cycle minus infinity values throughout the graph over v minus one iterations. The algorithm ensures the propagation of negative cycle minus infinity values throughout the graph over v minus one iterations.']}, {'end': 7724.171, 'start': 7482.667, 'title': 'Floyd warshall algorithm', 'summary': "Introduces the floyd warshall algorithm, an all pairs shortest path algorithm with a time complexity of o(v^3). it highlights the algorithm's suitability for small graphs, all pairs shortest path problems, and negative cycle detection.", 'duration': 241.504, 'highlights': ['The Floyd Warshall algorithm has a time complexity of O(v^3), making it ideal for graphs with no more than a couple 100 nodes. The time complexity of the Floyd Warshall algorithm is O(v^3), making it suitable for graphs with a limited number of nodes.', 'The algorithm shines in three areas: small graphs, solving the all pairs shortest path problem, and detecting negative cycles. The Floyd Warshall algorithm excels in small graphs, solving all pairs shortest path problems, and detecting negative cycles, making it well-suited for these scenarios.', 'The optimal way to represent the graph for Floyd Warshall is with a two-dimensional adjacency matrix denoted as M, with each cell representing the edge weight between nodes. The recommended representation for the graph in Floyd Warshall algorithm is a two-dimensional adjacency matrix, denoted as M, where each cell represents the edge weight between nodes.', "It's important to set the distance from a node to itself as zero, and when there is no edge between nodes i and j, the value in the matrix should be set to positive infinity. Setting the distance from a node to itself as zero and using positive infinity to indicate no direct connection between nodes are crucial steps in representing the graph using the adjacency matrix."]}], 'duration': 1339.063, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY6385108.jpg', 'highlights': ['The algorithm employs an optimization of returning early if the end node is reached, thereby ensuring efficiency in finding the shortest distance.', 'The start node is initialized with a distance value of zero, ensuring the algorithm begins with the correct distance.', 'The while loop forms the core of the Dijkstra algorithm, efficiently processing nodes in the priority queue based on their distance values.', 'Bellman Ford algorithm: a single source shortest path algorithm used to find the shortest path from a starting node to all other nodes on the graph, particularly useful in detecting negative cycles and its application in finance and economics.', "Bellman Ford has a much worse time complexity than Dijkstra's algorithm, running in a time complexity proportional to the product of the number of edges and the number of vertices, while Dijkstra's algorithm can do much better at around big O of E plus v log v with a binary heap.", "Bellman Ford's application in finance and economics is particularly evident when performing arbitrage between two or more markets, where negative cycles can manifest and lead to risk-free gains.", "Bellman Ford algorithm is used when Dijkstra's algorithm fails, particularly in graphs with negative edge weights, where the detection of negative cycles is of critical importance to avoid getting stuck in an infinite loop.", 'The algorithm initializes the distance array by setting every entry to positive infinity and then sets the distance of the starting node to zero.', 'The algorithm iterates v minus one times and relaxes each edge, updating the value from the starting node to the ending node.', 'To detect negative cycles, the algorithm runs a second time, checking for nodes that update to a better value and marking them as part of a negative cycle.', 'The Floyd Warshall algorithm has a time complexity of O(v^3), making it suitable for graphs with a limited number of nodes.', 'The Floyd Warshall algorithm excels in small graphs, solving all pairs shortest path problems, and detecting negative cycles, making it well-suited for these scenarios.', 'The recommended representation for the graph in Floyd Warshall algorithm is a two-dimensional adjacency matrix, denoted as M, where each cell represents the edge weight between nodes.', 'Setting the distance from a node to itself as zero and using positive infinity to indicate no direct connection between nodes are crucial steps in representing the graph using the adjacency matrix.']}, {'end': 9786.037, 'segs': [{'end': 7869.746, 'src': 'embed', 'start': 7809.298, 'weight': 12, 'content': [{'end': 7818.503, 'text': 'We are also not just limited to one intermediate node in between A and C and C and B, we can have several like in the graph below.', 'start': 7809.298, 'duration': 9.205}, {'end': 7824.911, 'text': 'Now the question comes up how do we actually compute all intermediate paths?', 'start': 7820.683, 'duration': 4.228}, {'end': 7830.441, 'text': 'The answer is we will use dynamic programming to cache previous optimal solutions.', 'start': 7825.572, 'duration': 4.869}, {'end': 7841.26, 'text': 'Let dp be a three dimensional matrix of size n by n by n, which acts as our memo table.', 'start': 7831.633, 'duration': 9.627}, {'end': 7854.39, 'text': "we're going to say that the cell dp at k i j in our table gives us the shortest path from node i to node j, routing through nodes zero through k.", 'start': 7841.26, 'duration': 13.13}, {'end': 7861.882, 'text': "what we'll do is start by computing k equals zero, then k equals one, then k equals two, and so on.", 'start': 7855.939, 'duration': 5.943}, {'end': 7869.746, 'text': 'This gradually builds up the optimal solution routing through zero, then all optimal solutions routing through zero and one,', 'start': 7862.462, 'duration': 7.284}], 'summary': 'Using dynamic programming to compute shortest paths through multiple nodes.', 'duration': 60.448, 'max_score': 7809.298, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY7809298.jpg'}, {'end': 7945.441, 'src': 'embed', 'start': 7921.618, 'weight': 10, 'content': [{'end': 7931.387, 'text': 'The left hand side of the recurrence simply says, reuse the best distance so far from itj routing through nodes zero to k minus one.', 'start': 7921.618, 'duration': 9.769}, {'end': 7939.014, 'text': "It's important to note that the solution using nodes zero to k minus one is a partial solution.", 'start': 7932.288, 'duration': 6.726}, {'end': 7940.795, 'text': 'It is not the whole picture.', 'start': 7939.414, 'duration': 1.381}, {'end': 7945.441, 'text': 'This is part of the dynamic programming aspect of the Floyd Warshall algorithm.', 'start': 7941.616, 'duration': 3.825}], 'summary': 'Floyd warshall algorithm uses partial solutions for dynamic programming.', 'duration': 23.823, 'max_score': 7921.618, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY7921618.jpg'}, {'end': 8162.613, 'src': 'embed', 'start': 8137.489, 'weight': 9, 'content': [{'end': 8142.674, 'text': 'inside the for loops, all I do is copy the input matrix into the DP matrix.', 'start': 8137.489, 'duration': 5.185}, {'end': 8148.019, 'text': 'Think of this as the base case, or rather the k equals zero case.', 'start': 8143.195, 'duration': 4.824}, {'end': 8159.77, 'text': 'For the next matrix, if the distance from i to j is not positive infinity, then the next node you want to go to from node i is node j by default.', 'start': 8148.66, 'duration': 11.11}, {'end': 8162.613, 'text': "Now we're back inside the Florida workshop function.", 'start': 8160.311, 'duration': 2.302}], 'summary': 'Copying input matrix into dp matrix for k=0 case, moving from i to j if distance is not infinity.', 'duration': 25.124, 'max_score': 8137.489, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY8137489.jpg'}, {'end': 8357.678, 'src': 'embed', 'start': 8334.334, 'weight': 13, 'content': [{'end': 8343.904, 'text': 'So, to identify whether or not the optimal path from i to j is affected by a negative cycle, rerun the Floyd Warshall algorithm.', 'start': 8334.334, 'duration': 9.57}, {'end': 8352.513, 'text': 'second time if the best distance is better than the already known best distance stored in our table dp,', 'start': 8343.904, 'duration': 8.609}, {'end': 8357.678, 'text': 'then set the value in the matrix from i to j to be negative infinity.', 'start': 8352.513, 'duration': 5.165}], 'summary': 'Rerun floyd warshall algorithm to identify negative cycle impact.', 'duration': 23.344, 'max_score': 8334.334, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY8334334.jpg'}, {'end': 8544.962, 'src': 'embed', 'start': 8489.741, 'weight': 2, 'content': [{'end': 8494.885, 'text': 'And I create our adjacency matrix by calling the create graph method.', 'start': 8489.741, 'duration': 5.144}, {'end': 8498.608, 'text': 'And if we look at up here, this is the create graph method.', 'start': 8495.265, 'duration': 3.343}, {'end': 8508.735, 'text': 'And all it does is it initializes a matrix of size n by n, it fills the matrix with the special constant positive infinity.', 'start': 8499.208, 'duration': 9.527}, {'end': 8518.102, 'text': 'And it also sets the diagonal to have all zero nodes by default, because I assume that this is the behavior you want.', 'start': 8509.876, 'duration': 8.226}, {'end': 8526.588, 'text': "If it's not, then that's not an issue, because you can just override it when you add some edge values to your adjacency matrix.", 'start': 8518.703, 'duration': 7.885}, {'end': 8530.992, 'text': 'Alright, so we created a matrix, we added some edge weights.', 'start': 8527.449, 'duration': 3.543}, {'end': 8537.036, 'text': "And then what you'll want to do is create an instance of the solver,", 'start': 8532.213, 'duration': 4.823}, {'end': 8544.962, 'text': 'give it our adjacency matrix and then call the get all pair shortest path matrix function,', 'start': 8537.036, 'duration': 7.926}], 'summary': 'The create graph method initializes an adjacency matrix of size n by n with positive infinity values and zero nodes on the diagonal, which can be overridden when adding edge values.', 'duration': 55.221, 'max_score': 8489.741, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY8489741.jpg'}, {'end': 8620.326, 'src': 'embed', 'start': 8590.935, 'weight': 1, 'content': [{'end': 8594.677, 'text': 'So here I want to reconstruct the shortest path between all pairs of nodes.', 'start': 8590.935, 'duration': 3.742}, {'end': 8598.579, 'text': 'So I loop through all pairs of nodes i and j.', 'start': 8595.558, 'duration': 3.021}, {'end': 8604.941, 'text': 'And then on the solver, again, I call reconstruct shortest path from i to j.', 'start': 8598.579, 'duration': 6.362}, {'end': 8608.342, 'text': 'And that returns a list of nodes.', 'start': 8604.941, 'duration': 3.401}, {'end': 8613.263, 'text': 'And here I just print three different options depending on when I get back.', 'start': 8608.362, 'duration': 4.901}, {'end': 8620.326, 'text': 'If the path is null, then there does not exist, or rather, sorry, there exists an infinite number of solutions.', 'start': 8613.983, 'duration': 6.343}], 'summary': 'Reconstruct shortest paths between all pairs of nodes using a solver, obtaining a list of nodes and handling different scenarios based on the result.', 'duration': 29.391, 'max_score': 8590.935, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY8590935.jpg'}, {'end': 8734.258, 'src': 'embed', 'start': 8701.621, 'weight': 3, 'content': [{'end': 8706.603, 'text': 'Okay, so looking at the constructor, again, you just pass in the input adjacency matrix.', 'start': 8701.621, 'duration': 4.982}, {'end': 8709.144, 'text': 'And then I do some initialization.', 'start': 8707.264, 'duration': 1.88}, {'end': 8722.771, 'text': "So simply allocate memory for our matrices that we're going to need, and then populate the DP matrix with whatever is given to us for our input.", 'start': 8709.505, 'duration': 13.266}, {'end': 8734.258, 'text': 'And Also, make sure to initialize the next matrix to contain j as the next value going from i to j.', 'start': 8723.351, 'duration': 10.907}], 'summary': 'Initializes adjacency matrix and populates dp matrix for input.', 'duration': 32.637, 'max_score': 8701.621, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY8701621.jpg'}, {'end': 8924.002, 'src': 'embed', 'start': 8897.999, 'weight': 0, 'content': [{'end': 8907.804, 'text': "Now, if we look at reconstructing the shortest path from the start node to some ending node, what we want to do is, if we haven't done so already,", 'start': 8897.999, 'duration': 9.805}, {'end': 8908.605, 'text': 'run the solver.', 'start': 8907.804, 'duration': 0.801}, {'end': 8915.111, 'text': 'and then initialize a value called path to an empty ArrayList.', 'start': 8910.086, 'duration': 5.025}, {'end': 8919.737, 'text': "Look at if it's even possible to reach the end node from the start node.", 'start': 8915.772, 'duration': 3.965}, {'end': 8924.002, 'text': "And if it's not, return an empty path,", 'start': 8920.638, 'duration': 3.364}], 'summary': 'Reconstruct shortest path from start to end, run solver, check possibility, return empty path if not possible.', 'duration': 26.003, 'max_score': 8897.999, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY8897999.jpg'}, {'end': 9621.051, 'src': 'embed', 'start': 9590.935, 'weight': 5, 'content': [{'end': 9594.577, 'text': 'Then we get into the actual depth first search traversal bit.', 'start': 9590.935, 'duration': 3.642}, {'end': 9602.861, 'text': "So we iterate over each nudge from the node we're at and attempt to go to each node which I've labeled to.", 'start': 9595.017, 'duration': 7.844}, {'end': 9615.488, 'text': 'Since this is an undirected graph, there is bound to be an edge that directly returns to the node we were just previously at,', 'start': 9604.702, 'duration': 10.786}, {'end': 9618.89, 'text': 'which is the parent node, which we want to avoid doing.', 'start': 9615.488, 'duration': 3.402}, {'end': 9621.051, 'text': 'So we continue on those cases.', 'start': 9619.01, 'duration': 2.041}], 'summary': 'The depth first search traversal avoids returning to the parent node in an undirected graph.', 'duration': 30.116, 'max_score': 9590.935, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY9590935.jpg'}, {'end': 9747.402, 'src': 'embed', 'start': 9710.257, 'weight': 8, 'content': [{'end': 9722.514, 'text': 'Continuing on our depth first search, which takes us downward And now we get to explore the other branch we have not visited.', 'start': 9710.257, 'duration': 12.257}, {'end': 9733.95, 'text': 'Again, we have an edge which reaches out to find a node with a lower ID.', 'start': 9728.886, 'duration': 5.064}, {'end': 9739.755, 'text': 'So we need to update our low link value for the current node, which is node eight.', 'start': 9734.491, 'duration': 5.264}, {'end': 9747.402, 'text': "Now we can update node seven's low link value to five on the callback of the depth first search method.", 'start': 9740.616, 'duration': 6.786}], 'summary': 'Depth first search updated low link values for nodes 8 and 7 to 5.', 'duration': 37.145, 'max_score': 9710.257, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY9710257.jpg'}], 'start': 7724.692, 'title': 'Floyd warshall algorithm and bridge finding', 'summary': 'Discusses the floyd warshall algorithm for finding all pairs shortest path, including its implementation, application, and space complexity. it also covers the algorithm for finding bridges in undirected graphs, emphasizing their significance, identification process, time complexity, and optimization.', 'chapters': [{'end': 8044.098, 'start': 7724.692, 'title': 'Floyd warshall algorithm', 'summary': 'Discusses the floyd warshall algorithm for finding the all pairs shortest path in a graph, using dynamic programming to compute intermediate paths, and optimizing space complexity to o(v^2).', 'duration': 319.406, 'highlights': ['The Floyd Warshall algorithm computes all intermediate routes between two nodes to find the optimal path, considering all possible intermediate paths between triplets of nodes.', 'Using dynamic programming, a three-dimensional matrix of size n by n by n, called the memo table, is used to cache previous optimal solutions, gradually building up the optimal solution routing through all nodes and solving the all pairs shortest path problem.', 'The algorithm initially uses big O of v^3 memory, but by computing the solution for k in place, the space complexity is reduced to big O of v^2.']}, {'end': 8432.121, 'start': 8044.359, 'title': 'Floyd warshall algorithm overview', 'summary': "Discusses the implementation of the floyd warshall algorithm, including the setup, main algorithm, handling of negative cycles, and path reconstruction, providing insights into the algorithm's functionality and usage.", 'duration': 387.762, 'highlights': ['The Floyd Warshall algorithm is used to find the shortest paths between all pairs of nodes in a weighted graph, and the chapter provides a detailed walkthrough of its implementation, including the setup, main algorithm, and handling of negative cycles.', 'The algorithm involves the use of a 2D memo table to store all pair shortest path solutions and another 2D table to reconstruct the shortest paths, emphasizing the importance of initializing the latter with null values.', "The chapter explains the process of gradually building up the best solutions by looping over k on the exterior loop, followed by a loop over all pairs of nodes i and j to test for the condition to improve the shortest path, and update the value at dp[i][j] if there's a better route through k.", 'The discussion delves into the identification of optimal paths affected by negative cycles, recommending the rerun of the Floyd Warshall algorithm to mark the affected paths with negative infinity and the corresponding indexes with -1 in the next matrix.', "The method for reconstructing the shortest path between any two pairs of nodes is outlined, including the handling of paths affected by negative cycles, providing a comprehensive understanding of the algorithm's functionality."]}, {'end': 8734.258, 'start': 8432.461, 'title': 'Floyd warshall algorithm', 'summary': 'Introduces the floyd warshall algorithm, demonstrating its source code implementation and its application to find all pairs shortest path in a graph with 7 nodes, showing the outcomes and reconstruction of shortest paths.', 'duration': 301.797, 'highlights': ['The chapter introduces the Floyd Warshall algorithm, demonstrating its source code implementation and its application to find all pairs shortest path in a graph with 7 nodes. The source code for the Floyd Warshall algorithm is presented, along with an example of its application to find all pairs shortest path in a graph with 7 nodes.', 'The outcomes of the algorithm can be a concrete shortest path, no path between two nodes (represented by infinity), or encountering a negative cycle (represented by negative infinity). The outcomes of the algorithm can be a concrete shortest path, no path between two nodes (represented by infinity), or encountering a negative cycle (represented by negative infinity).', 'The reconstruction of the shortest paths from all pairs of nodes is demonstrated, showcasing three possible outcomes: no solution if the path is null, no solution if the path has zero length, or a pretty formatting of the output for non-empty paths. The reconstruction of the shortest paths from all pairs of nodes is demonstrated, showcasing three possible outcomes: no solution if the path is null, no solution if the path has zero length, or a pretty formatting of the output for non-empty paths.']}, {'end': 8946.373, 'start': 8734.258, 'title': 'Floyd warshall algorithm in java', 'summary': 'Explains the implementation of the floyd warshall algorithm in java, including the methods for computing all pairs of shortest paths and identifying negative cycles.', 'duration': 212.115, 'highlights': ["The method 'get all pair shortest path matrix' is used to call the solver if the all pair shortest path problem has not been solved already, in order to avoid redundant calls, thus optimizing efficiency.", 'The process for computing all pairs of shortest paths involves iterating through nodes, checking for conditions, and updating the path values, aiming to identify negative cycles and mark the affected nodes with special constants in Java.', 'The algorithm also includes reconstructing the shortest path from the start node to an ending node, which involves running the solver, initializing a path value, and checking for the possibility of reaching the end node and negative cycles.']}, {'end': 9446.882, 'start': 8947.353, 'title': 'Algorithm for finding bridges in undirected graphs', 'summary': "Discusses the algorithm for finding bridges and articulation points in an undirected graph, highlighting their significance in real-world applications. it explains the process of identifying bridges and articulation points through depth-first search and low link values, emphasizing their relevance in graph theory and potential vulnerabilities. the algorithm's time complexity is also addressed, with a focus on optimizing the process to achieve linear time complexity.", 'duration': 499.529, 'highlights': ['The algorithm for finding bridges and articulation points in an undirected graph is explained, emphasizing their significance in graph theory and real-world applications. The chapter discusses the algorithm for finding bridges and articulation points in an undirected graph, highlighting their significance in real-world applications and graph theory.', 'The process of identifying bridges and articulation points through depth-first search and low link values is explained, showcasing their relevance in graph theory and potential vulnerabilities in real-world applications. The chapter explains the process of identifying bridges and articulation points through depth-first search and low link values, emphasizing their relevance in graph theory and potential vulnerabilities in real-world applications.', "The algorithm's time complexity is addressed, with a focus on optimizing the process to achieve linear time complexity. The chapter addresses the algorithm's time complexity and focuses on optimizing the process to achieve linear time complexity."]}, {'end': 9786.037, 'start': 9447.583, 'title': 'Bridges in graphs algorithm', 'summary': 'Explains the algorithm to find bridges in an undirected graph, including the variables used, the depth first search method, and the application of the min function to update low link values and identify bridge conditions.', 'duration': 338.454, 'highlights': ['The algorithm involves defining three variables in the global or class scope: ID for unique node labeling, an undirected graph G, and n for the number of nodes in the graph.', 'The find bridges function iterates over all unvisited nodes using depth first search to ensure finding all bridges in the graph, even in multiple connected components.', 'The depth first search method involves marking nodes as visited, incrementing the ID value, assigning default ID and low link values to the current node, and recursively calling itself if the next node is not visited.', 'The min function is used to propagate low link values and identify bridge conditions, updating the low link values and appending pairs of node IDs to the bridges array when the bridge condition is met.']}], 'duration': 2061.345, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY7724692.jpg', 'highlights': ['The Floyd Warshall algorithm computes all intermediate routes between two nodes to find the optimal path, considering all possible intermediate paths between triplets of nodes.', 'The algorithm initially uses big O of v^3 memory, but by computing the solution for k in place, the space complexity is reduced to big O of v^2.', 'The Floyd Warshall algorithm is used to find the shortest paths between all pairs of nodes in a weighted graph, and the chapter provides a detailed walkthrough of its implementation, including the setup, main algorithm, and handling of negative cycles.', 'The discussion delves into the identification of optimal paths affected by negative cycles, recommending the rerun of the Floyd Warshall algorithm to mark the affected paths with negative infinity and the corresponding indexes with -1 in the next matrix.', "The method for reconstructing the shortest path between any two pairs of nodes is outlined, including the handling of paths affected by negative cycles, providing a comprehensive understanding of the algorithm's functionality.", 'The chapter introduces the Floyd Warshall algorithm, demonstrating its source code implementation and its application to find all pairs shortest path in a graph with 7 nodes.', 'The outcomes of the algorithm can be a concrete shortest path, no path between two nodes (represented by infinity), or encountering a negative cycle (represented by negative infinity).', 'The reconstruction of the shortest paths from all pairs of nodes is demonstrated, showcasing three possible outcomes: no solution if the path is null, no solution if the path has zero length, or a pretty formatting of the output for non-empty paths.', "The method 'get all pair shortest path matrix' is used to call the solver if the all pair shortest path problem has not been solved already, in order to avoid redundant calls, thus optimizing efficiency.", 'The algorithm for finding bridges and articulation points in an undirected graph is explained, emphasizing their significance in graph theory and real-world applications.', 'The process of identifying bridges and articulation points through depth-first search and low link values is explained, showcasing their relevance in graph theory and potential vulnerabilities in real-world applications.', "The algorithm's time complexity is addressed, with a focus on optimizing the process to achieve linear time complexity.", 'The algorithm involves defining three variables in the global or class scope: ID for unique node labeling, an undirected graph G, and n for the number of nodes in the graph.', 'The find bridges function iterates over all unvisited nodes using depth first search to ensure finding all bridges in the graph, even in multiple connected components.', 'The depth first search method involves marking nodes as visited, incrementing the ID value, assigning default ID and low link values to the current node, and recursively calling itself if the next node is not visited.', 'The min function is used to propagate low link values and identify bridge conditions, updating the low link values and appending pairs of node IDs to the bridges array when the bridge condition is met.']}, {'end': 11995.239, 'segs': [{'end': 10692.285, 'src': 'embed', 'start': 10667.777, 'weight': 0, 'content': [{'end': 10676.381, 'text': 'I like to think of them as self contained cycles within a directed graph, where for every vertex in a given cycle,', 'start': 10667.777, 'duration': 8.604}, {'end': 10679.142, 'text': 'you can reach every other vertex in the same cycle.', 'start': 10676.381, 'duration': 2.761}, {'end': 10684.444, 'text': 'For example, in the graph below, there are four strongly connected components.', 'start': 10679.802, 'duration': 4.642}, {'end': 10688.402, 'text': "I've outlined them here in different colors.", 'start': 10685.86, 'duration': 2.542}, {'end': 10692.285, 'text': 'If you inspect each strongly connected component,', 'start': 10688.902, 'duration': 3.383}], 'summary': 'Graph contains 4 strongly connected components.', 'duration': 24.508, 'max_score': 10667.777, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY10667777.jpg'}, {'end': 10906.558, 'src': 'embed', 'start': 10878.678, 'weight': 2, 'content': [{'end': 10890.352, 'text': "This is where tarjans algorithm kicks in with its stacking variant to prevent strongly connected components from interfering with each other's low link values.", 'start': 10878.678, 'duration': 11.674}, {'end': 10902.095, 'text': "So, to cope with the random traversal order of the depth first search, Tarjan's algorithm maintains a set, often as a stack,", 'start': 10892.17, 'duration': 9.925}, {'end': 10906.558, 'text': 'of valid nodes from which to update low link values.', 'start': 10902.095, 'duration': 4.463}], 'summary': "Tarjan's algorithm uses stacking variant to prevent interference of strongly connected components with each other's low link values.", 'duration': 27.88, 'max_score': 10878.678, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY10878678.jpg'}, {'end': 11170.893, 'src': 'heatmap', 'start': 10921.607, 'weight': 1, 'content': [{'end': 10929.809, 'text': 'if the variables u and v are nodes in our graph and we are currently exploring node u,', 'start': 10921.607, 'duration': 8.202}, {'end': 10938.69, 'text': "then our new low link update condition is that to update node use loading value to node V's loading value,", 'start': 10929.809, 'duration': 8.881}, {'end': 10943.611, 'text': 'there has to be a path of edges from u to v.', 'start': 10938.69, 'duration': 4.921}, {'end': 10946.532, 'text': 'And node v must be on the stack.', 'start': 10943.611, 'duration': 2.921}, {'end': 10952.497, 'text': "Another small difference we're going to make to finding the correct low link values is that,", 'start': 10947.873, 'duration': 4.624}, {'end': 10963.705, 'text': "instead of finding all the low link values after the fact, we're going to update them, as we do our depth first search on the fly, if you will.", 'start': 10952.497, 'duration': 11.208}, {'end': 10967.388, 'text': 'this will allow us to obtain a linear time complexity.', 'start': 10963.705, 'duration': 3.683}, {'end': 10971.765, 'text': "We'll be doing an example in the following slides.", 'start': 10969.284, 'duration': 2.481}, {'end': 10973.746, 'text': "But this is Tarzan's algorithm.", 'start': 10971.885, 'duration': 1.861}, {'end': 10978.068, 'text': 'In a nutshell, start out and mark each node as unvisited.', 'start': 10973.766, 'duration': 4.302}, {'end': 10983.39, 'text': "Start the depth for search somewhere and don't stop until all the nodes are visited.", 'start': 10979.248, 'duration': 4.142}, {'end': 10988.292, 'text': 'Upon visiting a node, assign it an ID and a low link value.', 'start': 10984.17, 'duration': 4.122}, {'end': 10994.155, 'text': 'Additionally, also mark the node as visited and add it to the scene stack.', 'start': 10988.972, 'duration': 5.183}, {'end': 10998.66, 'text': 'on the depth first search callback after the recursion comes back.', 'start': 10995.055, 'duration': 3.605}, {'end': 11006.77, 'text': 'If the previous node is on the stack, then min the current nodes lowlink value with the last nodes lowlink value.', 'start': 10999.361, 'duration': 7.409}, {'end': 11012.857, 'text': 'This is essentially what will allow lowlink values to propagate throughout cycles.', 'start': 11007.571, 'duration': 5.286}, {'end': 11020.542, 'text': 'After visiting all nodes neighbors, if the current node started the strongly connected component,', 'start': 11013.758, 'duration': 6.784}, {'end': 11025.686, 'text': 'then pop off all nodes from the stack which are in the strongly connected component.', 'start': 11020.542, 'duration': 5.144}, {'end': 11032.51, 'text': 'You know, a node started a strongly connected component if its ID is equal to its loading value.', 'start': 11026.486, 'duration': 6.024}, {'end': 11034.491, 'text': "I'll let you think about that a bit more.", 'start': 11032.95, 'duration': 1.541}, {'end': 11036.533, 'text': "And it'll start making sense.", 'start': 11035.052, 'duration': 1.481}, {'end': 11039.075, 'text': "Let's do an example.", 'start': 11037.614, 'duration': 1.461}, {'end': 11047.625, 'text': "I'm going to mark unvisited nodes as blue nodes for which the depth first search is still exploring some neighbors, as orange,", 'start': 11039.075, 'duration': 8.55}, {'end': 11053.713, 'text': 'and nodes which the depth first search has explored all of its neighbors as gray.', 'start': 11047.625, 'duration': 6.088}, {'end': 11061.559, 'text': 'Note that if a node is orange or gray, then it is on the stack and we can update its loading value.', 'start': 11055.174, 'duration': 6.385}, {'end': 11066.243, 'text': 'I will also be tracking the nodes which are on the stack in the left column.', 'start': 11062.26, 'duration': 3.983}, {'end': 11068.765, 'text': 'So keep your eyes peeled on that as well.', 'start': 11066.463, 'duration': 2.302}, {'end': 11071.607, 'text': "So let's start our depth first search.", 'start': 11070.166, 'duration': 1.441}, {'end': 11074.53, 'text': 'So just randomly pick a node and start there.', 'start': 11072.128, 'duration': 2.402}, {'end': 11084.578, 'text': 'As we explore unvisited nodes, give each node an ID and a low link value equal to the ID.', 'start': 11077.212, 'duration': 7.366}, {'end': 11092.458, 'text': "So now we're at node two, and our only option is to now visit node zero.", 'start': 11086.693, 'duration': 5.765}, {'end': 11097.722, 'text': "Since node zero is already visited, we don't want to visit it again.", 'start': 11093.759, 'duration': 3.963}, {'end': 11102.026, 'text': 'So now we backtrack on the backtracking.', 'start': 11098.203, 'duration': 3.823}, {'end': 11111.294, 'text': 'Since node zero is on the stack, we take the minimum of the current nodes, low link value and node zeros low link value.', 'start': 11102.667, 'duration': 8.627}, {'end': 11120.844, 'text': 'Similarly, now min the low link value of the node we were just at, which is node one with node two.', 'start': 11113.2, 'duration': 7.644}, {'end': 11125.506, 'text': 'And also the same for node zero.', 'start': 11123.345, 'duration': 2.161}, {'end': 11134.126, 'text': "Upon returning back to node zero, we realize that we've actually finished a strongly connected component.", 'start': 11127.421, 'duration': 6.705}, {'end': 11140.751, 'text': 'Since we visited all the neighbors of node zero, and its ID is equal to its low length value.', 'start': 11134.726, 'duration': 6.025}, {'end': 11146.555, 'text': 'This means we need to remove all the nodes associated with a strongly connected component from the stack.', 'start': 11141.431, 'duration': 5.124}, {'end': 11153.04, 'text': "However, we're not done exploring the graph.", 'start': 11151.198, 'duration': 1.842}, {'end': 11155.922, 'text': 'So pick another node at random.', 'start': 11153.3, 'duration': 2.622}, {'end': 11159.583, 'text': "let's start at node three.", 'start': 11157.321, 'duration': 2.262}, {'end': 11161.905, 'text': 'And go right.', 'start': 11161.104, 'duration': 0.801}, {'end': 11166.489, 'text': 'Now, our only option is to go down.', 'start': 11163.246, 'duration': 3.243}, {'end': 11170.893, 'text': "Now we're at node five, let's take the edge to node zero.", 'start': 11167.149, 'duration': 3.744}], 'summary': "Tarzan's algorithm updates low link values during dfs, achieving linear time complexity.", 'duration': 249.286, 'max_score': 10921.607, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY10921607.jpg'}, {'end': 11140.751, 'src': 'embed', 'start': 11113.2, 'weight': 3, 'content': [{'end': 11120.844, 'text': 'Similarly, now min the low link value of the node we were just at, which is node one with node two.', 'start': 11113.2, 'duration': 7.644}, {'end': 11125.506, 'text': 'And also the same for node zero.', 'start': 11123.345, 'duration': 2.161}, {'end': 11134.126, 'text': "Upon returning back to node zero, we realize that we've actually finished a strongly connected component.", 'start': 11127.421, 'duration': 6.705}, {'end': 11140.751, 'text': 'Since we visited all the neighbors of node zero, and its ID is equal to its low length value.', 'start': 11134.726, 'duration': 6.025}], 'summary': 'Identified a strongly connected component after visiting all neighbors of node zero.', 'duration': 27.551, 'max_score': 11113.2, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY11113200.jpg'}, {'end': 11342.801, 'src': 'embed', 'start': 11287.384, 'weight': 7, 'content': [{'end': 11291.225, 'text': 'So it should be the start of a strongly connected component.', 'start': 11287.384, 'duration': 3.841}, {'end': 11296.667, 'text': 'However, we have not finished visiting all of node threes neighbors.', 'start': 11292.085, 'duration': 4.582}, {'end': 11300.608, 'text': 'So we cannot make that assessment just yet.', 'start': 11297.207, 'duration': 3.401}, {'end': 11305.535, 'text': 'Now see the downward edge to visit node seven.', 'start': 11302.873, 'duration': 2.662}, {'end': 11309.617, 'text': 'Now take the edge to node five.', 'start': 11307.016, 'duration': 2.601}, {'end': 11313.38, 'text': 'On the callback, notice that node five is not in the stack.', 'start': 11310.858, 'duration': 2.522}, {'end': 11315.921, 'text': "So we don't mean with its low link value.", 'start': 11313.48, 'duration': 2.441}, {'end': 11318.103, 'text': 'Now up to node three.', 'start': 11315.941, 'duration': 2.162}, {'end': 11324.467, 'text': 'On the callback, we can min with no threes low link since no three is on the stack.', 'start': 11319.404, 'duration': 5.063}, {'end': 11328.05, 'text': 'also men with node seven.', 'start': 11325.988, 'duration': 2.062}, {'end': 11336.416, 'text': "So now we've finished with the last strongly connected component, all we need to do is remove all associated nodes from the stack.", 'start': 11328.71, 'duration': 7.706}, {'end': 11342.801, 'text': "And that's how tires and algorithm works to find strongly connected components.", 'start': 11338.298, 'duration': 4.503}], 'summary': 'Explaining algorithm to find strongly connected components.', 'duration': 55.417, 'max_score': 11287.384, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY11287384.jpg'}, {'end': 11648.969, 'src': 'embed', 'start': 11620.53, 'weight': 4, 'content': [{'end': 11626.995, 'text': 'Last things are to stop popping off nodes from the stack once we reach the start of the strongly connected component.', 'start': 11620.53, 'duration': 6.465}, {'end': 11631.158, 'text': 'And also increment the strongly connected component count.', 'start': 11627.816, 'duration': 3.342}, {'end': 11636.202, 'text': 'If you want to track the number of strongly connected components that were found today,', 'start': 11631.698, 'duration': 4.504}, {'end': 11642.767, 'text': "we will be looking over some source code for Tarzan's strongly connected components algorithm.", 'start': 11636.202, 'duration': 6.565}, {'end': 11648.969, 'text': "here we are in the source code for Tarzan's algorithm to find strongly connected components.", 'start': 11643.527, 'duration': 5.442}], 'summary': 'Algorithm finds and counts strongly connected components in source code.', 'duration': 28.439, 'max_score': 11620.53, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY11620530.jpg'}], 'start': 9786.878, 'title': 'Graph articulation points and bridges', 'summary': 'Discusses finding articulation points in graphs, emphasizing the impact of bridges and cycles, and explains algorithms for finding articulation points, bridges, and strongly connected components with examples and specific complexities.', 'chapters': [{'end': 9884.552, 'start': 9786.878, 'title': 'Finding articulation points in graphs', 'summary': 'Discusses finding articulation points in graphs, demonstrating how bridges can be used to identify possible articulation points, but also highlighting that bridges alone are not sufficient to capture all articulation points, and that the presence of articulation points can be related to the cycles in the graph.', 'duration': 97.674, 'highlights': ['The presence of articulation points can be related to the cycles in the graph, as demonstrated by the case where node two is an articulation point because its removal would cause the graph to split into two components.', 'On a connected component with three or more vertices, if an edge UV is a bridge, then either U or V is an articulation point, which serves as a good starting point for identifying potential articulation points.']}, {'end': 10159.64, 'start': 9885.993, 'title': 'Finding articulation points and bridges', 'summary': 'Explains the algorithm to find articulation points and bridges in a graph, emphasizing the identification of cycles and the impact of the starting node on determining articulation points.', 'duration': 273.647, 'highlights': ['The presence of a cycle implies that the node is an articulation point, as it corresponds to a strongly connected component, and removing the node will sever the graph in two. The presence of a cycle implies that the node is an articulation point, as it corresponds to a strongly connected component, and removing the node will sever the graph in two.', 'The algorithm introduces changes to the finding bridges algorithm to identify articulation points, including tracking the number of outgoing edges and marking nodes as articulation points based on the count of outgoing edges. The algorithm introduces changes to the finding bridges algorithm to identify articulation points, including tracking the number of outgoing edges and marking nodes as articulation points based on the count of outgoing edges.', "The starting node's characteristics, such as having no outgoing edges or being part of a cycle with only one outgoing edge, impact its identification as an articulation point. The starting node's characteristics, such as having no outgoing edges or being part of a cycle with only one outgoing edge, impact its identification as an articulation point."]}, {'end': 10504.786, 'start': 10160.56, 'title': 'Finding bridges in undirected graph', 'summary': 'Explains how to find all bridges in an undirected graph using a class, including methods for creating the graph, adding edges, and finding bridges, with a specific example of a graph with nine nodes and the resulting list of bridges as pairs.', 'duration': 344.226, 'highlights': ["The method 'createGraph' initializes a graph with n nodes by creating a list of lists of type integer, representing an adjacency list with directed edges, and returns the graph.", "The 'addEdges' method is used to add directed edges into the graph, with the example of adding an edge from a node to another node and then to that node, and then passing the graph and the number of nodes into the class to find all bridges.", "The 'findBridges' method initializes variables and memory, populates the bridges array using a depth first search method for each node, initializing the low link value and the ID of that node, and recursively calling the depth first search method to keep probing for unvisited nodes in the undirected graph."]}, {'end': 11342.801, 'start': 10504.786, 'title': "Tarjan's algorithm for strongly connected components", 'summary': "Introduces tarjan's algorithm for finding strongly connected components, explaining low link values and the stacking variant to prevent interference with low link values, with a linear time complexity and an example showcasing the algorithm's execution.", 'duration': 838.015, 'highlights': ["Tarjan's algorithm maintains a set, often as a stack, of valid nodes from which to update low link values, ensuring a linear time complexity. Tarjan's algorithm uses a stacking variant to prevent strongly connected components from interfering with each other's low link values, maintaining a set of valid nodes to update low link values.", "Low link values are updated on the fly during the depth first search, allowing for linear time complexity. Instead of finding all the low link values after the fact, Tarjan's algorithm updates them during the depth first search, ensuring linear time complexity.", "Tarjan's algorithm marks each node as unvisited, starts the depth first search, assigns IDs and low link values, and maintains a stack to identify strongly connected components. Tarjan's algorithm marks nodes as unvisited, starts the depth first search, assigns IDs and low link values, and uses a stack to identify and manage strongly connected components."]}, {'end': 11995.239, 'start': 11343.562, 'title': "Tarzan's algorithm for strongly connected components", 'summary': "Introduces tarzan's algorithm for finding strongly connected components, which involves defining variables, initializing arrays, running depth first search, and using the algorithm to find and print the connected components.", 'duration': 651.677, 'highlights': ["The algorithm involves defining variables, initializing arrays, running depth first search, and using the algorithm to find and print the connected components. The chapter introduces Tarzan's algorithm for finding strongly connected components, which involves defining variables, initializing arrays, running depth first search, and using the algorithm to find and print the connected components.", 'The algorithm defines variables such as unvisited nodes, the number of nodes in the graph, and the adjacency list of directed edges. The algorithm involves defining a few variables including unvisited nodes, the number of nodes in the graph, and the adjacency list of directed edges.', 'The algorithm uses arrays to store auxiliary information about the nodes in the graph, such as IDs, low link values, and whether or not a node is on the stack. The algorithm utilizes arrays to store auxiliary information about the nodes in the graph, including IDs, low link values, and whether or not a node is on the stack.', 'The algorithm employs the depth first search method to start a depth first search on each node in the graph and return the array of low link values as the final output. The algorithm uses the depth first search method to start a depth first search on each node in the graph and return the array of low link values as the final output.', 'The algorithm tracks the number of strongly connected components and assigns an ID to each node, while ensuring that all nodes within the same component have the same ID. The algorithm tracks the number of strongly connected components, assigns an ID to each node, and ensures that all nodes within the same component have the same ID.']}], 'duration': 2208.361, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY9786878.jpg', 'highlights': ['The presence of articulation points can be related to the cycles in the graph, as demonstrated by the case where node two is an articulation point because its removal would cause the graph to split into two components.', 'On a connected component with three or more vertices, if an edge UV is a bridge, then either U or V is an articulation point, which serves as a good starting point for identifying potential articulation points.', "The starting node's characteristics, such as having no outgoing edges or being part of a cycle with only one outgoing edge, impact its identification as an articulation point.", 'The algorithm introduces changes to the finding bridges algorithm to identify articulation points, including tracking the number of outgoing edges and marking nodes as articulation points based on the count of outgoing edges.', "Tarjan's algorithm maintains a set, often as a stack, of valid nodes from which to update low link values, ensuring a linear time complexity.", 'Low link values are updated on the fly during the depth first search, allowing for linear time complexity.', 'The algorithm involves defining variables, initializing arrays, running depth first search, and using the algorithm to find and print the connected components.', 'The algorithm defines variables such as unvisited nodes, the number of nodes in the graph, and the adjacency list of directed edges.', 'The algorithm uses arrays to store auxiliary information about the nodes in the graph, such as IDs, low link values, and whether or not a node is on the stack.', 'The algorithm employs the depth first search method to start a depth first search on each node in the graph and return the array of low link values as the final output.', 'The algorithm tracks the number of strongly connected components and assigns an ID to each node, while ensuring that all nodes within the same component have the same ID.']}, {'end': 12964.472, 'segs': [{'end': 12079.726, 'src': 'embed', 'start': 12052.896, 'weight': 5, 'content': [{'end': 12059.878, 'text': 'In some other words, we can say that the problem is given a complete graph with weighted edges.', 'start': 12052.896, 'duration': 6.982}, {'end': 12070.341, 'text': 'What is the Hamiltonian cycle of minimum cost? A Hamiltonian cycle is simply a path which visits each node exactly once.', 'start': 12060.518, 'duration': 9.823}, {'end': 12079.726, 'text': 'In practice, you will probably want to represent whatever graph you have as an adjacency matrix for simplicity.', 'start': 12071.181, 'duration': 8.545}], 'summary': 'Finding minimum cost hamiltonian cycle in a complete graph with weighted edges.', 'duration': 26.83, 'max_score': 12052.896, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY12052896.jpg'}, {'end': 12440.873, 'src': 'embed', 'start': 12406.142, 'weight': 0, 'content': [{'end': 12416.85, 'text': 'The main idea is that if the ith node has been visited, we flip on the ith bit to a one in the binary representation of the integer.', 'start': 12406.142, 'duration': 10.708}, {'end': 12427.01, 'text': 'The advantage to this representation is that a 32 bit integer is compact, quick and allows for easy caching in a memo table.', 'start': 12417.791, 'duration': 9.219}, {'end': 12436.19, 'text': 'For example, on the leftmost graph, we have visited the zeroth and first nodes.', 'start': 12428.547, 'duration': 7.643}, {'end': 12440.873, 'text': 'So the binary representation is 0011.', 'start': 12436.771, 'duration': 4.102}], 'summary': 'Using binary representation of visited nodes in a compact 32-bit integer for quick caching and memoization.', 'duration': 34.731, 'max_score': 12406.142, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY12406142.jpg'}, {'end': 12503.854, 'src': 'embed', 'start': 12469.061, 'weight': 2, 'content': [{'end': 12478.626, 'text': 'What we want to do from our last node, which in this graph is no three is expand to visit all other unvisited nodes.', 'start': 12469.061, 'duration': 9.565}, {'end': 12487.01, 'text': 'These are the gray nodes one and two, to make our partial tour a little longer with three nodes.', 'start': 12479.246, 'duration': 7.764}, {'end': 12493.987, 'text': 'For this particular state, we were able to generate an additional two states.', 'start': 12489.164, 'duration': 4.823}, {'end': 12503.854, 'text': 'But we would also need to do this for all states with two nodes, not just this one with zero and three.', 'start': 12495.068, 'duration': 8.786}], 'summary': 'Expand from node three to visit unvisited nodes, generating two additional states.', 'duration': 34.793, 'max_score': 12469.061, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY12469061.jpg'}, {'end': 12597.622, 'src': 'embed', 'start': 12564.931, 'weight': 4, 'content': [{'end': 12571.878, 'text': "Just a heads up to everyone who's still a beginner, the following slides make use of advanced bit manipulation techniques.", 'start': 12564.931, 'duration': 6.947}, {'end': 12579.146, 'text': "So make sure you're comfortable with how binary shifts and ores and X ores work.", 'start': 12572.559, 'duration': 6.587}, {'end': 12584.736, 'text': "here's the function that solves the traveling salesman problem.", 'start': 12580.874, 'duration': 3.862}, {'end': 12586.977, 'text': 'It takes two inputs.', 'start': 12585.496, 'duration': 1.481}, {'end': 12597.622, 'text': 'The first is m, a two dimensional adjacency matrix representing the input graph, and s the index of the starting node.', 'start': 12587.417, 'duration': 10.205}], 'summary': 'Advanced bit manipulation techniques used in solving the traveling salesman problem with a two-dimensional adjacency matrix and starting node index.', 'duration': 32.691, 'max_score': 12564.931, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY12564931.jpg'}, {'end': 12676.274, 'src': 'embed', 'start': 12641.564, 'weight': 3, 'content': [{'end': 12650.849, 'text': 'It simply does what I illustrated a few slides ago by storing the optimal value from the start node to every other node.', 'start': 12641.564, 'duration': 9.285}, {'end': 12658.412, 'text': 'you loop through each node, skipping over the start node, and then you cache the optimal value from s to i.', 'start': 12650.849, 'duration': 7.563}, {'end': 12663.417, 'text': 'which can be found in the distance matrix.', 'start': 12660.353, 'duration': 3.064}, {'end': 12676.274, 'text': 'The DP state you store is the end node as I and the mask with bits s and I set to one, hence the double bit shift.', 'start': 12664.338, 'duration': 11.936}], 'summary': 'Stores optimal value from start node to every other node using dp.', 'duration': 34.71, 'max_score': 12641.564, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY12641564.jpg'}, {'end': 12934.999, 'src': 'embed', 'start': 12905.778, 'weight': 6, 'content': [{'end': 12916.067, 'text': "Afterwards, once we've considered all possible end nodes to connect to the next node, we store the best partial tour in the memo table.", 'start': 12905.778, 'duration': 10.289}, {'end': 12918.99, 'text': 'And this concludes the solve method.', 'start': 12916.368, 'duration': 2.622}, {'end': 12928.656, 'text': 'The only unanswered question in this slide is how the combinations method works.', 'start': 12920.313, 'duration': 8.343}, {'end': 12931.798, 'text': 'And I do not mean to leave this unanswered.', 'start': 12929.217, 'duration': 2.581}, {'end': 12934.999, 'text': "So let's see how this gets done.", 'start': 12932.018, 'duration': 2.981}], 'summary': 'Algorithm stores best partial tour in memo table, concludes solve method, and addresses combinations method.', 'duration': 29.221, 'max_score': 12905.778, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY12905778.jpg'}], 'start': 11996.201, 'title': 'Dynamic programming for traveling salesman problem', 'summary': "Discusses tarzan's algorithm for finding strongly connected components, the definition, complexities, and dynamic programming solution for the traveling salesman problem, reducing time complexity to o(n^2*2^n). it also emphasizes the need to store the optimal values, dynamic programming state, and provides pseudocode for solving the problem.", 'chapters': [{'end': 12224.86, 'start': 11996.201, 'title': 'Solving the traveling salesman problem', 'summary': "Discusses tarzan's algorithm for finding strongly connected components, the traveling salesman problem, its definition, complexities, and dynamic programming solution, which reduces the time complexity to o(n^2*2^n).", 'duration': 228.659, 'highlights': ['The dynamic programming solution reduces the complexity to O(n^2*2^N), making graphs with roughly 23 nodes feasible for modern home computers. The dynamic programming solution reduces the complexity to O(n^2*2^N), making graphs with roughly 23 nodes feasible for modern home computers.', 'The problem has been proven to be NP complete, and numerous approximation algorithms exist for finding an optimal solution for large inputs. The problem has been proven to be NP complete, and numerous approximation algorithms exist for finding an optimal solution for large inputs.', 'The brute force approach of computing all possible tours takes O(n!) time, which is very slow. The brute force approach of computing all possible tours takes O(n!) time, which is very slow.', 'The traveling salesman problem aims to find the shortest possible route that visits each city exactly once and then returns to the city of origin. The traveling salesman problem aims to find the shortest possible route that visits each city exactly once and then returns to the city of origin.']}, {'end': 12641.264, 'start': 12227.085, 'title': 'Dynamic programming for traveling salesman problem', 'summary': 'Discusses using dynamic programming to solve the traveling salesman problem, emphasizing the need to store the optimal values, the dynamic programming state, and the pseudocode for solving the problem.', 'duration': 414.179, 'highlights': ['The need to store the optimal value from the starting node to every other node is emphasized, solving the traveling salesman problem for all paths with exactly two nodes. The optimal value for paths with two nodes is given in the input through the adjacency matrix.', 'The importance of storing the dynamic programming state, including the set of visited nodes and the index of the last visited node, is highlighted. The dynamic programming state consists of the set of visited nodes and the index of the last visited node, forming a compact and quick representation using a single 32-bit integer.', 'The process of gradually expanding to visit all other unvisited nodes to make partial tours longer is explained, resulting in the generation of additional states for partial tours. Expanding to visit all other unvisited nodes gradually results in the generation of new states for partial tours, contributing to the solution of the traveling salesman problem.', 'The pseudocode for solving the traveling salesman problem is presented, requiring advanced bit manipulation techniques. The function for solving the traveling salesman problem takes two inputs: a two-dimensional adjacency matrix and the index of the starting node. It utilizes advanced bit manipulation techniques and functions such as setup, solve, find min cost, and find optimal tour.']}, {'end': 12964.472, 'start': 12641.564, 'title': 'Dynamic programming explanation', 'summary': 'Explains the dynamic programming aspect of the algorithm, illustrating the process of storing optimal values from the start node to every other node, iterating through each node and caching the optimal value from s to i, with an example of the solve method and the combinations method.', 'duration': 322.908, 'highlights': ['The algorithm stores the optimal value from the start node to every other node, iterating through each node and caching the optimal value from s to i, which can be found in the distance matrix.', 'The solve method loops through the number of nodes in a partial tour and generates all bit sets of a specified size with exactly the designated number of bits set to one, ensuring the starting node is part of the generated subset.', 'The algorithm includes a step to look up in the memo table to determine the best partial tour value when the next node is not yet in the subset, essential for the dynamic programming aspect of the algorithm.', 'The combinations method fills up the subsets array and returns the result, with the second combinations method further explained in a separate tutorial on backtracking the power set.']}], 'duration': 968.271, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY11996201.jpg', 'highlights': ['The dynamic programming solution reduces the complexity to O(n^2*2^N), making graphs with roughly 23 nodes feasible for modern home computers.', 'The problem has been proven to be NP complete, and numerous approximation algorithms exist for finding an optimal solution for large inputs.', 'The traveling salesman problem aims to find the shortest possible route that visits each city exactly once and then returns to the city of origin.', 'The need to store the optimal value from the starting node to every other node is emphasized, solving the traveling salesman problem for all paths with exactly two nodes.', 'The importance of storing the dynamic programming state, including the set of visited nodes and the index of the last visited node, is highlighted.', 'The process of gradually expanding to visit all other unvisited nodes to make partial tours longer is explained, resulting in the generation of additional states for partial tours.', 'The pseudocode for solving the traveling salesman problem is presented, requiring advanced bit manipulation techniques.', 'The algorithm stores the optimal value from the start node to every other node, iterating through each node and caching the optimal value from s to i, which can be found in the distance matrix.', 'The solve method loops through the number of nodes in a partial tour and generates all bit sets of a specified size with exactly the designated number of bits set to one, ensuring the starting node is part of the generated subset.', 'The algorithm includes a step to look up in the memo table to determine the best partial tour value when the next node is not yet in the subset, essential for the dynamic programming aspect of the algorithm.']}, {'end': 15742.414, 'segs': [{'end': 13978.404, 'src': 'embed', 'start': 13942.866, 'weight': 4, 'content': [{'end': 13945.689, 'text': 'So you check that bit is equal to zero.', 'start': 13942.866, 'duration': 2.823}, {'end': 13952.217, 'text': "Today we're going to talk about Eulerian paths and circuits from a computer science perspective.", 'start': 13946.309, 'duration': 5.908}, {'end': 13959.687, 'text': "we're going to start with discussing what Euler paths and circuits are, how to determine their existence, how to find them.", 'start': 13952.217, 'duration': 7.47}, {'end': 13963.572, 'text': "And lastly, we're going to look at some code to wrap things up.", 'start': 13960.008, 'duration': 3.564}, {'end': 13968.276, 'text': "Let's begin with what an Eulerian path is.", 'start': 13964.193, 'duration': 4.083}, {'end': 13978.404, 'text': 'an Euler path, also called an Eulerian trail, is a path of edges in a graph that visits every edge exactly once.', 'start': 13968.276, 'duration': 10.128}], 'summary': 'Eulerian paths and circuits in graphs, covering existence, finding, and code examples.', 'duration': 35.538, 'max_score': 13942.866, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY13942866.jpg'}, {'end': 14416.07, 'src': 'embed', 'start': 14386.346, 'weight': 3, 'content': [{'end': 14392.452, 'text': 'An additional requirement I have not yet mentioned is that, when finding paths and circuits,', 'start': 14386.346, 'duration': 6.106}, {'end': 14399.118, 'text': 'is that all vertices with non zero degree need to belong to a single connected component.', 'start': 14392.452, 'duration': 6.666}, {'end': 14402.321, 'text': 'And here we have two connected components.', 'start': 14399.558, 'duration': 2.763}, {'end': 14405.264, 'text': 'So we cannot have an or there in path or circuit.', 'start': 14402.761, 'duration': 2.503}, {'end': 14410.205, 'text': "Now let's have a look at an example with a directed graph.", 'start': 14406.021, 'duration': 4.184}, {'end': 14416.07, 'text': "Does the following graph have any or there in paths or circuits? I'll give you a moment to think about it.", 'start': 14410.385, 'duration': 5.685}], 'summary': 'Requirement: all vertices with non zero degree must belong to a single connected component.', 'duration': 29.724, 'max_score': 14386.346, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY14386346.jpg'}, {'end': 15231.979, 'src': 'embed', 'start': 15200.921, 'weight': 0, 'content': [{'end': 15205.583, 'text': 'This means that if we encounter a node with one extra outgoing edge,', 'start': 15200.921, 'duration': 4.662}, {'end': 15212.066, 'text': 'that this node must be the unique starting node and we can return that nodes index immediately.', 'start': 15205.583, 'duration': 6.483}, {'end': 15217.928, 'text': 'Otherwise, we just want to ensure that we begin on a node with an outgoing edge.', 'start': 15212.326, 'duration': 5.602}, {'end': 15222.291, 'text': 'our default node node zero might not have an outgoing edge.', 'start': 15218.308, 'duration': 3.983}, {'end': 15231.979, 'text': 'In fact, this check prevents us from starting the depth first search on a singleton node, then return the start node after the loop.', 'start': 15222.551, 'duration': 9.428}], 'summary': 'If a node has one extra outgoing edge, it is the unique starting node, else start on a node with an outgoing edge.', 'duration': 31.058, 'max_score': 15200.921, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY15200921.jpg'}, {'end': 15340.386, 'src': 'embed', 'start': 15309.514, 'weight': 1, 'content': [{'end': 15313.415, 'text': 'Once we exit the loop, append the current node to the front of the solution.', 'start': 15309.514, 'duration': 3.901}, {'end': 15326.899, 'text': 'Returning back to the main method, the last thing we need to do is check that we have actually found the correct number of edges for an Eulerian path.', 'start': 15314.575, 'duration': 12.324}, {'end': 15330.781, 'text': 'It might be the case that our graph is disconnected.', 'start': 15327.339, 'duration': 3.442}, {'end': 15340.386, 'text': "And we found an Eulerian path on one of the many connected components of our graph, in which case, it's impossible to actually have an Eulerian path.", 'start': 15330.961, 'duration': 9.425}], 'summary': 'Appended current node to solution, checked for correct number of edges for an eulerian path', 'duration': 30.872, 'max_score': 15309.514, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY15309514.jpg'}], 'start': 12964.472, 'title': 'Solving tsp and eulerian paths', 'summary': 'Covers solving the traveling salesman problem using dynamic programming, backtracking, and constructing end state lookup, as well as implementing tsp dynamic programming iterative. it also discusses eulerian paths and circuits, their definitions, existence, and algorithmic determination on directed and undirected graphs, and finding eulerian paths on a graph with a time complexity of o(e).', 'chapters': [{'end': 13234.166, 'start': 12964.472, 'title': 'Traveling salesman problem with dynamic programming', 'summary': 'Explains how to solve the traveling salesman problem using dynamic programming, including using backtracking and constructing a bit mask for end state lookup, with detailed explanation of the find optimal tour function.', 'duration': 269.694, 'highlights': ['The chapter explains how to solve the traveling salesman problem using dynamic programming It provides an overview of solving the traveling salesman problem using dynamic programming, showcasing the iterative implementation and mentioning the availability of a recursive implementation.', 'Constructing a bit mask for end state lookup It describes the process of constructing a bit mask for the end state and utilizing it for a lookup in the memo table to find the minimum tour value.', 'Detailed explanation of the find optimal tour function It provides a detailed explanation of the find optimal tour function, outlining the process of working backwards from the end state, performing lookups in the memo table, and tracking the next optimal node to find the optimal tour.', 'Mention of the availability of a recursive implementation It mentions the availability of a recursive implementation for the traveling salesman problem in the repository, providing flexibility for those interested in different implementation approaches.']}, {'end': 13942.406, 'start': 13234.806, 'title': 'Tsp dynamic programming iterative', 'summary': 'Discusses the implementation of tsp dynamic programming iterative, including the creation of the object, methods for obtaining the optimal tour and minimum tour cost, and the algorithm for solving the traveling salesman problem.', 'duration': 707.6, 'highlights': ['The solve method initializes a variable for the end state, creates a memo table of size n times 2 to the end, and adds all edges from the starting node to every other node, followed by the phase of creating tours of path that are one longer.', 'The algorithm calculates the minimum tour cost by looping and doing a lookup in the table for the end node, then minimizes and updates the minimum cost, returning the result as the minimum tour cost.', 'The method for finding the actual tour works by initializing a variable for the last index, then looping n minus one times to determine the next best node, adding the node index to the tour, and finally reversing the order of the tour to obtain the actual tour.']}, {'end': 14574.075, 'start': 13942.866, 'title': 'Eulerian paths and circuits', 'summary': 'Discusses eulerian paths and circuits, explaining their definitions, determining their existence, and finding them algorithmically on both directed and undirected graphs, emphasizing the conditions required for each variant and providing examples to illustrate their application.', 'duration': 631.209, 'highlights': ['Eulerian paths and circuits are paths of edges in a graph that visit every edge exactly once and start and end on the same vertex, respectively. Eulerian paths and circuits are paths of edges in a graph that visit every edge exactly once and start and end on the same vertex, respectively.', 'The conditions for the existence of Eulerian paths and circuits are determined by counting the in and out degrees of each node in the graph, with specific constraints for undirected and directed graphs. The conditions for the existence of Eulerian paths and circuits are determined by counting the in and out degrees of each node in the graph, with specific constraints for undirected and directed graphs.', 'An algorithm that finds an Eulerian path can also be used to find an Eulerian circuit by feeding the graph with the Eulerian circuit into the Eulerian path algorithm. An algorithm that finds an Eulerian path can also be used to find an Eulerian circuit by feeding the graph with the Eulerian circuit into the Eulerian path algorithm.']}, {'end': 15309.013, 'start': 14581.79, 'title': 'Finding eulerian path on a graph', 'summary': 'Explains how to find an eulerian path on a graph, including verifying the existence of an eulerian path, finding the valid starting node, and modifying the depth-first search algorithm to ensure all edges are visited, with a time complexity of o(e).', 'duration': 727.223, 'highlights': ['The chapter explains how to modify the depth first search algorithm to force the depth first search to visit all the edges of the graph. The depth first search algorithm is modified to ensure all edges of the graph are visited, thereby guaranteeing the discovery of a valid Eulerian path, leading to a comprehensive traversal of the graph.', 'The time complexity required to find an Eulerian path is O(E). The time complexity of finding an Eulerian path is determined to be O(E), where E represents the number of edges in the graph, showcasing the efficiency of the algorithm.', 'The process of verifying the existence of an Eulerian path involves counting the in and out degrees of each node and checking the preconditions for an Eulerian path. The verification process involves counting the in and out degrees of each node in the graph and subsequently checking the preconditions necessary for the existence of an Eulerian path, ensuring the graph is a suitable candidate for the algorithm.']}, {'end': 15742.414, 'start': 15309.514, 'title': 'Eulerian path algorithm', 'summary': 'Explains the eulerian path algorithm, including its source code and key steps such as setup method, graph verification, finding the starting node, and the depth-first search method.', 'duration': 432.9, 'highlights': ['The depth first search method is used to find the Eulerian path by recursively exploring nodes and adding the current node to the solution. The depth first search method recursively explores nodes, decreases the number of outgoing edges for the current node, and terminates when there are no more outgoing edges, adding the current node to the front of the solution.', 'The setup method loops through all the edges to increment in and out array degrees and compute the number of edges in the graph. The setup method increments in and out array degrees, computes the number of edges in the graph, and returns null if there are no edges to work with.', 'The graph verification step ensures that the graph has the correct number of start and end nodes for an Eulerian path to exist. The graph verification step ensures that no node has too many outgoing or incoming edges, tracks the start and end nodes, and returns a Boolean value based on the correct number of start and end nodes.']}], 'duration': 2777.942, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY12964472.jpg', 'highlights': ['The time complexity of finding an Eulerian path is determined to be O(E), showcasing the efficiency of the algorithm.', 'The chapter explains how to solve the traveling salesman problem using dynamic programming, showcasing the iterative implementation and mentioning the availability of a recursive implementation.', 'An algorithm that finds an Eulerian path can also be used to find an Eulerian circuit by feeding the graph with the Eulerian circuit into the Eulerian path algorithm.', 'The depth first search method recursively explores nodes, decreases the number of outgoing edges for the current node, and terminates when there are no more outgoing edges, adding the current node to the front of the solution.', 'The verification process involves counting the in and out degrees of each node in the graph and subsequently checking the preconditions necessary for the existence of an Eulerian path, ensuring the graph is a suitable candidate for the algorithm.']}, {'end': 17695.589, 'segs': [{'end': 15774.338, 'src': 'embed', 'start': 15744.085, 'weight': 11, 'content': [{'end': 15748.367, 'text': 'The next thing I do before returning the solution, which is optional,', 'start': 15744.085, 'duration': 4.282}, {'end': 15753.729, 'text': 'is simply to empty the contents of the linked list into a primitive integer array, just for convenience.', 'start': 15748.367, 'duration': 5.362}, {'end': 15759.512, 'text': "I do this because it's easier for the color to index an array than it is a linked list.", 'start': 15753.969, 'duration': 5.543}, {'end': 15768.876, 'text': 'The rest of this file are just helper methods for creating a directed graph and adding directed edges to the graph.', 'start': 15759.912, 'duration': 8.964}, {'end': 15774.338, 'text': 'I also provide two examples, one from the previous slides and another that I made up.', 'start': 15769.116, 'duration': 5.222}], 'summary': 'Convert linked list to array for easier indexing. includes helper methods for directed graph creation and examples.', 'duration': 30.253, 'max_score': 15744.085, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY15744085.jpg'}, {'end': 15939.847, 'src': 'embed', 'start': 15909.809, 'weight': 10, 'content': [{'end': 15914.392, 'text': 'This one is a bit of a trick question, because there is no minimum spanning tree.', 'start': 15909.809, 'duration': 4.583}, {'end': 15920.596, 'text': 'all the nodes must be connected on a single component for a spanning tree to exist.', 'start': 15914.392, 'duration': 6.204}, {'end': 15925.72, 'text': "Let's change focus and start talking about prims algorithm.", 'start': 15922.779, 'duration': 2.941}, {'end': 15932.443, 'text': 'prims is one of my favorite minimum spanning tree algorithms because of how simple and how intuitive it is.', 'start': 15926.12, 'duration': 6.323}, {'end': 15939.847, 'text': "By nature, it's a greedy algorithm, which always selects the next best edge and adds it to the minimum spanning tree.", 'start': 15933.164, 'duration': 6.683}], 'summary': "Minimum spanning tree requires all nodes to be connected on a single component. prim's algorithm is a simple and intuitive greedy algorithm.", 'duration': 30.038, 'max_score': 15909.809, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY15909809.jpg'}, {'end': 15981.051, 'src': 'embed', 'start': 15955.975, 'weight': 7, 'content': [{'end': 15962.118, 'text': "And it's slightly harder but not impossible to find the minimum spanning forest of a graph.", 'start': 15955.975, 'duration': 6.143}, {'end': 15966.4, 'text': 'There are two well known versions of prims I want to discuss.', 'start': 15963.257, 'duration': 3.143}, {'end': 15973.085, 'text': 'The first is the common lazy version, which runs in big O of E log E.', 'start': 15966.98, 'duration': 6.105}, {'end': 15981.051, 'text': "And then there's the improved eager version which runs in big O of E log V, but requires a slightly different data structure.", 'start': 15973.085, 'duration': 7.966}], 'summary': 'Finding minimum spanning forest: prims has 2 versions - lazy: o(e log e), eager: o(e log v)', 'duration': 25.076, 'max_score': 15955.975, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY15955975.jpg'}, {'end': 16605.204, 'src': 'embed', 'start': 16574.121, 'weight': 0, 'content': [{'end': 16577.926, 'text': 'If that edge points to an already visited node again,', 'start': 16574.121, 'duration': 3.805}, {'end': 16586.998, 'text': "edges can become stale or outdated in the priority queue if the node they're pointing at becomes visited via another path.", 'start': 16577.926, 'duration': 9.072}, {'end': 16593.66, 'text': 'Next, actually add the edge to the minimum spanning tree by adding it to the MST edges array.', 'start': 16588.058, 'duration': 5.602}, {'end': 16598.962, 'text': 'And while adding the edge to the tree also sum over the edge costs.', 'start': 16594.119, 'duration': 4.843}, {'end': 16605.204, 'text': 'The last thing we want to do is call the add edges method with the new current node.', 'start': 16599.883, 'duration': 5.321}], 'summary': 'Optimize edge handling in minimum spanning tree algorithm to prevent staleness and add edges efficiently.', 'duration': 31.083, 'max_score': 16574.121, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY16574121.jpg'}, {'end': 16677.864, 'src': 'embed', 'start': 16624.716, 'weight': 1, 'content': [{'end': 16629.939, 'text': "Today we're talking about finding minimum spanning trees with prims algorithm.", 'start': 16624.716, 'duration': 5.223}, {'end': 16635.524, 'text': 'The lazy implementation of prims inserts E edges into a priority queue.', 'start': 16630.461, 'duration': 5.063}, {'end': 16643.751, 'text': 'This results in each pole operation on the priority queue to be big O of log e in the eager version.', 'start': 16636.085, 'duration': 7.666}, {'end': 16650.498, 'text': 'we maintain the idea that instead of adding edges to the priority queue which can later become stale,', 'start': 16643.751, 'duration': 6.747}, {'end': 16660.187, 'text': 'that instead we should track node edge key value pairs that can easily be updated and pulled to determine the next best edge,', 'start': 16650.498, 'duration': 9.689}, {'end': 16662.73, 'text': 'we should add to the minimum spanning tree.', 'start': 16660.187, 'duration': 2.543}, {'end': 16666.774, 'text': "For this all to make sense, there's a key realization that needs to happen.", 'start': 16663.05, 'duration': 3.724}, {'end': 16677.864, 'text': 'And that is for any MST with directed edges, each node is paired with exactly one of its incoming edges that is except for the start node.', 'start': 16666.973, 'duration': 10.891}], 'summary': 'Prims algorithm creates minimum spanning trees by tracking node-edge key value pairs instead of adding edges to a priority queue, resulting in big o of log e in the eager version.', 'duration': 53.148, 'max_score': 16624.716, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY16624716.jpg'}, {'end': 16959.422, 'src': 'embed', 'start': 16926.625, 'weight': 4, 'content': [{'end': 16935.651, 'text': "Let's begin the algorithm on node zero, start by iterating over all the edges of zero and relax them during the relaxing process.", 'start': 16926.625, 'duration': 9.026}, {'end': 16940.434, 'text': 'Add a node edge pair to the index party queue if it does not exist yet.', 'start': 16936.031, 'duration': 4.403}, {'end': 16946.918, 'text': 'Otherwise, update the value if the new edge has a better cost than what already exists.', 'start': 16940.794, 'duration': 6.124}, {'end': 16959.422, 'text': 'The first node edge pair we add is node two with the incoming edge from zero to two with a cost of zero and similarly for the rest of zeros edges.', 'start': 16947.498, 'duration': 11.924}], 'summary': 'Iterate over edges of node zero, adding to queue if new edge is better.', 'duration': 32.797, 'max_score': 16926.625, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY16926625.jpg'}, {'end': 17251.909, 'src': 'embed', 'start': 17227.759, 'weight': 2, 'content': [{'end': 17237.986, 'text': 'And my guess is that the denser the graph, the fewer relaxation operations need to be performed, which is an expensive part of prims algorithm.', 'start': 17227.759, 'duration': 10.227}, {'end': 17245.408, 'text': 'Since the time to iterate over all the edges of a node is constant, but fewer relaxation operations are needed.', 'start': 17238.826, 'duration': 6.582}, {'end': 17247.648, 'text': 'performance should increase as a result.', 'start': 17245.408, 'duration': 2.24}, {'end': 17248.768, 'text': 'But I may be wrong.', 'start': 17247.888, 'duration': 0.88}, {'end': 17251.909, 'text': 'Even still, the results are interesting.', 'start': 17249.689, 'duration': 2.22}], 'summary': 'Denser graph -> fewer relaxation ops -> improved performance, though results are intriguing.', 'duration': 24.15, 'max_score': 17227.759, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY17227759.jpg'}, {'end': 17361.007, 'src': 'embed', 'start': 17331.435, 'weight': 13, 'content': [{'end': 17334.217, 'text': 'The first thing we do is mark the current node as visited.', 'start': 17331.435, 'duration': 2.782}, {'end': 17337.298, 'text': "So we don't visit it again in the future.", 'start': 17334.517, 'duration': 2.781}, {'end': 17344.14, 'text': 'Then I reach into our graph adjacency list and get all the edges going outwards from the current node.', 'start': 17338.078, 'duration': 6.062}, {'end': 17348.962, 'text': 'As we enter the loop and start iterating over all the outgoing edges,', 'start': 17345.321, 'duration': 3.641}, {'end': 17353.344, 'text': 'the first thing I do inside the loop is grab a reference to the destination node index.', 'start': 17348.962, 'duration': 4.382}, {'end': 17356.265, 'text': 'This is the node the edges pointing at.', 'start': 17354.084, 'duration': 2.181}, {'end': 17361.007, 'text': 'Next, skip edges which point at already visited nodes.', 'start': 17357.125, 'duration': 3.882}], 'summary': 'Mark current node as visited, get all outgoing edges, skip visited nodes.', 'duration': 29.572, 'max_score': 17331.435, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY17331435.jpg'}, {'end': 17450.237, 'src': 'embed', 'start': 17426.119, 'weight': 6, 'content': [{'end': 17432.807, 'text': 'But assuming that that is not the case, return the MST cost and the edges which make up the spanning tree.', 'start': 17426.119, 'duration': 6.688}, {'end': 17437.051, 'text': 'And that concludes the pseudocode for prims algorithm.', 'start': 17433.608, 'duration': 3.443}, {'end': 17441.813, 'text': 'All right, here we are in the source code for prims implemented in Java.', 'start': 17437.632, 'duration': 4.181}, {'end': 17450.237, 'text': 'At the top here, I posted some instructions on how to download and run the script in case you wanted to play around with it a little bit.', 'start': 17442.514, 'duration': 7.723}], 'summary': 'Pseudocode for prims algorithm discussed for spanning tree.', 'duration': 24.118, 'max_score': 17426.119, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY17426119.jpg'}, {'end': 17536.065, 'src': 'embed', 'start': 17503.761, 'weight': 14, 'content': [{'end': 17513.384, 'text': 'right here you can see that this particular minimum spanning tree has a cost of nine and it has these six edges.', 'start': 17503.761, 'duration': 9.623}, {'end': 17525.527, 'text': "If you were curious as to how the adjacency list gets initialized and how I add edges to the graph, here's the code that does exactly that.", 'start': 17516.705, 'duration': 8.822}, {'end': 17536.065, 'text': 'Next up is a class struct, which represents a directed edge used in the graph.', 'start': 17529.782, 'duration': 6.283}], 'summary': 'Minimum spanning tree cost: 9, 6 edges. code initializes adjacency list and adds edges to the graph.', 'duration': 32.304, 'max_score': 17503.761, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY17503761.jpg'}], 'start': 15744.085, 'title': 'Minimum spanning trees', 'summary': "Covers the concept of minimum spanning trees, explaining examples of finding minimum spanning trees, discussing prim's algorithm properties, versions, and implementation with priority queue, and emphasizes their uniqueness and algorithms' complexities. it also explores lazy implementation of prims algorithm, index priority queue in minimum spanning tree algorithm, and prims algorithm and graph representation, providing insights into their practical applications and performance comparisons.", 'chapters': [{'end': 16104.523, 'start': 15744.085, 'title': 'Minimum spanning trees', 'summary': "Covers the concept of minimum spanning trees, explaining what they are, providing examples of finding minimum spanning trees in weighted graphs, and discussing prim's algorithm, including its properties, versions, and implementation with priority queue. it emphasizes the importance of minimum spanning trees and provides insights into their uniqueness and algorithms' complexities.", 'duration': 360.438, 'highlights': ['A minimum spanning tree, or just MST for short, is a tree which spans the whole graph, connecting all nodes together while minimizing the total edge cost. Defines a minimum spanning tree as a tree that connects all nodes in a graph while minimizing the total edge cost.', 'One possible minimum spanning tree is the following edges highlighted in green, whose edge costs sum to nine. Provides an example of a minimum spanning tree with highlighted edges and their total cost.', 'Another possible minimum spanning tree is the following with a cost of 14. Presents another example of a minimum spanning tree with its total cost.', 'One possible answer with the minimum spanning tree highlighted in green with a cost of 39. Offers an example of a minimum spanning tree with highlighted edges and their total cost.', "prims is one of my favorite minimum spanning tree algorithms because of how simple and how intuitive it is. Highlights the simplicity and intuitiveness of prim's algorithm for minimum spanning trees."]}, {'end': 16437.705, 'start': 16104.523, 'title': 'Lazy implementation of prims', 'summary': "Explains the lazy implementation of prim's algorithm for finding the minimum spanning tree in a graph, demonstrating the steps involved and concluding with the total cost of the minimum spanning tree as 20.", 'duration': 333.182, 'highlights': ['The minimum spanning tree is complete when the number of edges in the tree is one less than the number of nodes in the graph. The completeness of the minimum spanning tree is determined by the number of edges being one less than the number of nodes in the graph.', 'The total cost of the minimum spanning tree is calculated as 20. The total cost of the minimum spanning tree is determined to be 20, achieved by summing up the cost of all the selected edges.', 'Edges are represented as triplets containing the start node, end node, and edge cost. The representation of edges as triplets includes the start node, end node, and the cost of the edge.']}, {'end': 16737.184, 'start': 16438.708, 'title': 'Lazy implementation of prims algorithm', 'summary': 'Discusses the lazy implementation of prims algorithm for finding minimum spanning trees, where e edges are inserted into a priority queue, resulting in each poll operation on the priority queue to be big o of log e, and emphasizes the realization that for any mst with directed edges, each node is paired with exactly one of its incoming edges, except for the start node.', 'duration': 298.476, 'highlights': ['For any MST with directed edges, each node is paired with exactly one of its incoming edges, except for the start node. The chapter emphasizes the realization that for any MST with directed edges, each node is paired with exactly one of its incoming edges, except for the start node.', 'The lazy implementation of Prims inserts E edges into a priority queue, resulting in each poll operation on the priority queue to be big O of log e. The lazy implementation of Prims inserts E edges into a priority queue, resulting in each poll operation on the priority queue to be big O of log e.', "In the eager version, we are trying to determine which of a node's incoming edges we should select to include in the MST. In the eager version, the goal is to determine which of a node's incoming edges should be selected to include in the MST."]}, {'end': 17052.147, 'start': 16737.563, 'title': 'Index priority queue in minimum spanning tree algorithm', 'summary': 'Explains the use of an index priority queue to efficiently update and retrieve node-edge pairs in the minimum spanning tree algorithm, reducing the time complexity from o(e log e) to o(e log v) and provides a detailed walkthrough of the eager version of the algorithm with practical examples.', 'duration': 314.584, 'highlights': ['The use of an Index Priority Queue reduces time complexity from O(E log E) to O(E log V) By utilizing an Index Priority Queue, the time complexity of the Minimum Spanning Tree algorithm is reduced from O(E log E) to O(E log V), as it can efficiently update and pull key value pairs in logarithmic time.', 'Explanation of the eager version of the Minimum Spanning Tree algorithm The transcript provides a detailed explanation of the eager version of the Minimum Spanning Tree algorithm, involving maintaining an IP queue of size v, sorting vertex-edge pairs based on minimum edge cost, and iterative relaxation of edges to form the Minimum Spanning Tree.', 'Practical walkthrough of the algorithm with a weighted undirected graph example The chapter offers a practical example of the eager version of the Minimum Spanning Tree algorithm, demonstrating the step-by-step execution of the algorithm on a weighted undirected graph, showcasing the efficient update and retrieval of node-edge pairs using an Index Priority Queue.']}, {'end': 17695.589, 'start': 17084.426, 'title': 'Prims algorithm and graph representation', 'summary': "Discusses the implementation of prims algorithm, comparing the performance of using adjacency list versus adjacency matrix, highlighting that adjacency list outperforms the matrix for graphs with fewer edges, and the matrix becomes the obvious choice as the edge density increases, impacting the algorithm's performance based on the graph's sparsity or density.", 'duration': 611.163, 'highlights': ["The performance comparison of using adjacency list versus adjacency matrix, highlighting that the adjacency list outperforms the matrix for graphs with fewer edges, and the matrix becomes the obvious choice as the edge density increases, impacting the algorithm's performance based on the graph's sparsity or density.", 'The explanation of the eager implementation of Prims algorithm, with a focus on the key details such as the use of index priority queue, graph representation, and the impact of graph sparsity or density on algorithm performance.', 'The pseudocode for the eager implementation of Prims algorithm, discussing the variables used, the relax edges at node method, and the overall algorithm logic.', 'The Java source code for implementing Prims algorithm, including instructions on how to download and run the script, setting up the graph, creating a minimum spanning tree solver, and obtaining the minimum spanning tree cost and edges.', 'The details of the minimum spanning tree solver class, including the variables used, the get MST method, get MST cost method, and the solve method, which initializes variables and memory allocation for the arrays used.']}], 'duration': 1951.504, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY15744085.jpg', 'highlights': ['The use of an Index Priority Queue reduces time complexity from O(E log E) to O(E log V)', 'The performance comparison of using adjacency list versus adjacency matrix, highlighting that the adjacency list outperforms the matrix for graphs with fewer edges', 'The lazy implementation of Prims inserts E edges into a priority queue, resulting in each poll operation on the priority queue to be big O of log e', 'The explanation of the eager implementation of Prims algorithm, with a focus on the key details such as the use of index priority queue, graph representation, and the impact of graph sparsity or density on algorithm performance', 'The completeness of the minimum spanning tree is determined by the number of edges being one less than the number of nodes in the graph', 'The chapter offers a practical example of the eager version of the Minimum Spanning Tree algorithm, demonstrating the step-by-step execution of the algorithm on a weighted undirected graph, showcasing the efficient update and retrieval of node-edge pairs using an Index Priority Queue', 'The total cost of the minimum spanning tree is calculated as 20', 'The representation of edges as triplets includes the start node, end node, and the cost of the edge', "The eager version, the goal is to determine which of a node's incoming edges should be selected to include in the MST", "The simplicity and intuitiveness of prim's algorithm for minimum spanning trees", 'The details of the minimum spanning tree solver class, including the variables used, the get MST method, get MST cost method, and the solve method, which initializes variables and memory allocation for the arrays used', 'The pseudocode for the eager implementation of Prims algorithm, discussing the variables used, the relax edges at node method, and the overall algorithm logic', 'The explanation of the eager version of the Minimum Spanning Tree algorithm', 'The Java source code for implementing Prims algorithm, including instructions on how to download and run the script, setting up the graph, creating a minimum spanning tree solver, and obtaining the minimum spanning tree cost and edges', 'The chapter emphasizes the realization that for any MST with directed edges, each node is paired with exactly one of its incoming edges, except for the start node']}, {'end': 18874.826, 'segs': [{'end': 18113.503, 'src': 'embed', 'start': 18089.787, 'weight': 4, 'content': [{'end': 18096.694, 'text': "When an edge becomes overcapacitated in some manner, it means that we've pushed the system past its limit.", 'start': 18089.787, 'duration': 6.907}, {'end': 18102.297, 'text': 'In the context of edges representing pipes with water, it means your pipe broke or it leaked.', 'start': 18096.994, 'duration': 5.303}, {'end': 18108.261, 'text': 'If your edge is a wire with electric current, it means your wire literally fried or melted,', 'start': 18102.637, 'duration': 5.624}, {'end': 18111.963, 'text': 'exploded or something bad happened to it because there was too much electric current.', 'start': 18108.261, 'duration': 3.702}, {'end': 18113.503, 'text': 'This is not good.', 'start': 18112.683, 'duration': 0.82}], 'summary': 'Overcapacitated edges lead to system failure, e.g. pipes breaking or wires melting.', 'duration': 23.716, 'max_score': 18089.787, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY18089787.jpg'}, {'end': 18296.612, 'src': 'embed', 'start': 18271.974, 'weight': 5, 'content': [{'end': 18277.578, 'text': "Effectively, we don't know which are the best or even correct augmenting paths to take.", 'start': 18271.974, 'duration': 5.604}, {'end': 18287.325, 'text': "So this mechanism enables us to freely find any augmenting paths without having to worry about whether or not we'll be able to achieve the maximum flow.", 'start': 18277.818, 'duration': 9.507}, {'end': 18296.612, 'text': 'it should be mentioned that residual edges become valid edges to take when finding an augmenting path in later iterations.', 'start': 18287.325, 'duration': 9.287}], 'summary': 'A mechanism enables freely finding augmenting paths without concerns about achieving maximum flow.', 'duration': 24.638, 'max_score': 18271.974, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY18271974.jpg'}, {'end': 18379.167, 'src': 'embed', 'start': 18349.737, 'weight': 0, 'content': [{'end': 18354.398, 'text': 'as the difference between the capacity and the flow of that edge.', 'start': 18349.737, 'duration': 4.661}, {'end': 18360.82, 'text': 'That is the difference between the capacity and the flow is the true remaining capacity for that edge.', 'start': 18354.918, 'duration': 5.902}, {'end': 18367.843, 'text': 'This ensures that the remaining capacity of an edge is always non negative, even if the flow can be negative.', 'start': 18361.18, 'duration': 6.663}, {'end': 18374.305, 'text': 'For example, in the residual edges we have right now, zero minus minus six is six.', 'start': 18368.083, 'duration': 6.222}, {'end': 18379.167, 'text': 'So we know that all our residual edges actually have a remaining capacity of six.', 'start': 18374.605, 'duration': 4.562}], 'summary': 'Remaining capacity of residual edges is always non-negative, with a value of six in the current example.', 'duration': 29.43, 'max_score': 18349.737, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY18349737.jpg'}, {'end': 18417.404, 'src': 'embed', 'start': 18393.595, 'weight': 1, 'content': [{'end': 18404.061, 'text': 'The key realization to make at this point is that the sum of the bottleneck values that we acquire with each augmenting paths will result in the maximum flow.', 'start': 18393.595, 'duration': 10.466}, {'end': 18406.741, 'text': "And that's the whole premise of this algorithm.", 'start': 18404.361, 'duration': 2.38}, {'end': 18415.883, 'text': "It doesn't matter so much how you find augmenting paths, but so long as you keep summing the bottleneck values which they produce,", 'start': 18406.882, 'duration': 9.001}, {'end': 18417.404, 'text': "you'll find the maximum flow.", 'start': 18415.883, 'duration': 1.521}], 'summary': 'The maximum flow is the sum of bottleneck values from augmenting paths, regardless of how they are found.', 'duration': 23.809, 'max_score': 18393.595, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY18393595.jpg'}, {'end': 18660.799, 'src': 'embed', 'start': 18636.269, 'weight': 2, 'content': [{'end': 18644.015, 'text': 'Instead, push relable algorithms maintain this concept of a pre flow, if you will, to find the maximum flow of a network.', 'start': 18636.269, 'duration': 7.746}, {'end': 18649.436, 'text': 'Please be mindful that the time complexity is posted here are very pessimistic.', 'start': 18644.255, 'duration': 5.181}, {'end': 18654.678, 'text': 'In practice, running maximum flow with any of these operates much faster.', 'start': 18649.796, 'duration': 4.882}, {'end': 18660.799, 'text': "So it's very hard to compare the performance of two flow algorithms solely based on the complexity.", 'start': 18654.978, 'duration': 5.821}], 'summary': 'Relable algorithms maintain pre flow concept for maximum network flow; time complexity may not reflect practical performance.', 'duration': 24.53, 'max_score': 18636.269, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY18636269.jpg'}, {'end': 18717.61, 'src': 'embed', 'start': 18686.153, 'weight': 3, 'content': [{'end': 18690.476, 'text': 'of the edges and achieving the maximum flow of 23..', 'start': 18686.153, 'duration': 4.323}, {'end': 18695.559, 'text': 'The source code and the example I have lined up for you today can both be found on GitHub.', 'start': 18690.476, 'duration': 5.083}, {'end': 18703.423, 'text': "There's a link in the description for today, I encourage you to check that out and also play along as we're going over the source code.", 'start': 18695.679, 'duration': 7.744}, {'end': 18706.564, 'text': 'Alright, here we are in the source code written in Java.', 'start': 18703.623, 'duration': 2.941}, {'end': 18717.61, 'text': 'This program has three main supporting classes, an edge class, a network flow solver base, and the Ford Fulkerson depth first search solver.', 'start': 18706.824, 'duration': 10.786}], 'summary': 'Source code in java with maximum flow of 23, available on github.', 'duration': 31.457, 'max_score': 18686.153, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY18686153.jpg'}], 'start': 17696.349, 'title': "Implementing prim's algorithm and ford fulkerson method", 'summary': "Covers the eager implementation of prim's algorithm for finding a minimum spanning tree using indexed priority queue, while also introducing the ford fulkerson method for maximum flow in a flow graph and its applications, with examples illustrating a maximum flow of seven and 23.", 'chapters': [{'end': 17890.749, 'start': 17696.349, 'title': "Eager implementation of prim's algorithm", 'summary': "Discusses the eager implementation of prim's algorithm for finding a minimum spanning tree, using an indexed priority queue and a while loop to add edges and build the tree until the loop breaks, ultimately determining the existence of a minimum spanning tree based on the edge count.", 'duration': 194.4, 'highlights': ['Indexed Priority Queue implemented using a binary heap The indexed priority queue of size n is implemented using a binary heap, providing efficient performance for the algorithm.', 'Relaxing edges and adding to priority queue The algorithm relaxes edges at each node and adds them to the index priority queue, contributing to the process of building the minimum spanning tree.', 'Determining existence of minimum spanning tree The algorithm determines the existence of a minimum spanning tree based on the edge count, where if the edge count is equal to the expected number of edges in the minimum spanning tree, a minimum spanning tree is successfully computed.']}, {'end': 18113.503, 'start': 17891.389, 'title': 'Max flow problem and ford fulkerson method', 'summary': 'Introduces the concept of maximum flow in a flow graph, where edges have capacities and flow values, and demonstrates its applications in network traffic, with the ford fulkerson method being used to find the maximum flow, illustrated by an example of a network with a maximum flow of seven.', 'duration': 222.114, 'highlights': ['The maximum flow problem asks with an infinite input source, how much flow can we push through the network without exceeding the capacity of any edge? The problem of finding the maximum flow involves determining the amount of flow that can be pushed through a network without exceeding the capacity of any edge.', 'The maximum flow for this particular network is seven. The specific network demonstrated as an example has a maximum flow of seven, determined after running the maximum flow algorithm.', 'Running a maximum flow algorithm is used to determine how much flow each edge should receive to achieve the overall maximum flow. The process of running a maximum flow algorithm is utilized to allocate flow to each edge in order to achieve the overall maximum flow for the network.']}, {'end': 18874.826, 'start': 18113.944, 'title': 'Ford fulkerson method for maximum flow', 'summary': "Explains the ford fulkerson method for finding the maximum flow and min cut in a flow graph, detailing the concept of augmenting paths, residual graphs, and bottleneck values, and the algorithm's time complexity and variations such as edmonds karp and push relabel. it also provides a demonstration of setting up a flow graph and running the ford fulkerson method to achieve a maximum flow of 23.", 'duration': 760.882, 'highlights': ['The Ford Fulkerson method finds augmenting paths through the residual graph and augments the flow until no more augmenting paths can be found. This is the fundamental process of the Ford Fulkerson method, iteratively improving the flow until the maximum flow is achieved.', 'An augmenting path is a path of edges in the residual graph with capacity greater than zero from the source s to the sink t. Understanding the concept of augmenting paths is crucial to the Ford Fulkerson method, as it determines the path for increasing the flow.', 'The algorithm derives its time complexity from how augmenting paths are found, with a complexity of O(F * E) if a depth first search is used. The time complexity is an important consideration for the efficiency of the algorithm, and it depends on the method used to find augmenting paths.']}], 'duration': 1178.477, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY17696349.jpg', 'highlights': ['Indexed Priority Queue implemented using a binary heap for efficient performance.', 'Relaxing edges and adding to priority queue contributes to building the minimum spanning tree.', 'Determining existence of minimum spanning tree based on edge count.', 'Maximum flow problem involves determining the amount of flow that can be pushed through a network without exceeding edge capacity.', 'Running a maximum flow algorithm allocates flow to each edge to achieve overall maximum flow.', 'Ford Fulkerson method iteratively improves flow until maximum flow is achieved.', 'Understanding augmenting paths is crucial to the Ford Fulkerson method.', 'Time complexity of Ford Fulkerson method depends on the method used to find augmenting paths.']}, {'end': 19623.673, 'segs': [{'end': 19053.151, 'src': 'embed', 'start': 19027.063, 'weight': 4, 'content': [{'end': 19036.134, 'text': "There's also the remaining capacity method, which can be used to determine the maximum amount of flow that we can push through this edge.", 'start': 19027.063, 'duration': 9.071}, {'end': 19040.198, 'text': 'This method works whether the flow is positive or negative.', 'start': 19036.354, 'duration': 3.844}, {'end': 19045.665, 'text': 'Next is the augment method which augments the flow for this edge alone.', 'start': 19040.639, 'duration': 5.026}, {'end': 19053.151, 'text': 'All it does is it increases the flow on the forward edge by the bottleneck value we found along the augmenting path.', 'start': 19045.985, 'duration': 7.166}], 'summary': 'Methods for determining maximum flow capacity and augmenting flow on an edge.', 'duration': 26.088, 'max_score': 19027.063, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY19027063.jpg'}, {'end': 19205.546, 'src': 'embed', 'start': 19182.637, 'weight': 2, 'content': [{'end': 19191.04, 'text': 'we can simply reset all the visited states of every node simultaneously by simply incrementing the visited token.', 'start': 19182.637, 'duration': 8.403}, {'end': 19199.464, 'text': "I know it's kind of hacky, but it's super efficient and really handy to have the alternative is actually to maintain a Boolean visitor array.", 'start': 19191.22, 'duration': 8.244}, {'end': 19205.546, 'text': 'And you fill that with false values every time just before you find an augmenting path.', 'start': 19199.724, 'duration': 5.822}], 'summary': 'Reset all visited states by incrementing the visited token, efficient and handy.', 'duration': 22.909, 'max_score': 19182.637, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY19182637.jpg'}, {'end': 19276.282, 'src': 'embed', 'start': 19251.483, 'weight': 3, 'content': [{'end': 19259.87, 'text': 'Looking at the constructor, we require the user to specify the number of nodes along with the index of the source and the sink nodes.', 'start': 19251.483, 'duration': 8.387}, {'end': 19267.476, 'text': 'Then, inside this method, I also take the opportunity to initialize the flow graph and as well,', 'start': 19260.43, 'duration': 7.046}, {'end': 19271.879, 'text': "allocate some memory for the visited array we'll be making use of later.", 'start': 19267.476, 'duration': 4.403}, {'end': 19276.282, 'text': 'In the initialize empty flow graph method.', 'start': 19272.86, 'duration': 3.422}], 'summary': 'Constructor initializes flow graph with specified nodes and allocates memory for visited array.', 'duration': 24.799, 'max_score': 19251.483, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY19251483.jpg'}, {'end': 19623.673, 'src': 'embed', 'start': 19597.913, 'weight': 0, 'content': [{'end': 19602.335, 'text': 'However, first check that the bottleneck value is greater than zero.', 'start': 19597.913, 'duration': 4.422}, {'end': 19607.478, 'text': 'it could be the case that we never actually made it to the sink and we hit a dead end.', 'start': 19602.335, 'duration': 5.143}, {'end': 19610.02, 'text': "Assuming that's not the case,", 'start': 19608.138, 'duration': 1.882}, {'end': 19620.19, 'text': 'simply argument the flow by increasing the flow in the forward edge by the bottleneck value and decreasing the flow in the residual edge by the bottleneck value.', 'start': 19610.02, 'duration': 10.17}, {'end': 19623.673, 'text': 'After that, simply return the bottleneck value.', 'start': 19620.871, 'duration': 2.802}], 'summary': 'Check bottleneck > 0, increase flow, return bottleneck value.', 'duration': 25.76, 'max_score': 19597.913, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY19597913.jpg'}], 'start': 18875.527, 'title': 'Compiling and executing java code and network flow solver', 'summary': 'Explains how to compile and execute java code, demonstrating the process of changing directory, compiling the file, and executing it. it also introduces classes such as edge, network flow solver base, and ford fulkerson depth first search solver, providing methods for creating and manipulating edges in a flow graph and implementing the solve method for finding the maximum flow in a flow graph.', 'chapters': [{'end': 18924.55, 'start': 18875.527, 'title': 'Compiling and executing java code', 'summary': 'Explains how to compile a java file and execute it, demonstrating the process of changing directory, compiling the file, and executing it to obtain the desired output.', 'duration': 49.023, 'highlights': ['The process involves changing directory into the algorithms folder and compiling the Ford for the person example dot Java file in the graph theory network flow examples package using Java C.', 'Executing the compiled Java file by typing Java followed by the name of the class produces the desired output.']}, {'end': 19623.673, 'start': 18924.89, 'title': 'Network flow solver', 'summary': 'Introduces the classes edge, network flow solver base, and ford fulkerson depth first search solver. the edge class defines methods for creating and manipulating edges in a flow graph, while the network flow solver base provides a generic base for max flow solvers. the ford fulkerson depth first search solver extends the network flow base solver to implement the solve method for finding the maximum flow in a flow graph.', 'duration': 698.783, 'highlights': ['The max flow value is 23, and the edge details include start and end nodes, flow amount, edge capacity, and a Boolean value indicating whether the edge is a residual edge or not. The max flow value is 23, and the edge details include start and end nodes, flow amount, edge capacity, and a Boolean value indicating whether the edge is a residual edge or not.', 'The Edge class includes instance variables for start and end nodes, flow, and capacity, with the capacity being constant and the flow being dynamic and adjustable. The Edge class includes instance variables for start and end nodes, flow, and capacity, with the capacity being constant and the flow being dynamic and adjustable.', 'The Network Flow Solver Base class contains variables for infinity, the number of nodes, source and sink indices, a visited token, and a Boolean variable indicating whether the solver has been run. The Network Flow Solver Base class contains variables for infinity, the number of nodes, source and sink indices, a visited token, and a Boolean variable indicating whether the solver has been run.']}], 'duration': 748.146, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY18875527.jpg', 'highlights': ['The process involves changing directory into the algorithms folder and compiling the Ford for the person example dot Java file in the graph theory network flow examples package using Java C.', 'Executing the compiled Java file by typing Java followed by the name of the class produces the desired output.', 'The max flow value is 23, and the edge details include start and end nodes, flow amount, edge capacity, and a Boolean value indicating whether the edge is a residual edge or not.', 'The Edge class includes instance variables for start and end nodes, flow, and capacity, with the capacity being constant and the flow being dynamic and adjustable.', 'The Network Flow Solver Base class contains variables for infinity, the number of nodes, source and sink indices, a visited token, and a Boolean variable indicating whether the solver has been run.']}, {'end': 21351.285, 'segs': [{'end': 19929.588, 'src': 'embed', 'start': 19902.193, 'weight': 0, 'content': [{'end': 19906.056, 'text': 'Now suppose we want to find the maximum cardinality bipartite matching.', 'start': 19902.193, 'duration': 3.863}, {'end': 19911.279, 'text': 'Or in other words, we want to match as many people with as many books as we can.', 'start': 19906.756, 'duration': 4.523}, {'end': 19915.081, 'text': "Let's try the greedy approach to this matching problem.", 'start': 19911.779, 'duration': 3.302}, {'end': 19918.222, 'text': "Let's start with person green.", 'start': 19916.081, 'duration': 2.141}, {'end': 19922.625, 'text': 'Their first edge connects to the second book on the right side.', 'start': 19919.143, 'duration': 3.482}, {'end': 19924.926, 'text': 'The second book is unallocated.', 'start': 19923.265, 'duration': 1.661}, {'end': 19929.588, 'text': 'So person green is matched with what is now book green.', 'start': 19925.026, 'duration': 4.562}], 'summary': 'Using a greedy approach to maximize bipartite matching.', 'duration': 27.395, 'max_score': 19902.193, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY19902193.jpg'}, {'end': 20016.332, 'src': 'embed', 'start': 19957.417, 'weight': 2, 'content': [{'end': 19964.603, 'text': "Now person red, red only has one edge, meaning that they're only willing to read that one book.", 'start': 19957.417, 'duration': 7.186}, {'end': 19969.768, 'text': 'However, that book has already been allocated to person orange.', 'start': 19965.003, 'duration': 4.765}, {'end': 19972.17, 'text': 'So person red cannot have it.', 'start': 19969.968, 'duration': 2.202}, {'end': 19974.812, 'text': 'Next up is person brown.', 'start': 19972.73, 'duration': 2.082}, {'end': 19979.697, 'text': 'They also want person oranges book, but they also cannot have it.', 'start': 19975.253, 'duration': 4.444}, {'end': 19983.959, 'text': "Fortunately, they have other options of books they're willing to read.", 'start': 19980.097, 'duration': 3.862}, {'end': 19986.1, 'text': 'So person brown gets one of those.', 'start': 19984.259, 'duration': 1.841}, {'end': 19992.663, 'text': 'So, in the end, the greedy approach only found a matching of four.', 'start': 19986.34, 'duration': 6.323}, {'end': 19995.964, 'text': 'only four people were able to be matched with books.', 'start': 19992.663, 'duration': 3.301}, {'end': 19997.145, 'text': 'but can we do any better?', 'start': 19995.964, 'duration': 1.181}, {'end': 20000.826, 'text': 'Is this the true maximum cardinality matching?', 'start': 19997.665, 'duration': 3.161}, {'end': 20006.309, 'text': "Turns out that it's not a greedy approach to the maximum matching.", 'start': 20001.347, 'duration': 4.962}, {'end': 20008.73, 'text': 'problem will not work, as we just saw.', 'start': 20006.309, 'duration': 2.421}, {'end': 20016.332, 'text': 'we need a more sophisticated approach to ensure that we are able to get that maximum matching.', 'start': 20009.19, 'duration': 7.142}], 'summary': 'Greedy approach found matching of only four people with books, need more sophisticated approach for maximum matching.', 'duration': 58.915, 'max_score': 19957.417, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY19957417.jpg'}, {'end': 20234.024, 'src': 'embed', 'start': 20207.689, 'weight': 4, 'content': [{'end': 20213.133, 'text': 'This effectively limits or controls the number of copies of a book.', 'start': 20207.689, 'duration': 5.444}, {'end': 20222.319, 'text': "let's change the capacity of those edges leading into the sink to allow having multiple copies of the same book and see what happens.", 'start': 20213.133, 'duration': 9.186}, {'end': 20227.481, 'text': 'If we rerun the max flow algorithm once again through the network,', 'start': 20223.079, 'duration': 4.402}, {'end': 20234.024, 'text': 'we see that we now have people matched with the same book multiple times because multiple copies exist.', 'start': 20227.481, 'duration': 6.543}], 'summary': 'Increasing edge capacity allows multiple copies of book, leading to people being matched with the same book multiple times.', 'duration': 26.335, 'max_score': 20207.689, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY20207689.jpg'}, {'end': 20346.704, 'src': 'embed', 'start': 20314.311, 'weight': 1, 'content': [{'end': 20314.831, 'text': "Let's have a look.", 'start': 20314.311, 'duration': 0.52}, {'end': 20322.056, 'text': "Suppose there are M mice out on a field and there's a hungry owl about to make a move.", 'start': 20315.331, 'duration': 6.725}, {'end': 20327.139, 'text': 'Assume that the owl can reach every single one of these mice.', 'start': 20322.456, 'duration': 4.683}, {'end': 20337.722, 'text': 'further suppose that there are H holes scattered across the ground and that each hole has a certain capacity for the number of mice that can hide in it.', 'start': 20327.819, 'duration': 9.903}, {'end': 20346.704, 'text': 'We also happen to know that every mouse is capable of running a distance of R in any direction before being caught by the owl.', 'start': 20338.202, 'duration': 8.502}], 'summary': 'Mice in field with owl, h holes with capacity, mice can run distance r.', 'duration': 32.393, 'max_score': 20314.311, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY20314311.jpg'}, {'end': 20544.381, 'src': 'embed', 'start': 20513.486, 'weight': 6, 'content': [{'end': 20520.55, 'text': "I've laid out some instructions on the top here in case you wanted to download the code and actually play around with it on your machine.", 'start': 20513.486, 'duration': 7.064}, {'end': 20530.555, 'text': 'This program also uses the Ford Fulkerson flow solver we saw two videos ago, So I highly recommend you go and watch that video before continuing.', 'start': 20520.811, 'duration': 9.744}, {'end': 20534.317, 'text': "I'll link to it in the description below just in case you haven't seen it.", 'start': 20530.775, 'duration': 3.542}, {'end': 20535.517, 'text': "So let's get started.", 'start': 20534.597, 'duration': 0.92}, {'end': 20544.381, 'text': 'The first thing I do here is I create a mouse class, which is essentially a wrapper around a point object.', 'start': 20535.938, 'duration': 8.443}], 'summary': 'Instruction provided for downloading code and using ford fulkerson flow solver.', 'duration': 30.895, 'max_score': 20513.486, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY20513486.jpg'}], 'start': 19624.114, 'title': 'Network flow and matching in graphs', 'summary': "Covers ford fulkerson method with a depth first search, introduces bipartite graphs, discusses maximum cardinality bipartite matching, outlines algorithms for finding maximum matching, demonstrates transforming matching problem into a network flow problem, and uses network flow to solve 'mice and owls' and 'mouse and hole' problems.", 'chapters': [{'end': 19986.1, 'start': 19624.114, 'title': 'Max flow and bipartite graph matching', 'summary': 'Explains the ford fulkerson method with a depth first search, introduces bipartite graphs, discusses maximum cardinality bipartite matching, and outlines various algorithms for finding maximum matching on bipartite and non-bipartite graphs.', 'duration': 361.986, 'highlights': ['The chapter explains the Ford Fulkerson method with a depth first search The Ford Fulkerson method is covered in the context of network flow, focusing on augmenting paths and bottleneck values.', 'Introduces bipartite graphs and maximum cardinality bipartite matching Definition of bipartite graphs is provided, and the concept of maximum cardinality bipartite matching is explained, highlighting its relevance in matching suitable candidates to jobs or surfers to surfboards.', 'Outlines various algorithms for finding maximum matching on bipartite and non-bipartite graphs Different algorithms such as flow problem setup, finding augmenting paths, Hopcroft CARP algorithm, min cost max flow algorithm, Hungarian algorithm, admins blossom algorithm, and dynamic programming for non-bipartite graphs are mentioned, emphasizing the complexity and variety of approaches based on graph characteristics.']}, {'end': 20290.013, 'start': 19986.34, 'title': 'Maximum matching and network flow', 'summary': 'Explains the limitations of a greedy approach in finding a maximum cardinality matching, introduces the concept of transforming the problem into a network flow problem to find the maximum flow, and demonstrates how to modify the flow network to allow for multiple copies of the same book, resulting in an improved matching.', 'duration': 303.673, 'highlights': ['The greedy approach only found a matching of four, indicating the limitation of this method in finding the true maximum cardinality matching. The greedy approach resulted in a matching of only four people with books, highlighting its limitations in finding the true maximum cardinality matching.', 'The chapter introduces the concept of transforming the problem into a network flow problem to find the maximum flow and demonstrates how to use any max flow algorithm to push flow through the network to identify the maximum matching. The chapter introduces the concept of transforming the problem into a network flow problem to find the maximum flow and demonstrates the use of any max flow algorithm to identify the maximum matching.', 'The demonstration shows how modifying the flow network can allow for multiple copies of the same book, resulting in an improved matching where each person can pick up multiple copies of a book, thereby enhancing the assignment of people to books. The demonstration illustrates how modifying the flow network allows for multiple copies of the same book, leading to an improved matching where each person can pick up multiple copies of a book, thereby enhancing the assignment of people to books.']}, {'end': 20535.517, 'start': 20290.573, 'title': 'Network flow for mice and owls problem', 'summary': "Discusses using network flow to solve the 'mice and owls' problem, determining the maximum number of mice that can safely hide from a hungry owl given certain constraints, and transforming the problem into a maximum flow problem.", 'duration': 244.944, 'highlights': ["The problem involves determining the maximum number of mice that can hide safely from a hungry owl, given the presence of holes with specific capacities and the mice's ability to run a distance of R. The problem involves M mice, H holes with specific capacities, and the ability of mice to run a distance of R. The question asks for the maximum number of mice that can hide safely before being caught.", 'Visualizing the problem by drawing a radius of R around each mouse and matching mice to holes to maximize overall safety by setting up a flow graph and running a maximum flow algorithm. Visualizing the problem by drawing a radius of R around each mouse and matching mice to holes to maximize overall safety. The key realization is that the graph is bipartite, leading to setting up a flow graph and running a maximum flow algorithm.', 'Transforming the problem into a maximum flow problem by creating mouse and hole nodes, setting up the edges with specific capacities, and running a maximum flow algorithm to determine the maximum number of mice that can be safe. Transforming the problem into a maximum flow problem by creating mouse and hole nodes, setting up the edges with specific capacities, and running a maximum flow algorithm to determine the maximum number of mice that can be safe.']}, {'end': 21351.285, 'start': 20535.938, 'title': 'Network flow for mouse and hole problem', 'summary': 'Discusses how to set up a flow graph for a bipartite network flow problem to find the maximum number of safe mice, and also introduces a similar elementary math problem that can be solved using a flow graph setup.', 'duration': 815.347, 'highlights': ['The max flow algorithm returns the number of safe mice, which for this configuration happens to be four. The max flow algorithm returns the number of safe mice, which is a quantifiable result demonstrating the effectiveness of the network flow setup.', 'The solution uses flow to handle the elementary math problem, which requires identifying a flow problem and setting it up correctly. The solution uses flow to handle the elementary math problem, demonstrating the practical application of flow graph setups.', 'The setup of the flow graph for the mouse and hole problem is explained, including the process of setting up a bipartite graph with input nodes on the left side and answer nodes on the right side. The detailed explanation of setting up a bipartite flow graph for the mouse and hole problem, showcasing the step-by-step process of creating the graph.']}], 'duration': 1727.171, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY19624114.jpg', 'highlights': ['The Ford Fulkerson method is covered in the context of network flow, focusing on augmenting paths and bottleneck values.', 'Definition of bipartite graphs is provided, and the concept of maximum cardinality bipartite matching is explained, highlighting its relevance in matching suitable candidates to jobs or surfers to surfboards.', 'Outlines various algorithms for finding maximum matching on bipartite and non-bipartite graphs, emphasizing the complexity and variety of approaches based on graph characteristics.', 'The demonstration illustrates how modifying the flow network allows for multiple copies of the same book, leading to an improved matching where each person can pick up multiple copies of a book, thereby enhancing the assignment of people to books.', 'The problem involves M mice, H holes with specific capacities, and the ability of mice to run a distance of R. The question asks for the maximum number of mice that can hide safely before being caught.', 'Visualizing the problem by drawing a radius of R around each mouse and matching mice to holes to maximize overall safety. The key realization is that the graph is bipartite, leading to setting up a flow graph and running a maximum flow algorithm.', 'Transforming the problem into a maximum flow problem by creating mouse and hole nodes, setting up the edges with specific capacities, and running a maximum flow algorithm to determine the maximum number of mice that can be safe.', 'The max flow algorithm returns the number of safe mice, which is a quantifiable result demonstrating the effectiveness of the network flow setup.', 'The solution uses flow to handle the elementary math problem, demonstrating the practical application of flow graph setups.', 'The detailed explanation of setting up a bipartite flow graph for the mouse and hole problem, showcasing the step-by-step process of creating the graph.']}, {'end': 24278.784, 'segs': [{'end': 22469.331, 'src': 'embed', 'start': 22443.596, 'weight': 3, 'content': [{'end': 22452.64, 'text': 'The algorithm will repeatedly find augmenting paths through the flow graph which have a remaining capacity greater than or equal to delta,', 'start': 22443.596, 'duration': 9.044}, {'end': 22455.622, 'text': 'until no more paths satisfy this criteria.', 'start': 22452.64, 'duration': 2.982}, {'end': 22464.227, 'text': 'Once this criteria is no longer met, what we do is decrease the value of delta by dividing it by two.', 'start': 22456.54, 'duration': 7.687}, {'end': 22469.331, 'text': 'And we repeat this process while delta is greater than zero.', 'start': 22464.747, 'duration': 4.584}], 'summary': "Algorithm finds paths with remaining capacity >= delta, reduces delta by half until it's greater than zero", 'duration': 25.735, 'max_score': 22443.596, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY22443596.jpg'}, {'end': 22547.888, 'src': 'embed', 'start': 22495.284, 'weight': 0, 'content': [{'end': 22502.006, 'text': 'which is basically Edmonds Karp, but with capacity scaling, although I have found that to be much slower.', 'start': 22495.284, 'duration': 6.722}, {'end': 22506.128, 'text': 'So I would recommend the depth first search if you are going to implement this.', 'start': 22502.227, 'duration': 3.901}, {'end': 22508.188, 'text': "Let's do an example.", 'start': 22507.188, 'duration': 1}, {'end': 22513.51, 'text': "Let's find the maximum flow of the following flow graph using capacity scaling.", 'start': 22508.428, 'duration': 5.082}, {'end': 22518.691, 'text': 'First compute u as the maximum of all initial capacity values.', 'start': 22514.15, 'duration': 4.541}, {'end': 22523.972, 'text': 'And this example u is the maximum of 614 157 1011 and 12, which happens to be 14.', 'start': 22519.171, 'duration': 4.801}, {'end': 22526.473, 'text': 'Next compute the starting value for delta.', 'start': 22523.972, 'duration': 2.501}, {'end': 22537.659, 'text': 'which is the smallest power of two less than or equal to u, which we know is 14.', 'start': 22532.754, 'duration': 4.905}, {'end': 22543.624, 'text': 'Therefore, the starting value of delta is eight since the next power of two after eight is 16.', 'start': 22537.659, 'duration': 5.965}, {'end': 22547.888, 'text': 'But 16 is larger than 14.', 'start': 22543.624, 'duration': 4.264}], 'summary': 'Capacity scaling for maximum flow computation using example with maximum initial capacity values of 614, 157, 1011, and 12, resulting in a starting value of delta as eight.', 'duration': 52.604, 'max_score': 22495.284, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY22495284.jpg'}, {'end': 23115.187, 'src': 'embed', 'start': 23086.92, 'weight': 4, 'content': [{'end': 23093.266, 'text': 'So I create a flow solver, I add all the edges, and then I push some flow through it and get the maximum flow.', 'start': 23086.92, 'duration': 6.346}, {'end': 23099.031, 'text': 'I also display the resulting flow graph after the flow algorithm has been executed.', 'start': 23093.826, 'duration': 5.205}, {'end': 23100.413, 'text': 'So you can see what happened.', 'start': 23099.151, 'duration': 1.262}, {'end': 23103.496, 'text': "Awesome That's all I wanted to cover for capacity scaling.", 'start': 23100.973, 'duration': 2.523}, {'end': 23105.938, 'text': "Today, we're still talking about network flow.", 'start': 23103.816, 'duration': 2.122}, {'end': 23115.187, 'text': "And in particular, we're looking at finding the maximum flow and a new very efficient method of solving the unweighted bipartite matching problem.", 'start': 23106.218, 'duration': 8.969}], 'summary': 'Created flow solver, pushed flow through graph to find maximum flow. discussed efficient method for unweighted bipartite matching.', 'duration': 28.267, 'max_score': 23086.92, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY23086920.jpg'}], 'start': 21356.198, 'title': 'Various network flow algorithms', 'summary': 'Explores the edmonds karp algorithm with a time complexity of o(v * e^2), the implementation of capacity scaling algorithm with a time complexity of big o of e squared log q, the capacity scaling algorithm resulting in a maximum flow value of 1054, the denix algorithm reducing runtime and achieving a maximum flow of 30, and the implementation of dinnits algorithm in java aiming to compute a maximum flow, resulting in a maximum flow of 30.', 'chapters': [{'end': 21932.627, 'start': 21356.198, 'title': 'Edmonds karp algorithm', 'summary': 'Explores the edmonds karp algorithm, a maximum flow algorithm that utilizes breadth first search to find augmenting paths, resulting in a time complexity of o(v * e^2), independent of the maximum flow, offering a significantly improved approach compared to the ford fulkerson method with depth first search.', 'duration': 576.429, 'highlights': ['The Edmonds Karp algorithm utilizes breadth first search to find the shortest augmenting paths, significantly reducing the chance of long augmenting paths and potential bottleneck values, resulting in a time complexity of O(v * e^2), independent of the maximum flow, which is a major improvement compared to the Ford Fulkerson method with depth first search.', 'The time complexity of the Ford Fulkerson method with a depth first search is O(e * f), where e is the number of edges and f is the maximum flow, while the Edmonds Karp algorithm offers a time complexity of O(v * e^2), where v is the number of vertices and e is the number of edges, demonstrating a better approach in terms of time complexity.', 'The Edmonds Karp algorithm ensures that the time complexity does not depend on the capacity value of any edge in the flow graph, making it a strongly polynomial algorithm, which was a revolutionary advancement during its time and contributes to its significance in solving a variety of problems.', 'The algorithm focuses on finding the maximum flow on a flow graph, which is essential for tasks such as finding bipartite matchings and solving a wide range of problems, underlining the practical significance and usefulness of the Edmonds Karp algorithm.', 'The implementation of the Edmonds Karp algorithm involves finding augmenting paths through breadth first search, ensuring the discovery of the shortest path from the source to the sink, effectively minimizing the length of augmenting paths and the potential for bottleneck values, leading to improved efficiency and performance.']}, {'end': 22547.888, 'start': 21933.367, 'title': 'Network flow: capacity scaling algorithm', 'summary': 'Covers the implementation of capacity scaling algorithm in network flow, emphasizing the heuristic to push flow through largest edges first, achieving a better runtime and time complexity of big o of e squared log q and in big o of e times v log u.', 'duration': 614.521, 'highlights': ['The chapter covers the implementation of capacity scaling algorithm in network flow It explains the heuristic to push flow through largest edges first, achieving a better runtime and time complexity of big O of E squared log q and in big O of E times v log u.', 'The capacity scaling algorithm is pretty straightforward. It defines the variables u and delta, prioritizes edges with remaining capacity greater than or equal to delta, and decreases delta by dividing it by two when no more paths satisfy the criteria.', 'The depth first search runs in big O of E squared log q and in big O of E times v log u It provides the time complexity of capacity scaling with a depth first search, highlighting its performance compared to Edmonds Karp.']}, {'end': 23177.094, 'start': 22547.888, 'title': 'Capacity scaling algorithm', 'summary': 'Describes the capacity scaling algorithm for finding the maximum flow in a network, highlighting its key steps and the resulting maximum flow value of 1054. the algorithm uses a decreasing parameter delta to determine accepted edges, and the implementation includes a depth-first search and the concept of augmenting paths.', 'duration': 629.206, 'highlights': ['The algorithm results in a maximum flow value of 1054. The transcript concludes with the computation of the maximum flow value, which is determined to be 1054.', 'Capacity scaling algorithm aims to achieve better runtime by pushing flow through larger edges first. The capacity scaling algorithm is described as an approach to pushing flow through larger edges first, aiming for improved runtime.', 'The algorithm uses a decreasing parameter delta to determine accepted edges and rejected edges based on their remaining capacity. The algorithm employs a decreasing parameter delta to decide which edges should be accepted or rejected based on their remaining capacity.', 'The implementation includes a depth-first search and the concept of augmenting paths. The implementation of the capacity scaling algorithm involves a depth-first search and the concept of augmenting paths.', 'The algorithm has a strongly polynomial time complexity of big O, of square root v times e for bipartite graphs. The capacity scaling algorithm is noted for its strongly polynomial time complexity of big O, of square root v times e specifically for bipartite graphs.']}, {'end': 23726.859, 'start': 23177.574, 'title': 'Denix algorithm for maximum flow', 'summary': "Introduces the denix algorithm, which was conceived in 1969 by ephraim dennett's and popularized by shimon evan, used to solve maximum flow algorithms by guiding augmenting paths from the source to the sink using a level graph, greatly reducing the runtime, with a maximum flow of 30 being achieved in the provided example.", 'duration': 549.285, 'highlights': ["Denix algorithm conceived in 1969 by Ephraim Dennett's and popularized by Shimon Evan, used to solve maximum flow algorithms The Denix algorithm, conceived in 1969 and popularized by Shimon Evan, is utilized to solve maximum flow algorithms, providing an efficient approach to guide augmenting paths from the source to the sink using a level graph.", 'Algorithm employs a level graph, reducing runtime The algorithm utilizes a level graph to guide augmenting paths from the source to the sink, effectively reducing the runtime of the maximum flow algorithm.', 'Maximum flow achieved in the example is 30 The provided example demonstrates the achievement of a maximum flow of 30 by employing the Denix algorithm to guide augmenting paths from the source to the sink using a level graph.']}, {'end': 24278.784, 'start': 23727.623, 'title': 'Dinnits algorithm in java', 'summary': 'Covers the implementation of dinnits algorithm in java, including building a level graph, finding a blocking flow, and optimizing dead end pruning, aiming to compute the maximum flow, with a demonstration resulting in a maximum flow of 30.', 'duration': 551.161, 'highlights': ['Building the Level Graph Describing the process of building a level graph using a breadth-first search, marking nodes as unvisited, initializing a queue, and determining the reachable sink node during the search.', 'Depth First Search Method and Next Array Explaining the usage of the next array to efficiently prune dead ends, detailing the process of depth-first search recursively to find augmenting paths and compute the maximum flow.', 'Optimizing Dead End Pruning Detailing the strategy of using the next array to track the next edge to traverse, enabling the efficient pruning of dead ends by avoiding previously forbidden edges.']}], 'duration': 2922.586, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/09_LlHjoEiY/pics/09_LlHjoEiY21356198.jpg', 'highlights': ['The Edmonds Karp algorithm ensures that the time complexity does not depend on the capacity value of any edge in the flow graph, making it a strongly polynomial algorithm, which was a revolutionary advancement during its time and contributes to its significance in solving a variety of problems.', 'The capacity scaling algorithm is pretty straightforward. It defines the variables u and delta, prioritizes edges with remaining capacity greater than or equal to delta, and decreases delta by dividing it by two when no more paths satisfy the criteria.', 'The algorithm results in a maximum flow value of 1054.', "Denix algorithm conceived in 1969 by Ephraim Dennett's and popularized by Shimon Evan, used to solve maximum flow algorithms The Denix algorithm, conceived in 1969 and popularized by Shimon Evan, is utilized to solve maximum flow algorithms, providing an efficient approach to guide augmenting paths from the source to the sink using a level graph.", 'Building the Level Graph Describing the process of building a level graph using a breadth-first search, marking nodes as unvisited, initializing a queue, and determining the reachable sink node during the search.']}], 'highlights': ['The Floyd Warshall algorithm has a time complexity of O(v^3), making it suitable for graphs with a limited number of nodes.', 'The algorithm for finding bridges and articulation points in an undirected graph is explained, emphasizing their significance in graph theory and real-world applications.', 'The dynamic programming solution reduces the complexity to O(n^2*2^N), making graphs with roughly 23 nodes feasible for modern home computers.', 'The Ford Fulkerson method is covered in the context of network flow, focusing on augmenting paths and bottleneck values.', 'The Edmonds Karp algorithm ensures that the time complexity does not depend on the capacity value of any edge in the flow graph, making it a strongly polynomial algorithm, which was a revolutionary advancement during its time and contributes to its significance in solving a variety of problems.']}