title

DP 48. Matrix Chain Multiplication | MCM | Partition DP Starts ðŸ”¥

description

Lecture Notes/C++/Java Codes: https://takeuforward.org/dynamic-programming/striver-dp-series-dynamic-programming-problems/
Problem Link: https://bit.ly/3nXqfce
Pre-req for this Series: https://www.youtube.com/watch?v=yVdKa8dnKiE&list=PLgUwDviBIf0rGlzIn_7rsaR2FQ5e6ZOL9
a
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Full Playlist: https://www.youtube.com/watch?v=FfXoiwwnxFw&list=PLgUwDviBIf0qUlt5H_kiKYaNSqJ81PMMY
In this video, we solve the MCM Dp, this is the first problem on the pattern Partition DP.
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detail

{'title': 'DP 48. Matrix Chain Multiplication | MCM | Partition DP Starts ðŸ”¥', 'heatmap': [{'end': 2073.699, 'start': 1959.3, 'weight': 0.839}], 'summary': 'Delves into mastering partition dp, matrix multiplication optimization, understanding matrix chain multiplication, and recursive solutions, with quantifiable data of 4500 vs. 27000 operations, aiming to provide a comprehensive understanding of these concepts and their practical applications.', 'chapters': [{'end': 275.179, 'segs': [{'end': 106.225, 'src': 'embed', 'start': 47.336, 'weight': 0, 'content': [{'end': 55.503, 'text': 'But over here I will be solving I think four or five or six problems on this particular partition DP so that this gets right into your head.', 'start': 47.336, 'duration': 8.167}, {'end': 60.446, 'text': "So let's understand what's this partition DP all about.", 'start': 56.603, 'duration': 3.843}, {'end': 74.066, 'text': "So generally in all the problems where you'll be said To solve a problem in a particular way, remember this, solve a problem in a pattern.", 'start': 60.927, 'duration': 13.139}, {'end': 85.089, 'text': 'Now, what is the pattern? For an example, whenever you do mathematical calculations, right?', 'start': 77.503, 'duration': 7.586}, {'end': 91.914, 'text': 'So assume you have been given something like 1 plus 2 plus 3 into 5, right?', 'start': 85.429, 'duration': 6.485}, {'end': 96.118, 'text': 'But if you put it like this, then the answer is different.', 'start': 92.195, 'duration': 3.923}, {'end': 99.24, 'text': 'But if you put it like this, then the answer is different.', 'start': 96.698, 'duration': 2.542}, {'end': 106.225, 'text': "Don't you agree? So, whenever you will find questions similar to these, like if you solve.", 'start': 99.28, 'duration': 6.945}], 'summary': 'Solving 4-6 problems on partition dp, understanding the pattern in mathematical calculations.', 'duration': 58.889, 'max_score': 47.336, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y47336.jpg'}, {'end': 205.682, 'src': 'embed', 'start': 176.332, 'weight': 5, 'content': [{'end': 181.879, 'text': 'So you will have to try out, given an entire array.', 'start': 176.332, 'duration': 5.547}, {'end': 185.223, 'text': 'And generally, the starting point will be known as the i.', 'start': 182.319, 'duration': 2.904}, {'end': 190.072, 'text': 'and this will be the ending point.', 'start': 187.27, 'duration': 2.802}, {'end': 192.613, 'text': 'okay, and the partitions.', 'start': 190.072, 'duration': 2.541}, {'end': 196.596, 'text': 'partitions can be this when you solve this and this.', 'start': 192.613, 'duration': 3.983}, {'end': 203.42, 'text': 'so, or the partition can be this, where you can solve this and this, or the partition can be this, where you can solve this and this.', 'start': 196.596, 'duration': 6.824}, {'end': 205.682, 'text': 'so there can be a lot of partitions.', 'start': 203.42, 'duration': 2.262}], 'summary': 'Explaining the process of trying out an entire array with known starting and ending points, and multiple possible partitions.', 'duration': 29.35, 'max_score': 176.332, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y176332.jpg'}, {'end': 253.151, 'src': 'embed', 'start': 227.005, 'weight': 6, 'content': [{'end': 232.829, 'text': 'so, out of all these partitions, whichever will yield you the best answer, you have to tell us that.', 'start': 227.005, 'duration': 5.824}, {'end': 246.097, 'text': 'so, generally, all the problems which are like uh, in a pattern that can be solved using partition dp, and you have to, basically, i can say,', 'start': 232.829, 'duration': 13.268}, {'end': 253.151, 'text': 'whenever you try to partition it in this way, have to solve this portion which you can call from i to k.', 'start': 246.097, 'duration': 7.054}], 'summary': 'Using partition dp for problem-solving, finding the best answer from different partitions.', 'duration': 26.146, 'max_score': 227.005, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y227005.jpg'}], 'start': 0.189, 'title': 'Mastering partition dp', 'summary': 'Introduces and explains partition dynamic programming, aiming to make learners comfortable by solving multiple problems and ensuring mastery of the pattern. it covers identification of partitions, selection of the best solution, and various patterns and arrays.', 'chapters': [{'end': 106.225, 'start': 0.189, 'title': 'Partition dp: mastering a new pattern', 'summary': 'Introduces a challenging pattern of dynamic programming called partition dp and aims to make learners comfortable by solving multiple problems, ensuring mastery and understanding of the pattern.', 'duration': 106.036, 'highlights': ['The chapter focuses on introducing the challenging pattern of partition DP and aims to make learners comfortable by solving multiple problems, ensuring mastery and understanding of the pattern.', 'The instructor guarantees increased comfort and proficiency in partition DP after solving all the problems presented, similar to previous DP patterns covered.', 'The approach includes solving four to six problems on partition DP to ensure a deep understanding and comfort with this pattern.', 'The chapter emphasizes the significance of understanding patterns in problem-solving, illustrated through an example of mathematical calculations.']}, {'end': 275.179, 'start': 107.318, 'title': 'Partition dynamic programming', 'summary': 'Explains how to solve partition dynamic programming problems, involving the identification of multiple partitions and the selection of the best solution, based on various patterns and arrays.', 'duration': 167.861, 'highlights': ['Partition DP involves solving problems by identifying multiple partitions and selecting the best solution, usually involving arrays and different patterns.', "In partition DP problems, the array's starting point is known as 'i' and the ending point as 'j', with various partitions that need to be tried out to determine the best answer.", 'The approach to solving partition DP problems involves identifying and solving portions of the array, such as f(i to k) and k+1 to j, which leads to the best answer.']}], 'duration': 274.99, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y189.jpg', 'highlights': ['The chapter focuses on introducing the challenging pattern of partition DP and aims to make learners comfortable by solving multiple problems, ensuring mastery and understanding of the pattern.', 'The instructor guarantees increased comfort and proficiency in partition DP after solving all the problems presented, similar to previous DP patterns covered.', 'The approach includes solving four to six problems on partition DP to ensure a deep understanding and comfort with this pattern.', 'Partition DP involves solving problems by identifying multiple partitions and selecting the best solution, usually involving arrays and different patterns.', 'The chapter emphasizes the significance of understanding patterns in problem-solving, illustrated through an example of mathematical calculations.', "In partition DP problems, the array's starting point is known as 'i' and the ending point as 'j', with various partitions that need to be tried out to determine the best answer.", 'The approach to solving partition DP problems involves identifying and solving portions of the array, such as f(i to k) and k+1 to j, which leads to the best answer.']}, {'end': 1194.525, 'segs': [{'end': 379.723, 'src': 'embed', 'start': 327.137, 'weight': 1, 'content': [{'end': 337.769, 'text': 'why? because if you have done maths in your school, you know, if this guy, this, these guys are equal,', 'start': 327.137, 'duration': 10.632}, {'end': 346.433, 'text': 'if the column of the first matrix is equal to the row of the second matrix, then the multiplication is possible.', 'start': 337.769, 'duration': 8.664}, {'end': 349.035, 'text': 'and how many operations do you require?', 'start': 346.433, 'duration': 2.602}, {'end': 352.536, 'text': 'is, how many operations do you require in order to multiply this?', 'start': 349.035, 'duration': 3.501}, {'end': 353.877, 'text': "that's very simple.", 'start': 352.536, 'duration': 1.341}, {'end': 360.419, 'text': 'you take this guy 10, you take the common guy 30 and you take the other guy 5.', 'start': 353.877, 'duration': 6.542}, {'end': 370.401, 'text': 'so the number of operations or the number of multiplications required in order to multiply both the matrices will be 10, cross 30 into 5,', 'start': 360.419, 'duration': 9.982}, {'end': 374.222, 'text': 'which is somewhere around 1500 operations.', 'start': 370.401, 'duration': 3.821}, {'end': 375.762, 'text': 'if i have done the calculations correctly.', 'start': 374.222, 'duration': 1.54}, {'end': 377.663, 'text': 'yeah, i have done it correctly.', 'start': 375.762, 'duration': 1.901}, {'end': 379.723, 'text': 'now, what is these operations now?', 'start': 377.663, 'duration': 2.06}], 'summary': 'Matrix multiplication requires 1500 operations if columns and rows are equal.', 'duration': 52.586, 'max_score': 327.137, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y327137.jpg'}, {'end': 671.245, 'src': 'embed', 'start': 638.644, 'weight': 0, 'content': [{'end': 643.887, 'text': 'so which way of multiplying these three matrices was a better way.', 'start': 638.644, 'duration': 5.243}, {'end': 650.75, 'text': 'i can say, If I multiply this way, I will take 27000 multiplication operations.', 'start': 643.887, 'duration': 6.863}, {'end': 657.012, 'text': "But if I multiply first AB and the resultant with C, then I'll just take 4500 operations.", 'start': 651.39, 'duration': 5.622}, {'end': 659.533, 'text': 'Is there any other way? Obviously not.', 'start': 657.752, 'duration': 1.781}, {'end': 663.114, 'text': 'So what I figured out was the minimal operations required.', 'start': 659.993, 'duration': 3.121}, {'end': 665.835, 'text': 'Yes, the minimal operations required.', 'start': 663.134, 'duration': 2.701}, {'end': 671.245, 'text': 'was 4500.', 'start': 667.984, 'duration': 3.261}], 'summary': 'Multiplying three matrices using a different method reduced operations from 27000 to 4500.', 'duration': 32.601, 'max_score': 638.644, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y638644.jpg'}], 'start': 276.055, 'title': 'Matrix multiplication optimization', 'summary': 'Discusses matrix multiplication basics, understanding matrix chain multiplication, and optimizing matrix multiplication. it covers the number of operations required for specific matrix dimensions, quantifies 4500 operations for a specific multiplication method, and compares 27,000 operations for one approach with 4,500 operations for another, utilizing partition dp. additionally, it explains dynamic programming by breaking a problem into smaller subproblems and returning the best possible two partitions to solve the problem.', 'chapters': [{'end': 437.256, 'start': 276.055, 'title': 'Understanding matrix chain multiplication', 'summary': 'Discusses the concept of matrix multiplication and explains the number of operations required for multiplying matrices, highlighting the criteria for matrix multiplication and the number of operations required for specific matrix dimensions.', 'duration': 161.201, 'highlights': ['Matrix multiplication criteria: The chapter explains that for two matrices to be multiplied, the number of columns in the first matrix should be equal to the number of rows in the second matrix, illustrating this with an example of three matrices with dimensions 10x30, 30x5, and 5x60.', 'Number of operations for matrix multiplication: It highlights that for the given example of matrices with dimensions 10x30 and 30x5, the number of required multiplications is 10x30x5, which equals 1500 operations.', 'Detailed explanation of operations: The chapter provides a detailed explanation of the operations involved in matrix multiplication, using a specific example of 2x2 and 2x1 matrices to demonstrate the four multiplications required.']}, {'end': 572.851, 'start': 437.256, 'title': 'Matrix multiplication basics', 'summary': 'Covers the basics of matrix multiplication, including the number of operations involved in multiplying three matrices and the quantifiable data of 4500 operations for a specific multiplication method.', 'duration': 135.595, 'highlights': ['The number of operations involved in multiplying three matrices using a specific method is 4500, which is a significant amount.', 'The basics of matrix multiplication involve understanding the dimensions of the matrices and the number of operations required for multiplication.', 'Explaining the number of operations involved in multiplying specific matrix dimensions (10x30, 30x5, 5x60) results in a total of 4500 operations.']}, {'end': 942.175, 'start': 572.851, 'title': 'Optimizing matrix multiplication', 'summary': 'Discusses the optimal way to multiply matrices by identifying the minimal number of operations required, with a comparison of 27,000 operations for one approach and 4,500 operations for another, utilizing partition dp.', 'duration': 369.324, 'highlights': ['The minimal number of operations required for matrix multiplication was found to be 4,500, which is significantly lower than the 27,000 operations required for an alternative approach.', 'The dimensions of the matrices were provided as an array, representing the dimensions of n-1 matrices, with the dimension of each matrix calculated as a[i-1] * a[i], where i is the index of the matrix.', 'The problem of determining the minimum number of operations for multiplying matrices was solved using partition DP, with a focus on identifying different patterns and rules to optimize the process.']}, {'end': 1194.525, 'start': 943.558, 'title': 'Dynamic programming explained', 'summary': 'Explains the concept of dynamic programming using an example of breaking a problem into smaller subproblems, trying all partitions, and returning the best possible two partitions to solve the problem.', 'duration': 250.967, 'highlights': ['Start with the entire problem and break it down into smaller subproblems, trying every possible smaller problem and returning the best possible partition.', "Mark the start and end points of the problem as 'i' and 'j', then try all partitions by running a loop to test every possible way to solve it.", 'Return the best possible two partitions after breaking down the problem into smaller subproblems.']}], 'duration': 918.47, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y276055.jpg', 'highlights': ['The minimal number of operations required for matrix multiplication was found to be 4,500, significantly lower than the 27,000 operations for an alternative approach.', 'The number of operations involved in multiplying three matrices using a specific method is 4500, a significant amount.', 'For the given example of matrices with dimensions 10x30 and 30x5, the number of required multiplications is 10x30x5, which equals 1500 operations.', 'The chapter explains that for two matrices to be multiplied, the number of columns in the first matrix should be equal to the number of rows in the second matrix.']}, {'end': 1784.723, 'segs': [{'end': 1264.045, 'src': 'embed', 'start': 1194.525, 'weight': 0, 'content': [{'end': 1197.066, 'text': "that's the straight funda.", 'start': 1194.525, 'duration': 2.541}, {'end': 1201.7, 'text': 'okay, so Something we have understood.', 'start': 1197.066, 'duration': 4.634}, {'end': 1204.201, 'text': 'That we have to start with f of i, j.', 'start': 1201.74, 'duration': 2.461}, {'end': 1208.424, 'text': "Right? That's for sure.", 'start': 1204.201, 'duration': 4.223}, {'end': 1214.288, 'text': 'Now, what was given to us? Like, we knew that we had a, b, c, d.', 'start': 1209.745, 'duration': 4.543}, {'end': 1217.13, 'text': 'But we cannot think in terms of a, b, c, d.', 'start': 1214.288, 'duration': 2.842}, {'end': 1218.771, 'text': 'You have to think in terms of the array.', 'start': 1217.13, 'duration': 1.641}, {'end': 1219.832, 'text': "Let's understand.", 'start': 1219.312, 'duration': 0.52}, {'end': 1221.113, 'text': 'You have to think in terms of the array.', 'start': 1219.852, 'duration': 1.261}, {'end': 1223.595, 'text': 'And if you remember, the array was something like.', 'start': 1221.693, 'duration': 1.902}, {'end': 1238.301, 'text': '10 20 30 40 50 correct and can i say this a was nothing but 10 into 20.', 'start': 1225.795, 'duration': 12.506}, {'end': 1239.761, 'text': 'the second matrix was 20 into 30.', 'start': 1238.301, 'duration': 1.46}, {'end': 1243.883, 'text': 'the third matrix was 30 into 40.', 'start': 1239.761, 'duration': 4.122}, {'end': 1245.524, 'text': 'the this matrix was 40 into 50.', 'start': 1243.883, 'duration': 1.641}, {'end': 1257.64, 'text': "can i say this i can Thereby, what I can say is, hey, listen, if I'm starting with, I can say this A, I can say this B, I can say this D.", 'start': 1245.524, 'duration': 12.116}, {'end': 1264.045, 'text': "Where, where, where? If I'm looking for A's size, it's this guy and the previous guy.", 'start': 1257.64, 'duration': 6.405}], 'summary': 'Understanding matrix multiplication with array and matrix sizes.', 'duration': 69.52, 'max_score': 1194.525, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y1194525.jpg'}, {'end': 1360.286, 'src': 'embed', 'start': 1318.275, 'weight': 1, 'content': [{'end': 1340.579, 'text': 'in what way returns the minimum multiplications to multiply matrix 1 to matrix 4?', 'start': 1318.275, 'duration': 22.304}, {'end': 1351.317, 'text': 'remember this It is the minimum multiplications to multiply matrix 1 to matrix 4..', 'start': 1340.579, 'duration': 10.738}, {'end': 1355.481, 'text': 'So we have got our start point which is 1 and which is n minus 1.', 'start': 1351.317, 'duration': 4.164}, {'end': 1360.286, 'text': 'So every time the problem will start over here with f of 1 comma n minus 1.', 'start': 1355.481, 'duration': 4.805}], 'summary': 'Determine minimum multiplications to multiply matrices 1 to 4.', 'duration': 42.011, 'max_score': 1318.275, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y1318275.jpg'}, {'end': 1471.563, 'src': 'embed', 'start': 1440.835, 'weight': 2, 'content': [{'end': 1446.057, 'text': 'you shrink it, you shrink it, you shrink it and you have came down to a portion.', 'start': 1440.835, 'duration': 5.222}, {'end': 1450.098, 'text': 'where is it is just a single matrix.', 'start': 1446.057, 'duration': 4.041}, {'end': 1452.078, 'text': 'then the number of operations.', 'start': 1450.098, 'duration': 1.98}, {'end': 1453.279, 'text': 'what are the number of operations?', 'start': 1452.078, 'duration': 1.201}, {'end': 1456.341, 'text': "ask yourself it's zero.", 'start': 1453.279, 'duration': 3.062}, {'end': 1458.563, 'text': 'the number of operations required is zero.', 'start': 1456.341, 'duration': 2.222}, {'end': 1461.707, 'text': 'so we figured out the smallest base case.', 'start': 1458.563, 'duration': 3.144}, {'end': 1471.563, 'text': 'the smallest case is when we just have a single piece of matrix and thereby I can say we can easily return zero,', 'start': 1461.707, 'duration': 9.856}], 'summary': 'Shrinking matrix to single piece reduces operations to zero.', 'duration': 30.728, 'max_score': 1440.835, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y1440835.jpg'}, {'end': 1574.746, 'src': 'embed', 'start': 1545.489, 'weight': 3, 'content': [{'end': 1549.704, 'text': 'if i run a loop from i to j minus 1 For the first time,', 'start': 1545.489, 'duration': 4.215}, {'end': 1559.112, 'text': 'can I say it will be something like the first partition will be i to k and the other partition will be k plus 1 to j.', 'start': 1549.704, 'duration': 9.408}, {'end': 1560.874, 'text': 'We need to understand why j minus 1.', 'start': 1559.112, 'duration': 1.762}, {'end': 1561.835, 'text': 'I will explain you that.', 'start': 1560.874, 'duration': 0.961}, {'end': 1562.835, 'text': 'Let us understand this.', 'start': 1562.115, 'duration': 0.72}, {'end': 1571.66, 'text': 'Let us take this example and try to write the values 0, 1, 2, 3.', 'start': 1565.618, 'duration': 6.042}, {'end': 1572.704, 'text': 'So where is k??', 'start': 1571.663, 'duration': 1.041}, {'end': 1574.746, 'text': 'k is trying to go from 0 to.', 'start': 1573.485, 'duration': 1.261}], 'summary': 'Explaining loop iteration and partitioning with example', 'duration': 29.257, 'max_score': 1545.489, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y1545489.jpg'}], 'start': 1194.525, 'title': 'Matrix multiplication', 'summary': 'Explains the fundamental concept of matrix multiplication and emphasizes the need to think in terms of arrays. it also discusses optimizing matrix multiplication, including identifying starting and ending points, determining minimum multiplications, exploring all possible partitions, and understanding the base case for minimal operations.', 'chapters': [{'end': 1287.96, 'start': 1194.525, 'title': 'Matrix multiplication fundamentals', 'summary': "Explains the fundamental concept of matrix multiplication, emphasizing the need to think in terms of arrays and the key formula of 'previous matrix size into current matrix size'.", 'duration': 93.435, 'highlights': ['The need to think in terms of arrays rather than individual elements when dealing with matrix multiplication, such as representing the given elements a, b, c, d as an array.', "The crucial formula for matrix multiplication, 'previous matrix size into current matrix size', emphasizing the relationship between the sizes of consecutive matrices being multiplied."]}, {'end': 1784.723, 'start': 1288.961, 'title': 'Matrix multiplication optimization', 'summary': 'Discusses optimizing matrix multiplication, including identifying the starting and ending points, determining minimum multiplications, exploring all possible partitions, and understanding the base case for minimal operations.', 'duration': 495.762, 'highlights': ['Identifying the starting point as 1 and the ending point as n minus 1, the chapter explores the minimum multiplications required to multiply matrix 1 to matrix 4.', 'Exploring all possible partitions by running a loop from i to j minus 1 and understanding the significance of partitioning for minimizing operations.', 'Understanding the base case for minimal operations where a single matrix requires zero operations, thus forming the smallest case.', 'Discussing the shrinking nature of the matrix operation, highlighting the progression from an entire block to a single matrix and the consequent reduction in operations.']}], 'duration': 590.198, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y1194525.jpg', 'highlights': ["The crucial formula for matrix multiplication, 'previous matrix size into current matrix size', emphasizing the relationship between the sizes of consecutive matrices being multiplied.", 'Identifying the starting point as 1 and the ending point as n minus 1, the chapter explores the minimum multiplications required to multiply matrix 1 to matrix 4.', 'Understanding the base case for minimal operations where a single matrix requires zero operations, thus forming the smallest case.', 'Exploring all possible partitions by running a loop from i to j minus 1 and understanding the significance of partitioning for minimizing operations.', 'The need to think in terms of arrays rather than individual elements when dealing with matrix multiplication, such as representing the given elements a, b, c, d as an array.', 'Discussing the shrinking nature of the matrix operation, highlighting the progression from an entire block to a single matrix and the consequent reduction in operations.']}, {'end': 2232.945, 'segs': [{'end': 1917.19, 'src': 'embed', 'start': 1887.546, 'weight': 3, 'content': [{'end': 1894.412, 'text': 'No matter in which way you do the multiplication, the resultant matrix will be of size 20 cross 30 only.', 'start': 1887.546, 'duration': 6.866}, {'end': 1901.704, 'text': 'So I can say 20 cross, rather 50, my bad, 50.', 'start': 1894.813, 'duration': 6.891}, {'end': 1904.745, 'text': 'So, that is the resultant size of BCD.', 'start': 1901.704, 'duration': 3.041}, {'end': 1909.987, 'text': 'So, if A is this, BCD is this?', 'start': 1905.245, 'duration': 4.742}, {'end': 1914.469, 'text': 'what are the number of operations that you will be requiring in order to multiply A and BCD?', 'start': 1909.987, 'duration': 4.482}, {'end': 1917.19, 'text': 'Because A was a part, BCD was a part.', 'start': 1914.709, 'duration': 2.481}], 'summary': 'Resultant matrix size is 20x50. number of operations for a x bcd?', 'duration': 29.644, 'max_score': 1887.546, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y1887546.jpg'}, {'end': 2106.662, 'src': 'heatmap', 'start': 1959.3, 'weight': 0, 'content': [{'end': 1967.606, 'text': 'so the number of steps required will be a of i, minus 1, multiplied with.', 'start': 1959.3, 'duration': 8.306}, {'end': 1975.651, 'text': "let's see, let's come back where's 20?", 'start': 1967.606, 'duration': 8.045}, {'end': 1976.992, 'text': 'can i say 20 would be here.', 'start': 1975.651, 'duration': 1.341}, {'end': 1980.291, 'text': "You'll be like it's I.", 'start': 1978.49, 'duration': 1.801}, {'end': 1982.952, 'text': "It's actually not I, it's K.", 'start': 1980.291, 'duration': 2.661}, {'end': 1984.952, 'text': "Try taking some other partition, I'll show you that.", 'start': 1982.952, 'duration': 2}, {'end': 1988.454, 'text': "It's actually K.", 'start': 1985.373, 'duration': 3.081}, {'end': 1991.155, 'text': 'So what you can do is, you can easily write array of K now.', 'start': 1988.454, 'duration': 2.701}, {'end': 1997.177, 'text': "How's the other one? The last one is 50.", 'start': 1994.016, 'duration': 3.161}, {'end': 1999.858, 'text': "Where's 50? At J.", 'start': 1997.177, 'duration': 2.681}, {'end': 2001.759, 'text': 'So the last one is array of J.', 'start': 1999.858, 'duration': 1.901}, {'end': 2013.092, 'text': 'basically, that signifies is you had two paths, so when you combine them, the number of steps required will be this right now you had two partitions.', 'start': 2004.128, 'duration': 8.964}, {'end': 2017.433, 'text': 'what are the two partitions f of i k and f of k plus 1 j?', 'start': 2013.092, 'duration': 4.341}, {'end': 2023.336, 'text': 'because bcd in itself will take some number of steps.', 'start': 2017.433, 'duration': 5.903}, {'end': 2027.697, 'text': 'bcd in itself will take some number of steps to get multiplied.', 'start': 2023.336, 'duration': 4.361}, {'end': 2033.305, 'text': 'so i can say okay, The i comma k will take some steps.', 'start': 2027.697, 'duration': 5.608}, {'end': 2035.246, 'text': 'Let the function give me that.', 'start': 2034.125, 'duration': 1.121}, {'end': 2037.548, 'text': 'Because I want the minimal number.', 'start': 2036.067, 'duration': 1.481}, {'end': 2038.749, 'text': 'Function will give me that.', 'start': 2037.888, 'duration': 0.861}, {'end': 2041.611, 'text': 'Or k plus 1 comma j will also.', 'start': 2039.409, 'duration': 2.202}, {'end': 2045.113, 'text': 'The left guy will take some steps.', 'start': 2042.892, 'duration': 2.221}, {'end': 2046.514, 'text': 'Give me the minimal.', 'start': 2045.914, 'duration': 0.6}, {'end': 2048.355, 'text': 'The right guy will give some steps.', 'start': 2046.894, 'duration': 1.461}, {'end': 2049.136, 'text': 'Give me the minimal.', 'start': 2048.514, 'duration': 0.622}, {'end': 2051.016, 'text': 'And these will be the steps.', 'start': 2049.956, 'duration': 1.06}, {'end': 2053.498, 'text': 'If I do a partition on k.', 'start': 2051.777, 'duration': 1.721}, {'end': 2055.219, 'text': 'If I do a partition on k.', 'start': 2053.498, 'duration': 1.721}, {'end': 2056.721, 'text': 'Right And what I can say is.', 'start': 2055.219, 'duration': 1.502}, {'end': 2058.963, 'text': 'Okay I am the minimal of all.', 'start': 2056.88, 'duration': 2.083}, {'end': 2073.699, 'text': 'probably I can store a mini equal to a bigger value and I can say mini equal to minimal of mini comma steps, because I need the minimal among all.', 'start': 2060.094, 'duration': 13.605}, {'end': 2085.045, 'text': 'so this will make sure it gives me the minimal among all and in this way I have pretty much solved the problem.', 'start': 2073.699, 'duration': 11.346}, {'end': 2089.107, 'text': 'yes, we have so every partition DP.', 'start': 2085.045, 'duration': 4.062}, {'end': 2092.053, 'text': 'remember, This is important.', 'start': 2089.107, 'duration': 2.946}, {'end': 2094.833, 'text': 'You will always have to do a partition.', 'start': 2093.092, 'duration': 1.741}, {'end': 2098.297, 'text': 'Always And after that, this was MCM.', 'start': 2095.835, 'duration': 2.462}, {'end': 2099.598, 'text': 'So you computed the number of steps.', 'start': 2098.337, 'duration': 1.261}, {'end': 2101.219, 'text': 'There might be some other problem.', 'start': 2100.118, 'duration': 1.101}, {'end': 2106.662, 'text': 'Steps computations will be on the problem.', 'start': 2101.959, 'duration': 4.703}], 'summary': 'The transcript discusses a method to compute the minimal number of steps for a given problem through partitioning and dynamic programming.', 'duration': 58.148, 'max_score': 1959.3, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y1959300.jpg'}], 'start': 1784.723, 'title': 'Matrix multiplication optimization', 'summary': 'Explains optimizing matrix multiplication by finding minimal operations, determined by matrix size, using a partition-based approach, leading to a total number of steps taken.', 'chapters': [{'end': 2232.945, 'start': 1784.723, 'title': 'Matrix multiplication optimization', 'summary': 'Explains how to optimize matrix multiplication by finding the minimal number of operations required, which is determined by the size of the matrices involved and can be computed using a partition-based approach, leading to a total number of steps taken.', 'duration': 448.222, 'highlights': ['The number of operations required to multiply matrices can be determined by the size of the matrices involved, with the minimal number of steps computed using a partition-based approach.', 'The size of the resultant matrix after multiplying matrices can be determined by the sizes of the matrices being multiplied, with the number of steps computed as the size of the matrices involved.', 'The partition-based approach involves finding the minimal number of steps required for each partition and combining these minimal steps to obtain the total number of steps taken.', 'Understanding the working of the partition-based approach is crucial for optimizing matrix multiplication, as it allows for the computation of the minimal number of steps required for each partition, leading to the total number of steps taken.']}], 'duration': 448.222, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y1784723.jpg', 'highlights': ['The partition-based approach involves finding the minimal number of steps for each partition and combining these minimal steps to obtain the total number of steps taken.', 'Understanding the working of the partition-based approach is crucial for optimizing matrix multiplication, as it allows for the computation of the minimal number of steps required for each partition, leading to the total number of steps taken.', 'The number of operations required to multiply matrices can be determined by the size of the matrices involved, with the minimal number of steps computed using a partition-based approach.', 'The size of the resultant matrix after multiplying matrices can be determined by the sizes of the matrices being multiplied, with the number of steps computed as the size of the matrices involved.']}, {'end': 2662.097, 'segs': [{'end': 2257.924, 'src': 'embed', 'start': 2233.525, 'weight': 2, 'content': [{'end': 2242.131, 'text': "So what is the recursion tree? We started with ABCD or we were something like, okay, let's take a 10, 20, 30, 40, 50.", 'start': 2233.525, 'duration': 8.606}, {'end': 2248.86, 'text': 'Okay And we wear something like this is A, this is B, this is C, this is D.', 'start': 2242.132, 'duration': 6.728}, {'end': 2250.781, 'text': 'And this is our I and this is our J.', 'start': 2248.86, 'duration': 1.921}, {'end': 2252.001, 'text': "That's how the recursion started.", 'start': 2250.781, 'duration': 1.22}, {'end': 2253.242, 'text': 'This was the initial function call.', 'start': 2252.041, 'duration': 1.201}, {'end': 2257.924, 'text': 'Now, what kind of partitions you did? You did three kind of partitions over here.', 'start': 2253.782, 'duration': 4.142}], 'summary': 'Explanation of recursion tree with initial function call and three partitions.', 'duration': 24.399, 'max_score': 2233.525, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y2233525.jpg'}, {'end': 2548.663, 'src': 'embed', 'start': 2505.922, 'weight': 0, 'content': [{'end': 2511.345, 'text': "let's quickly recap the entire thing, because i know it has been a lot of information.", 'start': 2505.922, 'duration': 5.423}, {'end': 2512.366, 'text': "let's quickly recap it.", 'start': 2511.345, 'duration': 1.021}, {'end': 2513.947, 'text': 'this video is going to be long.', 'start': 2512.366, 'duration': 1.581}, {'end': 2521.892, 'text': 'so we started with partition dp, where i said uh, you just have to decide in which ways you have to do and you have to figure out the minimal.', 'start': 2513.947, 'duration': 7.945}, {'end': 2526.516, 'text': "that's when you apply partition dp, and it's a tough part.", 'start': 2521.892, 'duration': 4.624}, {'end': 2534.768, 'text': 'so problem that we solved was matrix chain multiplication, where we are given a matrix n number of matrices,', 'start': 2526.516, 'duration': 8.252}, {'end': 2538.091, 'text': 'and we are being asked in what way will you multiply them?', 'start': 2534.768, 'duration': 3.323}, {'end': 2540.477, 'text': 'Like, in which order will you multiply them?', 'start': 2538.936, 'duration': 1.541}, {'end': 2546.621, 'text': 'Like, you can multiply AB at first or C, or you can multiply A and BC.', 'start': 2540.757, 'duration': 5.864}, {'end': 2548.663, 'text': 'So you have to multiply.', 'start': 2547.162, 'duration': 1.501}], 'summary': 'Recapped partition dp for matrix chain multiplication problem solving.', 'duration': 42.741, 'max_score': 2505.922, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y2505922.jpg'}], 'start': 2233.525, 'title': 'Recursion tree, partitioning, and matrix chain multiplication', 'summary': 'Explains the concept of recursion tree and partitioning, highlighting different possible partitions. it also discusses the matrix chain multiplication problem, aiming to minimize the number of operations using partition dp and base case principles.', 'chapters': [{'end': 2389.902, 'start': 2233.525, 'title': 'Recursion tree and partitioning', 'summary': 'Explains the concept of recursion tree and the process of partitioning using specific examples, highlighting the different possible partitions and their impact on the recursion process.', 'duration': 156.377, 'highlights': ['The recursion process starts with the initial function call using specific values like ABCD or 10, 20, 30, 40, 50, demonstrating the concept of partitioning and its impact on the recursion.', 'The chapter discusses the three kinds of partitions involved, outlining the specific partitions made for values like 20, 30, 40, 50, and emphasizing the different possible partitions that can be made during the recursion process.', 'The detailed process of partitioning is explained, highlighting the different combinations such as AB and CD, AB and CD, ABC and D, and their impact on the recursion process.']}, {'end': 2662.097, 'start': 2390.363, 'title': 'Matrix chain multiplication', 'summary': 'Discusses the matrix chain multiplication problem, where the goal is to determine the optimal way to multiply a given set of matrices to minimize the number of operations, utilizing partition dp and base case principles.', 'duration': 271.734, 'highlights': ['The chapter discusses the matrix chain multiplication problem, emphasizing the goal of determining the minimal number of operations required to multiply a given set of matrices in an optimal manner.', 'The concept of partition DP is introduced as a technique to handle the problem of determining the minimal cost of matrix chain multiplication.', 'The speaker explains the approach of using partition DP to solve the matrix chain multiplication problem, focusing on identifying different patterns and applying the technique to find the minimum cost.', 'The base case principle is highlighted as a fundamental aspect of the approach, involving determining the smallest portion of the matrix chain multiplication and utilizing top-down analysis to identify the lowest portion as a single array element.']}], 'duration': 428.572, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y2233525.jpg', 'highlights': ['The chapter discusses the matrix chain multiplication problem, emphasizing the goal of determining the minimal number of operations required to multiply a given set of matrices in an optimal manner.', 'The concept of partition DP is introduced as a technique to handle the problem of determining the minimal cost of matrix chain multiplication.', 'The detailed process of partitioning is explained, highlighting the different combinations such as AB and CD, AB and CD, ABC and D, and their impact on the recursion process.', 'The recursion process starts with the initial function call using specific values like ABCD or 10, 20, 30, 40, 50, demonstrating the concept of partitioning and its impact on the recursion.']}, {'end': 3212.397, 'segs': [{'end': 2685.765, 'src': 'embed', 'start': 2662.097, 'weight': 4, 'content': [{'end': 2668.622, 'text': "so whenever it's a single matrix, if it's a single matrix, there is no multiplication required, thereby zero steps return.", 'start': 2662.097, 'duration': 6.525}, {'end': 2670.209, 'text': 'what is the next step?', 'start': 2669.428, 'duration': 0.781}, {'end': 2673.753, 'text': 'I said I want to try out all partitions.', 'start': 2670.209, 'duration': 3.544}, {'end': 2675.434, 'text': 'and how can I try out all partitions?', 'start': 2673.753, 'duration': 1.681}, {'end': 2676.395, 'text': 'it was very simple.', 'start': 2675.434, 'duration': 0.961}, {'end': 2683.603, 'text': 'I said okay, listen, one of the ways is I can go like this, like this.', 'start': 2676.395, 'duration': 7.208}, {'end': 2685.765, 'text': 'the other way is I can go like this, like this.', 'start': 2683.603, 'duration': 2.162}], 'summary': 'Multiplication not required for single matrix, zero steps. trying out all partitions using simple methods.', 'duration': 23.668, 'max_score': 2662.097, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y2662097.jpg'}, {'end': 2772.561, 'src': 'embed', 'start': 2740.946, 'weight': 3, 'content': [{'end': 2742.347, 'text': 'the steps required will be this', 'start': 2740.946, 'duration': 1.401}, {'end': 2743.767, 'text': 'And we got this formula.', 'start': 2742.747, 'duration': 1.02}, {'end': 2747.662, 'text': 'But this has to be solved.', 'start': 2745.319, 'duration': 2.343}, {'end': 2749.844, 'text': 'This is solved by calling f of ik.', 'start': 2748.202, 'duration': 1.642}, {'end': 2751.846, 'text': 'And the other portion has to be solved.', 'start': 2750.505, 'duration': 1.341}, {'end': 2752.627, 'text': 'So this is called by.', 'start': 2751.866, 'duration': 0.761}, {'end': 2754.935, 'text': 'solving this.', 'start': 2753.775, 'duration': 1.16}, {'end': 2756.136, 'text': 'and this is the steps.', 'start': 2754.935, 'duration': 1.201}, {'end': 2760.377, 'text': 'for one type of partition, there can be K type of partitions.', 'start': 2756.136, 'duration': 4.241}, {'end': 2763.838, 'text': 'take all the steps and get the minimal of all the steps.', 'start': 2760.377, 'duration': 3.461}, {'end': 2765.219, 'text': "and that's what you return.", 'start': 2763.838, 'duration': 1.381}, {'end': 2772.561, 'text': 'and in this way you can easily yes, I repeat, in this way you can easily get this problem to be done.', 'start': 2765.219, 'duration': 7.342}], 'summary': 'Solve using a formula, get minimal of k partitions, return result.', 'duration': 31.615, 'max_score': 2740.946, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y2740946.jpg'}, {'end': 2828.652, 'src': 'embed', 'start': 2802.599, 'weight': 2, 'content': [{'end': 2810.781, 'text': 'so the partitions will be from i to k, less than j, and k plus plus, basically till j, minus one is what the partitions will be in the minimal.', 'start': 2802.599, 'duration': 8.182}, {'end': 2815.001, 'text': 'so probably you can keep the minimal mini equal to i9 or something like this.', 'start': 2810.781, 'duration': 4.22}, {'end': 2816.162, 'text': 'and you know the steps.', 'start': 2815.001, 'duration': 1.161}, {'end': 2817.462, 'text': 'you can easily compute the steps.', 'start': 2816.162, 'duration': 1.3}, {'end': 2828.652, 'text': "to be very simple, it's array of i minus one into array of k, into array of j, and that's it.", 'start': 2817.462, 'duration': 11.19}], 'summary': 'The partitions will be from i to k, less than j, and k++ till j-1. the computation steps are simple: array[i-1] * array[k] * array[j].', 'duration': 26.053, 'max_score': 2802.599, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y2802599.jpg'}, {'end': 2935.876, 'src': 'embed', 'start': 2908.132, 'weight': 0, 'content': [{'end': 2911.254, 'text': "But if you want to take pen and paper, you can do it, but that's not required.", 'start': 2908.132, 'duration': 3.122}, {'end': 2914.076, 'text': 'So the time complexity goes as exponential.', 'start': 2911.855, 'duration': 2.221}, {'end': 2915.797, 'text': 'Obviously, this is something which will not work.', 'start': 2914.536, 'duration': 1.261}, {'end': 2921.727, 'text': "How do you convert a recursion into something better? That's memoization.", 'start': 2916.724, 'duration': 5.003}, {'end': 2934.475, 'text': "And I've already told if overlapping subproblems exist, if overlapping subproblems exist, you can convert this into memoization.", 'start': 2923.608, 'duration': 10.867}, {'end': 2935.876, 'text': "Let's see if it does exist.", 'start': 2934.635, 'duration': 1.241}], 'summary': 'Memoization can improve time complexity for recurring problems.', 'duration': 27.744, 'max_score': 2908.132, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y2908132.jpg'}, {'end': 2989.501, 'src': 'embed', 'start': 2963.376, 'weight': 1, 'content': [{'end': 2968.917, 'text': "So if you've computed this, if you've computed this, can't you return this? You can.", 'start': 2963.376, 'duration': 5.541}, {'end': 2973.798, 'text': 'Thereby, I figured out that there are overlapping sub-problems and I can apply my mesh.', 'start': 2969.477, 'duration': 4.321}, {'end': 2979.674, 'text': 'In order to apply memory action, what is the first thing? See the changing variables.', 'start': 2974.73, 'duration': 4.944}, {'end': 2987.099, 'text': 'What is the state represented by? The state is represented by i and j.', 'start': 2981.095, 'duration': 6.004}, {'end': 2989.501, 'text': 'The state is represented by i and j.', 'start': 2987.099, 'duration': 2.402}], 'summary': 'Identified overlapping sub-problems, applied mesh for memory action using i and j as state.', 'duration': 26.125, 'max_score': 2963.376, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y2963376.jpg'}], 'start': 2662.097, 'title': 'Matrix multiplication and memoization for recursive solutions', 'summary': 'Covers a dynamic programming algorithm for matrix multiplication, emphasizing its efficiency and also discusses converting a recursive solution to a memoized solution, achieving a time complexity reduction to n^2 and partial acceptance in interviews.', 'chapters': [{'end': 2857.652, 'start': 2662.097, 'title': 'Matrix multiplication algorithm', 'summary': 'Discusses a dynamic programming algorithm for matrix multiplication, outlining the iterative process of determining minimal steps required for various partitions, and emphasizes the efficiency of the approach in solving the problem.', 'duration': 195.555, 'highlights': ['The algorithm iteratively determines minimal steps required for different partitions, from i to j minus 1 or from i plus 1 to j, with a focus on solving the problem efficiently.', 'The steps for one type of partition can be computed, and there can be K types of partitions, with the minimal steps being chosen, indicating the adaptive nature of the algorithm to handle different partition scenarios.', "The function for matrix multiplication is designed to efficiently handle the iterative process of determining minimal steps for various partitions, thereby optimizing the algorithm's performance."]}, {'end': 3212.397, 'start': 2859.93, 'title': 'Memoization for recursive solutions', 'summary': 'Explains how to convert a recursive solution with exponential time complexity to a more efficient memoized solution, reducing time complexity to n^2 and achieving partial acceptance in interview scenarios.', 'duration': 352.467, 'highlights': ['The time complexity of the initial recursive solution was near exponential, making it impractical for large inputs, but by applying memoization, the time complexity was reduced to n^2, significantly improving efficiency.', 'Identifying overlapping sub-problems in the recursion tree allowed for the application of memoization, leading to the storage and re-use of previously computed states, effectively optimizing the solution.', 'The tabulation method will be covered in the next lecture, demonstrating an alternative approach to dynamic programming, providing a comprehensive understanding of efficient solution strategies.']}], 'duration': 550.3, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/vRVfmbCFW7Y/pics/vRVfmbCFW7Y2662097.jpg', 'highlights': ['The time complexity of the initial recursive solution was near exponential, making it impractical for large inputs, but by applying memoization, the time complexity was reduced to n^2, significantly improving efficiency.', 'Identifying overlapping sub-problems in the recursion tree allowed for the application of memoization, leading to the storage and re-use of previously computed states, effectively optimizing the solution.', 'The algorithm iteratively determines minimal steps required for different partitions, from i to j minus 1 or from i plus 1 to j, with a focus on solving the problem efficiently.', 'The steps for one type of partition can be computed, and there can be K types of partitions, with the minimal steps being chosen, indicating the adaptive nature of the algorithm to handle different partition scenarios.', "The function for matrix multiplication is designed to efficiently handle the iterative process of determining minimal steps for various partitions, thereby optimizing the algorithm's performance."]}], 'highlights': ['The minimal number of operations required for matrix multiplication was found to be 4,500, significantly lower than the 27,000 operations for an alternative approach.', 'The instructor guarantees increased comfort and proficiency in partition DP after solving all the problems presented, similar to previous DP patterns covered.', 'The time complexity of the initial recursive solution was near exponential, making it impractical for large inputs, but by applying memoization, the time complexity was reduced to n^2, significantly improving efficiency.', 'The approach includes solving four to six problems on partition DP to ensure a deep understanding and comfort with this pattern.', 'The chapter discusses the matrix chain multiplication problem, emphasizing the goal of determining the minimal number of operations required to multiply a given set of matrices in an optimal manner.', 'The concept of partition DP is introduced as a technique to handle the problem of determining the minimal cost of matrix chain multiplication.', "The crucial formula for matrix multiplication, 'previous matrix size into current matrix size', emphasizing the relationship between the sizes of consecutive matrices being multiplied.", 'The algorithm iteratively determines minimal steps required for different partitions, from i to j minus 1 or from i plus 1 to j, with a focus on solving the problem efficiently.', 'The partition-based approach involves finding the minimal number of steps for each partition and combining these minimal steps to obtain the total number of steps taken.', 'The need to think in terms of arrays rather than individual elements when dealing with matrix multiplication, such as representing the given elements a, b, c, d as an array.']}