title

Amazon Coding Interview Question - Recursive Staircase Problem

description

Amazon coding interview question and answer - recursive staircase problem!
For daily coding problems like this one, I’d recommend this website called Daily Coding Problem. You can find it here: https://csdojo.io/daily
(That’s a referral link, and you can get a 10% discount through that link. Their free option and blog articles are good, too, though.)
Outline (check my comment for the clickable outline):
0:07: Problem description
1:14: A variation of the problem
2:15: Thinking about simple cases
4:18: Finding a pattern
5:24: Relabeling the steps
6:41: Revisiting the pattern with the new labels
7:53: The pattern we’ve found - recap.
8:11: The recursive relationship we’ve found
8:50: What about when N = 0?
9:40: Writing a naive recursive solution
10:39: Why this solution is not efficient
11:24: How to fix it with dynamic programming (bottom-up)
12:27: The bottom-up solution in code
13:34: How to make it more efficient in terms of space
14:19: Solution to the variation of the problem
14:49: The recursive relationship for this problem (the variation)
15:08: A naive, INCORRECT recursive solution to this problem
15:50: A naive, CORRECT recursive solution to this problem
16:17: A naive, correct recursive solution in code
17:11: A dynamic programming / bottom-up approach
19:17: How to get daily coding problems like this one (go to https://csdojo.io/daily)
Also, keep in touch on Facebook: https://www.facebook.com/entercsdojo

detail

{'title': 'Amazon Coding Interview Question - Recursive Staircase Problem', 'heatmap': [{'end': 871.938, 'start': 852.309, 'weight': 1}, {'end': 948.874, 'start': 924.298, 'weight': 0.836}, {'end': 1170.759, 'start': 1148.725, 'weight': 0.768}], 'summary': 'Addresses the amazon coding interview question of finding the ways to climb a staircase with one or two steps, analyzing paths for n=1 to 4, with n=4 having five paths, and discusses dynamic programming for efficient solutions and the comparison of recursive and bottom-up approaches.', 'chapters': [{'end': 274.314, 'segs': [{'end': 103.665, 'src': 'embed', 'start': 24.881, 'weight': 0, 'content': [{'end': 28.362, 'text': 'And suppose that you can only take one or two steps at a time.', 'start': 24.881, 'duration': 3.481}, {'end': 34.964, 'text': 'So if you start at the bottom, you can only go from the bottom to here or here.', 'start': 29.042, 'duration': 5.922}, {'end': 41.646, 'text': 'And if you decide to go here, you can only go from here to here or to here.', 'start': 35.804, 'duration': 5.842}, {'end': 47.108, 'text': 'And the problem here is writing a function called numways,', 'start': 42.146, 'duration': 4.962}, {'end': 53.97, 'text': 'which takes the positive integer n and returns the number of ways you can go from the bottom to the top.', 'start': 47.108, 'duration': 6.862}, {'end': 60.494, 'text': "So if the given n is 2 instead of 4, there's only two ways of going from the bottom to the top.", 'start': 54.81, 'duration': 5.684}, {'end': 68.859, 'text': 'The first way would be taking one step at a time, and the second way would be taking two steps from bottom to the top.', 'start': 61.034, 'duration': 7.825}, {'end': 74.643, 'text': 'So your function numways should return 2 if the given n is 2.', 'start': 69.5, 'duration': 5.143}, {'end': 77.906, 'text': "And once you're done with this problem, think about this variation.", 'start': 74.643, 'duration': 3.263}, {'end': 84.15, 'text': "What if, in addition to n, you're also given x, which is a set of positive integers?", 'start': 78.426, 'duration': 5.724}, {'end': 90.014, 'text': "And this is going to be the set of numbers of steps you're allowed to take, instead of just one or two steps.", 'start': 84.95, 'duration': 5.064}, {'end': 97.94, 'text': 'So, in this particular case, you can go from the bottom to over here by taking one step at a time,', 'start': 90.815, 'duration': 7.125}, {'end': 103.665, 'text': 'or you can go from the bottom to over here by taking one, two, three steps.', 'start': 97.94, 'duration': 5.725}], 'summary': "Function 'numways' finds ways to climb stairs with given steps.", 'duration': 78.784, 'max_score': 24.881, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5o-kdjv7FD0/pics/5o-kdjv7FD024881.jpg'}, {'end': 223.376, 'src': 'embed', 'start': 198.38, 'weight': 3, 'content': [{'end': 207.966, 'text': "so that's This path right here 0, 2, 1, and then I'm gonna write down here 1 as well to show that we have only one path.", 'start': 198.38, 'duration': 9.586}, {'end': 211.488, 'text': "and let's do the same thing for when n equals 3.", 'start': 207.966, 'duration': 3.522}, {'end': 216.332, 'text': "first of all, let's label these stairs as 0, 1, 2 and 3.", 'start': 211.488, 'duration': 4.844}, {'end': 223.376, 'text': 'the first path we can immediately find is this one 0, 1, 2 and then 3.', 'start': 216.332, 'duration': 7.044}], 'summary': "The path for n=2 has 1 path. for n=3, there's 1 path from 0 to 3.", 'duration': 24.996, 'max_score': 198.38, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5o-kdjv7FD0/pics/5o-kdjv7FD0198380.jpg'}, {'end': 280.475, 'src': 'embed', 'start': 252.743, 'weight': 4, 'content': [{'end': 258.408, 'text': "i'm just gonna skip over this just to save time, but you'll find that there are five paths here.", 'start': 252.743, 'duration': 5.665}, {'end': 263.011, 'text': "okay. so the question here is can you find any pattern for what we've found so far?", 'start': 258.408, 'duration': 4.603}, {'end': 271.093, 'text': "It's actually kind of hard if you're not used to this kind of thing, but I think the easiest way to find a good pattern here is by visually.", 'start': 263.852, 'duration': 7.241}, {'end': 274.314, 'text': "So let's go back to when n is equal to three.", 'start': 271.714, 'duration': 2.6}, {'end': 280.475, 'text': 'In that case, you ask yourself how can we go from step zero to step three?', 'start': 274.854, 'duration': 5.621}], 'summary': 'Analyzing five paths to find patterns visually, starting from n=3.', 'duration': 27.732, 'max_score': 252.743, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5o-kdjv7FD0/pics/5o-kdjv7FD0252743.jpg'}], 'start': 0.389, 'title': 'Staircase climbing problem', 'summary': 'Discusses amazon coding interview question involving finding the number of ways to climb a staircase of n steps by taking one or two steps at a time, and counting the number of paths to the top of a staircase for n=1 to 4, identifying the number of ways for each value, with n=4 having five paths.', 'chapters': [{'end': 125.311, 'start': 0.389, 'title': 'Amazon coding interview question', 'summary': 'Discusses a coding interview question asked by companies like amazon, involving finding the number of ways to climb a staircase of n steps by taking one or two steps at a time and considering a variation where the set of allowed step sizes is given, prompting the consideration of multiple step sizes.', 'duration': 124.922, 'highlights': ['The problem involves finding the number of ways to climb a staircase of n steps by taking one or two steps at a time. This problem entails determining the number of ways to ascend a staircase with n steps, where each step allows taking one or two steps at a time.', 'The function numways should return the number of ways to climb the staircase. The function numways should take a positive integer n as input and return the number of ways to ascend the staircase of n steps.', 'A variation of the problem involves considering a set of allowed step sizes. The variation introduces the concept of a set of positive integers x, representing the allowed step sizes, thereby requiring the consideration of multiple step sizes in addition to the original problem of taking one or two steps at a time.']}, {'end': 274.314, 'start': 125.891, 'title': 'Counting paths to the top', 'summary': 'Discusses counting the number of paths to the top of a staircase for different values of n, ranging from 1 to 4, and identifies the number of ways for each value, with n=4 having five paths, and encourages finding a pattern visually.', 'duration': 148.423, 'highlights': ['The chapter discusses counting the number of paths to the top of a staircase for different values of n, ranging from 1 to 4, and identifies the number of ways for each value, with n=4 having five paths.', 'The narrator emphasizes finding a pattern visually as the easiest way to identify a good pattern for the given paths.', 'For n=4, the number of paths to the top of the staircase is five, showcasing the increasing complexity of the problem with higher values of n.']}], 'duration': 273.925, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5o-kdjv7FD0/pics/5o-kdjv7FD0389.jpg', 'highlights': ['The problem involves finding the number of ways to climb a staircase of n steps by taking one or two steps at a time.', 'The function numways should return the number of ways to climb the staircase.', 'A variation of the problem involves considering a set of allowed step sizes.', 'The chapter discusses counting the number of paths to the top of a staircase for different values of n, ranging from 1 to 4.', 'The narrator emphasizes finding a pattern visually as the easiest way to identify a good pattern for the given paths.', 'For n=4, the number of paths to the top of the staircase is five.']}, {'end': 855.89, 'segs': [{'end': 352.511, 'src': 'embed', 'start': 274.854, 'weight': 0, 'content': [{'end': 280.475, 'text': 'In that case, you ask yourself how can we go from step zero to step three?', 'start': 274.854, 'duration': 5.621}, {'end': 288.777, 'text': 'And to answer that question, you need to ask yourself should I go over just one step or two steps from the bottom?', 'start': 281.095, 'duration': 7.682}, {'end': 298.164, 'text': 'And if you decided to take just one step from the bottom, then you needed to ask yourself how can we go from step one to step three?', 'start': 289.597, 'duration': 8.567}, {'end': 306.13, 'text': "But if you look at only this part up here, that's actually just a staircase with two steps.", 'start': 298.884, 'duration': 7.246}, {'end': 314.557, 'text': "So when you ask yourself, how can we go from step one to step three? That's actually a problem that we've already solved when n is equal to two.", 'start': 306.65, 'duration': 7.907}, {'end': 323.68, 'text': 'Because in this case too, we asked ourselves how can we go from step zero to step two or go over a staircase of two steps?', 'start': 315.177, 'duration': 8.503}, {'end': 329.502, 'text': 'And I think this pattern is going to be more clear once we relabel these steps.', 'start': 324.48, 'duration': 5.022}, {'end': 335.024, 'text': "So let's label these steps not from the bottom, but from the top instead.", 'start': 330.302, 'duration': 4.722}, {'end': 346.508, 'text': 'So the top step is going to be 0 here, and then this step is going to be 1, 2, and this last step, the bottom step, is going to be called 3.', 'start': 335.564, 'duration': 10.944}, {'end': 352.511, 'text': 'And once we relabel all the other steps as well, these steps is going to look like this.', 'start': 346.508, 'duration': 6.003}], 'summary': 'Explaining a staircase problem-solving pattern using relabeling and steps from 0 to 3.', 'duration': 77.657, 'max_score': 274.854, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5o-kdjv7FD0/pics/5o-kdjv7FD0274854.jpg'}, {'end': 518.884, 'src': 'embed', 'start': 492.146, 'weight': 4, 'content': [{'end': 495.869, 'text': 'And based on that we can write that numways of 3,', 'start': 492.146, 'duration': 3.723}, {'end': 504.414, 'text': 'or the number of ways of climbing over a staircase of 3 steps is equal to numways of 2 plus numways of 1..', 'start': 495.869, 'duration': 8.545}, {'end': 512.62, 'text': 'And actually in general, this should be numways of n is equal to numways of n-1 plus numways of n-2.', 'start': 504.414, 'duration': 8.206}, {'end': 518.884, 'text': 'So you can check that, for example, for when n is equal to 4.', 'start': 514.321, 'duration': 4.563}], 'summary': 'The number of ways to climb a staircase of 3 steps is equal to the sum of the number of ways to climb 2 steps and 1 step, following a general pattern of numways of n = numways of n-1 + numways of n-2.', 'duration': 26.738, 'max_score': 492.146, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5o-kdjv7FD0/pics/5o-kdjv7FD0492146.jpg'}, {'end': 704.404, 'src': 'embed', 'start': 678.291, 'weight': 3, 'content': [{'end': 682.492, 'text': 'and this problem gets worse and worse as n becomes larger and larger.', 'start': 678.291, 'duration': 4.201}, {'end': 684.493, 'text': "So let's see how we can fix that.", 'start': 682.973, 'duration': 1.52}, {'end': 690.896, 'text': 'We can fix it with dynamic programming, and one way to implement it would be with a bottom-up approach.', 'start': 684.953, 'duration': 5.943}, {'end': 693.597, 'text': 'So let me just quickly explain the idea here.', 'start': 691.616, 'duration': 1.981}, {'end': 700.221, 'text': "For the bottom-up approach, we're going to use this table of n and numways of n.", 'start': 694.138, 'duration': 6.083}, {'end': 704.404, 'text': "And at the beginning, we don't have the values for numways of n.", 'start': 700.221, 'duration': 4.183}], 'summary': 'Dynamic programming resolves problem for increasing n values.', 'duration': 26.113, 'max_score': 678.291, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5o-kdjv7FD0/pics/5o-kdjv7FD0678291.jpg'}], 'start': 274.854, 'title': 'Staircase problem-solving and dynamic programming', 'summary': 'Discusses problem-solving for navigating a staircase with two steps and addresses the question of going from step zero to step three. it also involves relabeling steps, analyzing paths, and implementing a recursive function and dynamic programming approach for efficient solutions.', 'chapters': [{'end': 323.68, 'start': 274.854, 'title': 'Staircase problem solving', 'summary': 'Discusses the problem-solving approach for navigating a staircase with two steps, addressing the question of going from step zero to step three and the solutions when n is equal to two.', 'duration': 48.826, 'highlights': ["The problem of going from step one to step three is a problem that we've already solved when n is equal to two, addressing how to go from step zero to step two or over a staircase of two steps.", 'The approach involves asking whether to go over just one step or two steps from the bottom, providing a structured problem-solving method for navigating the staircase.']}, {'end': 855.89, 'start': 324.48, 'title': 'Staircase path analysis and dynamic programming', 'summary': 'Involves relabeling steps, analyzing paths, and using recursive relationship to find a solution, leading to the implementation of a recursive function and a dynamic programming approach to solve the staircase problem efficiently.', 'duration': 531.41, 'highlights': ['Relabeling of Steps The speaker relabels the steps from top to bottom, illustrating the relabeling process and showing that the paths for n = 1, n = 2, and n = 3 change accordingly.', 'Recursive Relationship for Staircase Problem The speaker establishes the recursive relationship for the number of ways to climb n steps, demonstrating its application for n = 3 and n = 4, and discussing the base case for n = 0.', 'Inefficiency of Recursive Function The speaker points out the inefficiency of the recursive function for finding the number of ways to climb n steps, highlighting the repeated computation and its worsening impact as n increases.', 'Bottom-Up Dynamic Programming Approach The speaker explains the bottom-up dynamic programming approach using an array to store the number of ways to climb n steps, and provides a code implementation that avoids repeated computations and optimizes space usage.']}], 'duration': 581.036, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5o-kdjv7FD0/pics/5o-kdjv7FD0274854.jpg', 'highlights': ['The approach involves asking whether to go over just one step or two steps from the bottom, providing a structured problem-solving method for navigating the staircase.', "The problem of going from step one to step three is a problem that we've already solved when n is equal to two, addressing how to go from step zero to step two or over a staircase of two steps.", 'Relabeling of Steps The speaker relabels the steps from top to bottom, illustrating the relabeling process and showing that the paths for n = 1, n = 2, and n = 3 change accordingly.', 'Bottom-Up Dynamic Programming Approach The speaker explains the bottom-up dynamic programming approach using an array to store the number of ways to climb n steps, and provides a code implementation that avoids repeated computations and optimizes space usage.', 'Recursive Relationship for Staircase Problem The speaker establishes the recursive relationship for the number of ways to climb n steps, demonstrating its application for n = 3 and n = 4, and discussing the base case for n = 0.', "The problem of going from step one to step three is a problem that we've already solved when n is equal to two, addressing how to go from step zero to step two or over a staircase of two steps."]}, {'end': 1200.061, 'segs': [{'end': 888.347, 'src': 'embed', 'start': 856.431, 'weight': 0, 'content': [{'end': 858.672, 'text': "Let's take a look at the variation I told you about.", 'start': 856.431, 'duration': 2.241}, {'end': 869.137, 'text': "As I explained earlier, in this variation of the problem, you're given n as well as x, which is a set of numbers of steps you can take at a time.", 'start': 859.292, 'duration': 9.845}, {'end': 871.938, 'text': "And let's solve this problem recursively as well.", 'start': 869.597, 'duration': 2.341}, {'end': 879.382, 'text': "And we're going to base our recursive solution to this problem on the recursive solution that we saw earlier to the previous problem.", 'start': 872.499, 'duration': 6.883}, {'end': 882.884, 'text': 'So we have the function we saw earlier, numways.', 'start': 879.982, 'duration': 2.902}, {'end': 888.347, 'text': "The only difference from what we saw earlier is the fact that we don't have an else statement.", 'start': 883.344, 'duration': 5.003}], 'summary': 'Exploring variation of problem solving using recursive approach', 'duration': 31.916, 'max_score': 856.431, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5o-kdjv7FD0/pics/5o-kdjv7FD0856431.jpg'}, {'end': 948.874, 'src': 'heatmap', 'start': 924.298, 'weight': 0.836, 'content': [{'end': 935.785, 'text': 'and this says the base case when n is equal to 0, we should return 1, And otherwise we should return the sum of numways of n-1, n-3, and n-5..', 'start': 924.298, 'duration': 11.487}, {'end': 937.986, 'text': 'But this function is not gonna work.', 'start': 936.225, 'duration': 1.761}, {'end': 948.874, 'text': "So when n is, for example, 2, numways of n-3 would be numways of minus 1, and numways of minus 1 isn't defined well.", 'start': 938.507, 'duration': 10.367}], 'summary': 'The function to calculate numways is flawed due to undefined values for n-3 and n-1.', 'duration': 24.576, 'max_score': 924.298, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5o-kdjv7FD0/pics/5o-kdjv7FD0924298.jpg'}, {'end': 1195.457, 'src': 'heatmap', 'start': 1148.725, 'weight': 1, 'content': [{'end': 1153.749, 'text': 'then you just need to replace this with x and then have this function.', 'start': 1148.725, 'duration': 5.024}, {'end': 1155.75, 'text': 'take x as an extra argument.', 'start': 1153.749, 'duration': 2.001}, {'end': 1161.953, 'text': 'Okay now, this problem actually came from this website called Daily Coding Problem.', 'start': 1157.31, 'duration': 4.643}, {'end': 1166.696, 'text': "It's a website where you can get a daily coding problem just like this one.", 'start': 1162.834, 'duration': 3.862}, {'end': 1170.759, 'text': "And if you want more problems like this one, I'd actually highly recommend it.", 'start': 1167.317, 'duration': 3.442}, {'end': 1175.362, 'text': "It's actually a website that's made by a friend of mine who I used to work with at Google too.", 'start': 1171.279, 'duration': 4.083}, {'end': 1180.546, 'text': 'and if you use my referral link, csdojo.io daily,', 'start': 1176.162, 'duration': 4.384}, {'end': 1188.192, 'text': "it'll let them know that you came from here and with that link you can get a 10 discount on their premium subscription as well.", 'start': 1180.546, 'duration': 7.646}, {'end': 1195.457, 'text': 'but personally i would say that even their free option and their blog articles on the site have a lot of value too.', 'start': 1188.192, 'duration': 7.265}], 'summary': 'Recommend daily coding problem website for daily coding challenges and a 10% discount on premium subscription using referral link csdojo.io daily.', 'duration': 46.732, 'max_score': 1148.725, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5o-kdjv7FD0/pics/5o-kdjv7FD01148725.jpg'}], 'start': 856.431, 'title': 'Recursive vs bottom-up approach for staircase problem', 'summary': 'Explores the efficiency and adaptability of recursive and bottom-up approaches for solving a staircase problem, demonstrated through the iterative solution, as derived from a daily coding problem website.', 'chapters': [{'end': 1200.061, 'start': 856.431, 'title': 'Recursive vs bottom-up approach for staircase problem', 'summary': "Explores the recursive and bottom-up approaches for solving a staircase problem, demonstrating the iterative solution's efficiency and adaptability, originating from a daily coding problem website.", 'duration': 343.63, 'highlights': ["Iterative solution's efficiency and adaptability The chapter demonstrates the bottom-up approach as an efficient and adaptable solution to the staircase problem.", 'Daily Coding Problem website The problem discussed originates from the Daily Coding Problem website, offering daily coding challenges and valuable resources.', "Referral link benefit The referral link 'csdojo.io daily' provides a 10% discount on the website's premium subscription, recommended for its free options and blog articles' value."]}], 'duration': 343.63, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5o-kdjv7FD0/pics/5o-kdjv7FD0856431.jpg', 'highlights': ["Iterative solution's efficiency and adaptability demonstrated as an efficient and adaptable solution to the staircase problem", 'Daily Coding Problem website offers daily coding challenges and valuable resources', "Referral link 'csdojo.io daily' provides a 10% discount on the website's premium subscription"]}], 'highlights': ['The problem involves finding the number of ways to climb a staircase of n steps by taking one or two steps at a time.', 'The function numways should return the number of ways to climb the staircase.', 'The chapter discusses counting the number of paths to the top of a staircase for different values of n, ranging from 1 to 4.', 'For n=4, the number of paths to the top of the staircase is five.', 'The approach involves asking whether to go over just one step or two steps from the bottom, providing a structured problem-solving method for navigating the staircase.', 'Relabeling of Steps The speaker relabels the steps from top to bottom, illustrating the relabeling process and showing that the paths for n = 1, n = 2, and n = 3 change accordingly.', 'Bottom-Up Dynamic Programming Approach The speaker explains the bottom-up dynamic programming approach using an array to store the number of ways to climb n steps, and provides a code implementation that avoids repeated computations and optimizes space usage.', 'Recursive Relationship for Staircase Problem The speaker establishes the recursive relationship for the number of ways to climb n steps, demonstrating its application for n = 3 and n = 4, and discussing the base case for n = 0.', "Iterative solution's efficiency and adaptability demonstrated as an efficient and adaptable solution to the staircase problem", 'Daily Coding Problem website offers daily coding challenges and valuable resources', "Referral link 'csdojo.io daily' provides a 10% discount on the website's premium subscription"]}