title

Introduction to Algorithms - Problem Session 1: Asymptotic Behavior of Functions and Double-ended...

description

MIT 6.006 Introduction to Algorithms, Spring 2020
Instructor: Jason Ku
View the complete course: https://ocw.mit.edu/6-006S20
YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP63EdVPNLG3ToM6LaEUuStEY
Four examples of worked problems on the asymptotic behavior of functions and double-ended sequence operations.
License: Creative Commons BY-NC-SA
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{'title': 'Introduction to Algorithms - Problem Session 1: Asymptotic Behavior of Functions and Double-ended...', 'heatmap': [{'end': 369.428, 'start': 254.667, 'weight': 0.746}, {'end': 474.025, 'start': 415.798, 'weight': 0.867}, {'end': 990.206, 'start': 884.271, 'weight': 0.728}, {'end': 2812.455, 'start': 2753.951, 'weight': 1}, {'end': 3690.772, 'start': 3631.463, 'weight': 0.785}, {'end': 4116.83, 'start': 4048.64, 'weight': 0.753}, {'end': 4213.229, 'start': 4153.421, 'weight': 0.913}], 'summary': "An introduction to the 6.006 problem session aims to enhance students' approach to problem sets and foster a safe learning environment. it covers analyzing functions' asymptotic behavior, algorithm abstraction, data structure operations, recursion, dynamic arrays, dynamic array operations, reversing linked lists, and code cleaning with live coding in latex.", 'chapters': [{'end': 136.677, 'segs': [{'end': 51.368, 'src': 'embed', 'start': 13.001, 'weight': 0, 'content': [{'end': 13.541, 'text': "I'm Jason.", 'start': 13.001, 'duration': 0.54}, {'end': 15.202, 'text': 'Hopefully, you guys saw me on Tuesday.', 'start': 13.681, 'duration': 1.521}, {'end': 22.527, 'text': "This is our first ever 6.006 problem session that we'll be having on Fridays this term.", 'start': 16.023, 'duration': 6.504}, {'end': 24.529, 'text': "It's really an experiment.", 'start': 22.988, 'duration': 1.541}, {'end': 26.29, 'text': "We've never done this before.", 'start': 25.109, 'duration': 1.181}, {'end': 36.036, 'text': 'But one of the things that we were discussed while preparing for this class is that we have two different methods of instruction,', 'start': 27.271, 'duration': 8.765}, {'end': 42.401, 'text': 'formally usually in this class a lecture, which is there to present you with the fundamental material,', 'start': 36.036, 'duration': 6.365}, {'end': 51.368, 'text': 'the data structures and the algorithms that are the base, the foundation of how you will be approaching problems in this class.', 'start': 43.181, 'duration': 8.187}], 'summary': 'Jason introduces the first 6.006 problem session on fridays, emphasizing the importance of lectures for fundamental material.', 'duration': 38.367, 'max_score': 13.001, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k13001.jpg'}, {'end': 136.677, 'src': 'embed', 'start': 61.365, 'weight': 1, 'content': [{'end': 69.154, 'text': "But there's usually a much different feel between those problems that we'll give you than the underlying foundational material.", 'start': 61.365, 'duration': 7.789}, {'end': 72.518, 'text': 'So the application of that material will feel very different.', 'start': 69.555, 'duration': 2.963}, {'end': 75.762, 'text': 'And a lot of times, there are tricks to approaching the problems.', 'start': 72.919, 'duration': 2.843}, {'end': 85.168, 'text': 'or ways of approaching the problems that you kind of just have to figure out by working on the problems, sometimes going to office hours.', 'start': 76.903, 'duration': 8.265}, {'end': 99.655, 'text': "But what we wanted to do this term was to since we had the opportunity to be recorded by OCW was to record us going through some problems that we've had on problem sets in the past.", 'start': 86.288, 'duration': 13.367}, {'end': 107.359, 'text': "so you could see how we would approach working on these problems that you'll be working on at least in a similar vein.", 'start': 99.655, 'duration': 7.704}, {'end': 110.041, 'text': "So that's the goal of these problem sessions.", 'start': 107.68, 'duration': 2.361}, {'end': 114.303, 'text': 'In the past, for OCW, we have recorded a recitation.', 'start': 110.321, 'duration': 3.982}, {'end': 123.208, 'text': 'But we felt that that was a little less useful to you guys because recitation is meant for interaction, questions, one-on-one questions.', 'start': 114.984, 'duration': 8.224}, {'end': 126.75, 'text': 'We wanted to be a safe space for you guys to interact with the material.', 'start': 123.228, 'duration': 3.522}, {'end': 131.493, 'text': 'in a smaller environment that might not be recorded.', 'start': 127.75, 'duration': 3.743}, {'end': 136.677, 'text': "So that's the goal of these sessions that we'll be doing on Fridays.", 'start': 131.593, 'duration': 5.084}], 'summary': 'Recording problem-solving sessions for ocw to help students approach problems effectively and interact with foundational material.', 'duration': 75.312, 'max_score': 61.365, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k61365.jpg'}], 'start': 13.001, 'title': '6.006 problem session introduction', 'summary': "Introduces the first-ever 6.006 problem session on fridays to complement foundational material, enhancing students' approach to problem sets, fostering a safe interactive learning environment.", 'chapters': [{'end': 136.677, 'start': 13.001, 'title': '6.006 problem session introduction', 'summary': "Introduces the first-ever 6.006 problem session on fridays aimed at providing recorded problem-solving methods to complement the foundational material presented in lectures, enhancing students' approach to problem sets and fostering a safe interactive learning environment.", 'duration': 123.676, 'highlights': ['The chapter introduces the first-ever 6.006 problem session on Fridays, aiming to provide recorded problem-solving methods to complement the foundational material presented in lectures.', 'The problem sets are applications of the fundamental material, and there is usually a different feel between the problems and the underlying foundational material.', "The goal of these problem sessions is to show how to approach working on problems similar to those in the problem sets, aiming to enhance students' problem-solving skills.", 'The sessions aim to provide a safe space for students to interact with the material in a smaller environment, fostering interaction and one-on-one questions.']}], 'duration': 123.676, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k13001.jpg', 'highlights': ['The chapter introduces the first-ever 6.006 problem session on Fridays, aiming to provide recorded problem-solving methods to complement the foundational material presented in lectures.', "The goal of these problem sessions is to show how to approach working on problems similar to those in the problem sets, aiming to enhance students' problem-solving skills.", 'The problem sets are applications of the fundamental material, and there is usually a different feel between the problems and the underlying foundational material.', 'The sessions aim to provide a safe space for students to interact with the material in a smaller environment, fostering interaction and one-on-one questions.']}, {'end': 1388.261, 'segs': [{'end': 161.854, 'src': 'embed', 'start': 137.798, 'weight': 0, 'content': [{'end': 149.41, 'text': "Any questions about what we're going to be doing today? OK, so we do have a handout up here by the door, which we may or may not have in the future.", 'start': 137.798, 'duration': 11.612}, {'end': 153.572, 'text': "This is all an experiment, so you'll have to work with us as we figure this stuff out.", 'start': 149.45, 'duration': 4.122}, {'end': 158.993, 'text': 'They were posted on LMOD about an hour before this session.', 'start': 154.272, 'duration': 4.721}, {'end': 161.854, 'text': "We'll try to keep that as a standard.", 'start': 159.013, 'duration': 2.841}], 'summary': 'Experimental handout may be available in future sessions, posted an hour before on lmod.', 'duration': 24.056, 'max_score': 137.798, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k137798.jpg'}, {'end': 242.516, 'src': 'embed', 'start': 212.904, 'weight': 1, 'content': [{'end': 217.005, 'text': "I've omitted a b and a c that was on last term's problem set.", 'start': 212.904, 'duration': 4.101}, {'end': 219.506, 'text': 'And it has five functions each.', 'start': 217.485, 'duration': 2.021}, {'end': 222.027, 'text': "And you're trying to order them.", 'start': 220.326, 'duration': 1.701}, {'end': 226.094, 'text': 'increasing based on their asymptotic behavior.', 'start': 223.248, 'duration': 2.846}, {'end': 230.243, 'text': 'So here are the functions that we have.', 'start': 226.375, 'duration': 3.868}, {'end': 232.568, 'text': "Maybe I'll stick it up instead.", 'start': 231.025, 'duration': 1.543}, {'end': 242.516, 'text': 'All right, so we have a few sets of functions, and we just want to order them.', 'start': 238.353, 'duration': 4.163}], 'summary': 'Sorting 5 functions based on asymptotic behavior.', 'duration': 29.612, 'max_score': 212.904, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k212904.jpg'}, {'end': 369.428, 'src': 'heatmap', 'start': 254.667, 'weight': 0.746, 'content': [{'end': 264.494, 'text': 'So what we have as an example are three functions n, root n and n plus root n.', 'start': 254.667, 'duration': 9.827}, {'end': 266.575, 'text': 'This is 1, 2, 3.', 'start': 264.494, 'duration': 2.081}, {'end': 272.679, 'text': "And what we're going to ask you to do is order those functions based on their asymptotic complexity.", 'start': 266.575, 'duration': 6.104}, {'end': 274.86, 'text': 'So hopefully you guys can get this one.', 'start': 272.799, 'duration': 2.061}, {'end': 280.463, 'text': "Which one's the slowest growth in term? Root n.", 'start': 275.46, 'duration': 5.003}, {'end': 281.504, 'text': 'So number 2.', 'start': 280.463, 'duration': 1.041}, {'end': 285.246, 'text': 'So if we say f of 2 will be our first one.', 'start': 281.504, 'duration': 3.742}, {'end': 289.593, 'text': "And then how about the other two? They're the same.", 'start': 285.386, 'duration': 4.207}, {'end': 292.075, 'text': "They're both order n.", 'start': 289.653, 'duration': 2.422}, {'end': 298.678, 'text': 'So we would put in set brackets F1 and F3 on your problem set.', 'start': 292.075, 'duration': 6.603}, {'end': 302.52, 'text': 'If you put just 2 and 1 and 3 here, that would probably be fine.', 'start': 298.778, 'duration': 3.742}, {'end': 313.746, 'text': "But if you were to put 2 over here or not have these curly braces around here, those would not be correct, and you'd get points marked off.", 'start': 304.181, 'duration': 9.565}, {'end': 324.133, 'text': "Does that make sense? We're going to approach the first set of functions, which is a little different than the second set of functions.", 'start': 314.146, 'duration': 9.987}, {'end': 325.554, 'text': "Hopefully, this one's a little easier.", 'start': 324.173, 'duration': 1.381}, {'end': 335.601, 'text': 'One of the common approaches that I have in going through these things, some of these are in a form that is hard for me to tell.', 'start': 328.196, 'duration': 7.405}, {'end': 338.687, 'text': 'how they would compare to other things.', 'start': 337.147, 'duration': 1.54}, {'end': 341.508, 'text': 'Actually, most of these are fine.', 'start': 339.568, 'duration': 1.94}, {'end': 354.071, 'text': 'But in general, can anyone just by eyeballing tell me an order that works for them? Yeah? OK.', 'start': 341.648, 'duration': 12.423}, {'end': 358.411, 'text': 'JASON KU- This is a little difficult to do on the spot with five functions.', 'start': 354.511, 'duration': 3.9}, {'end': 359.812, 'text': 'Yeah, a little idiot on f1.', 'start': 358.692, 'duration': 1.12}, {'end': 360.832, 'text': 'JASON KU- OK.', 'start': 360.192, 'duration': 0.64}, {'end': 369.428, 'text': 'OK OK, great.', 'start': 366.465, 'duration': 2.963}], 'summary': 'Order functions based on asymptotic complexity: root n, n, n plus root n.', 'duration': 114.761, 'max_score': 254.667, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k254667.jpg'}, {'end': 302.52, 'src': 'embed', 'start': 275.46, 'weight': 2, 'content': [{'end': 280.463, 'text': "Which one's the slowest growth in term? Root n.", 'start': 275.46, 'duration': 5.003}, {'end': 281.504, 'text': 'So number 2.', 'start': 280.463, 'duration': 1.041}, {'end': 285.246, 'text': 'So if we say f of 2 will be our first one.', 'start': 281.504, 'duration': 3.742}, {'end': 289.593, 'text': "And then how about the other two? They're the same.", 'start': 285.386, 'duration': 4.207}, {'end': 292.075, 'text': "They're both order n.", 'start': 289.653, 'duration': 2.422}, {'end': 298.678, 'text': 'So we would put in set brackets F1 and F3 on your problem set.', 'start': 292.075, 'duration': 6.603}, {'end': 302.52, 'text': 'If you put just 2 and 1 and 3 here, that would probably be fine.', 'start': 298.778, 'duration': 3.742}], 'summary': 'The slowest growth term is root n, with f(2) as the first one, and the other two being the same (order n).', 'duration': 27.06, 'max_score': 275.46, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k275460.jpg'}, {'end': 474.025, 'src': 'heatmap', 'start': 415.798, 'weight': 0.867, 'content': [{'end': 425.007, 'text': 'I guess that A is less than asymptotically.', 'start': 415.798, 'duration': 9.209}, {'end': 434.955, 'text': 'this polynomial, this log N to any power, is asymptotically less than any polynomial for any positive A and B.', 'start': 425.007, 'duration': 9.948}, {'end': 439.719, 'text': "And in particular, there's actually a stronger thing you can say, which is Little o.", 'start': 434.955, 'duration': 4.764}, {'end': 443.881, 'text': 'Did you guys talk about little o in recitation at all? Probably not.', 'start': 439.719, 'duration': 4.162}, {'end': 449.404, 'text': "It's kind of the same as big O, except it's big O minus theta.", 'start': 444.481, 'duration': 4.923}, {'end': 455.346, 'text': 'So things that are asymptotically equivalent are not going to be included in this set.', 'start': 449.924, 'duration': 5.422}, {'end': 461.469, 'text': 'So actually, these things grow strictly asymptotically slower than any polynomial.', 'start': 455.386, 'duration': 6.083}, {'end': 462.149, 'text': 'Does that make sense?', 'start': 461.509, 'duration': 0.64}, {'end': 470.383, 'text': 'So, knowing this identity or this relation, Can we say anything about f1?', 'start': 463.23, 'duration': 7.153}, {'end': 474.025, 'text': 'Someone, maybe someone else?', 'start': 470.403, 'duration': 3.622}], 'summary': 'Polynomial with log n to any power is asymptotically less than any polynomial for any positive values of a and b. little o is also discussed.', 'duration': 58.227, 'max_score': 415.798, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k415798.jpg'}, {'end': 649.647, 'src': 'embed', 'start': 623.233, 'weight': 4, 'content': [{'end': 633.263, 'text': 'This thing is what? Does anyone remember? What does n choose k mean? It means that, yeah? The number of ways to choose k objects from n.', 'start': 623.233, 'duration': 10.03}, {'end': 637.564, 'text': 'Yeah, the number of ways to choose k objects from n things.', 'start': 634.062, 'duration': 3.502}, {'end': 639.665, 'text': 'I never remember this formula.', 'start': 637.764, 'duration': 1.901}, {'end': 644.307, 'text': 'And probably a lot of you have memorized this formula.', 'start': 642.046, 'duration': 2.261}, {'end': 646.986, 'text': "I'm not going to ask you to do it.", 'start': 645.886, 'duration': 1.1}, {'end': 649.647, 'text': "I'm going to tell you how I kind of think about this.", 'start': 647.006, 'duration': 2.641}], 'summary': 'N choose k represents the number of ways to choose k objects from n.', 'duration': 26.414, 'max_score': 623.233, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k623233.jpg'}, {'end': 708.516, 'src': 'embed', 'start': 677.411, 'weight': 5, 'content': [{'end': 686.378, 'text': "But then essentially in here and in here, k and n minus k, I don't really care what their order is.", 'start': 677.411, 'duration': 8.967}, {'end': 691.542, 'text': "So I'm going to divide out the permutations of this stuff and this stuff.", 'start': 686.999, 'duration': 4.543}, {'end': 702.231, 'text': 'Does that make sense? So the formula here, as I remember it, that hopefully is correct, is n minus k factorial.', 'start': 692.363, 'duration': 9.868}, {'end': 708.516, 'text': "So I'm getting all of the permutations of the whole thing divided by their constituents.", 'start': 702.351, 'duration': 6.165}], 'summary': 'The formula for permutations is n minus k factorial.', 'duration': 31.105, 'max_score': 677.411, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k677411.jpg'}, {'end': 990.206, 'src': 'heatmap', 'start': 873.183, 'weight': 7, 'content': [{'end': 877.506, 'text': "So when we're inside a logarithm, multiplication, we can split it out.", 'start': 873.183, 'duration': 4.323}, {'end': 878.907, 'text': 'It becomes addition.', 'start': 877.526, 'duration': 1.381}, {'end': 883.05, 'text': 'Division becomes subtraction.', 'start': 879.447, 'duration': 3.603}, {'end': 890.935, 'text': 'And this thing grows faster than all these other things, so we can ignore them when we add them out asymptotically.', 'start': 884.271, 'duration': 6.664}, {'end': 893.937, 'text': 'And so what we end up getting is this n to the n.', 'start': 891.315, 'duration': 2.622}, {'end': 897.8, 'text': "The n comes out on the logarithm, and you get something that's theta.", 'start': 893.937, 'duration': 3.863}, {'end': 902.425, 'text': "Oh, that's fun.", 'start': 899.301, 'duration': 3.124}, {'end': 905.269, 'text': 'This is something we might use later on in the class.', 'start': 902.445, 'duration': 2.824}, {'end': 917.104, 'text': "But when we are comparing these functions, one of the nice things to do is convert them into something that's familiar to us,", 'start': 908.834, 'duration': 8.27}, {'end': 919.025, 'text': 'so that we can compare them easily.', 'start': 917.104, 'duration': 1.921}, {'end': 929.409, 'text': 'So here, this thing is whatever that thing is, roughly square root n, n over e to the n.', 'start': 919.465, 'duration': 9.944}, {'end': 936.131, 'text': "This is, I'm going to say, theta.", 'start': 929.409, 'duration': 6.722}, {'end': 937.432, 'text': "That's a little bit more precise.", 'start': 936.231, 'duration': 1.201}, {'end': 942.283, 'text': "All right, then what about these two things? Let's start with the bottom one.", 'start': 938.882, 'duration': 3.401}, {'end': 945.624, 'text': 'Can someone tell me what this is asymptotically? Yeah.', 'start': 942.623, 'duration': 3.001}, {'end': 947.325, 'text': 'N cubed.', 'start': 946.905, 'duration': 0.42}, {'end': 947.625, 'text': 'N cubed.', 'start': 947.345, 'duration': 0.28}, {'end': 958.589, 'text': 'Why is that? Well, if we plug this stuff into that definition here, we have n factorial over 3 factorial n minus 3 factorial.', 'start': 947.645, 'duration': 10.944}, {'end': 970.356, 'text': 'n factorial over n minus 3 factorial just leaves us with an n, an n minus 1 and an n minus 2 over 6.', 'start': 961.252, 'duration': 9.104}, {'end': 974.738, 'text': 'And if you multiply all that out, the leading term is an n cubed.', 'start': 970.356, 'duration': 4.382}, {'end': 980.201, 'text': 'So this thing is asymptotically n cubed.', 'start': 975.959, 'duration': 4.242}, {'end': 983.663, 'text': 'I kind of skipped some steps, but hopefully you can follow that.', 'start': 980.461, 'duration': 3.202}, {'end': 990.206, 'text': 'And then the last thing to remain is this one right there.', 'start': 985.844, 'duration': 4.362}], 'summary': 'Logarithmic operations, asymptotic comparison, and computation of n cubed.', 'duration': 29.242, 'max_score': 873.183, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k873183.jpg'}, {'end': 1069.725, 'src': 'embed', 'start': 997.782, 'weight': 6, 'content': [{'end': 1006.67, 'text': "What we can do is we can stick it into this formula and then apply Stirling's approximation to replace the factorials.", 'start': 997.782, 'duration': 8.888}, {'end': 1012.095, 'text': "That make sense? OK, so what I'm going to do is let's do this in two steps.", 'start': 1007.071, 'duration': 5.024}, {'end': 1015.479, 'text': 'This is going to be n factorial over.', 'start': 1012.936, 'duration': 2.543}, {'end': 1019.491, 'text': 'What is this? n over 2 factorial.', 'start': 1017.81, 'duration': 1.681}, {'end': 1023.514, 'text': "And then what is n minus n over 2? That's also n over 2.", 'start': 1019.531, 'duration': 3.983}, {'end': 1030.157, 'text': 'So this is going to be n over 2 factorial squared.', 'start': 1023.514, 'duration': 6.643}, {'end': 1038.983, 'text': "Is that OK? Yeah? Now let's replace this stuff with Sterling's approximation and see if we can simplify.", 'start': 1030.798, 'duration': 8.185}, {'end': 1042.086, 'text': 'So on the top, we have 2 pi n.', 'start': 1039.304, 'duration': 2.782}, {'end': 1050.216, 'text': "n over e to the n over, and then we've got a square here, pi n.", 'start': 1043.29, 'duration': 6.926}, {'end': 1052.658, 'text': "It's canceled the 2.", 'start': 1050.216, 'duration': 2.442}, {'end': 1060.544, 'text': 'n over 2 over e to the n over 2.', 'start': 1052.658, 'duration': 7.886}, {'end': 1066.202, 'text': "Did I do that right? OK, I can't spell, and a lot of times I make arithmetic errors.", 'start': 1060.544, 'duration': 5.658}, {'end': 1069.725, 'text': 'So catch me if I am doing one.', 'start': 1066.222, 'duration': 3.503}], 'summary': "Applying stirling's approximation to simplify factorial calculations and arithmetic operations in a formula.", 'duration': 71.943, 'max_score': 997.782, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k997782.jpg'}, {'end': 1286.192, 'src': 'embed', 'start': 1252.557, 'weight': 3, 'content': [{'end': 1259.061, 'text': "So it's going to be slower asymptotically than the first one right there, right?", 'start': 1252.557, 'duration': 6.504}, {'end': 1260.242, 'text': 'So you got it right.', 'start': 1259.141, 'duration': 1.101}, {'end': 1262.544, 'text': 'What is it? F3, F1, and F4.', 'start': 1261.303, 'duration': 1.241}, {'end': 1262.724, 'text': 'OK? Cool.', 'start': 1262.564, 'duration': 0.16}, {'end': 1277.29, 'text': "So it's a little complicated, but just applying some logarithm and exponent rules,", 'start': 1270.529, 'duration': 6.761}, {'end': 1286.192, 'text': 'understanding that logarithmic factors grow slower than polynomial ones and again grow slower than exponential ones,', 'start': 1277.29, 'duration': 8.902}], 'summary': 'Comparing functions f3, f1, and f4, noting logarithmic growth is slower than polynomial and exponential.', 'duration': 33.635, 'max_score': 1252.557, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k1252557.jpg'}, {'end': 1388.261, 'src': 'embed', 'start': 1356.073, 'weight': 10, 'content': [{'end': 1366.544, 'text': 'So this is, if I erase the 2 and the pi, This thing is theta of that, 2 to the n over a polynomial factor.', 'start': 1356.073, 'duration': 10.471}, {'end': 1370.968, 'text': "It's over n to the 1 half.", 'start': 1367.285, 'duration': 3.683}, {'end': 1375.992, 'text': 'n to the 1 half grows non-trivially.', 'start': 1371.468, 'duration': 4.524}, {'end': 1381.436, 'text': 'And so this is going to decrease the running time of this thing by a polynomial factor.', 'start': 1376.432, 'duration': 5.004}, {'end': 1386.68, 'text': 'You could think about where multiplying this by n to the minus 1 half as well.', 'start': 1381.636, 'duration': 5.044}, {'end': 1388.261, 'text': "That's another way of thinking about it.", 'start': 1387.12, 'duration': 1.141}], 'summary': 'Theta of 2^n over a polynomial factor decreases running time non-trivially.', 'duration': 32.188, 'max_score': 1356.073, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k1356073.jpg'}], 'start': 137.798, 'title': "Analyzing functions' asymptotic behavior", 'summary': "Covers ordering functions by asymptotic complexity, comparing functions based on growth rates, understanding binomial coefficients, and analyzing functions' asymptotic behavior, emphasizing interactive learning and experimental approach to materials dissemination.", 'chapters': [{'end': 302.52, 'start': 137.798, 'title': 'Problem set order by asymptotic complexity', 'summary': 'Covers the process of ordering functions based on their asymptotic behavior, with a focus on identifying the slowest growth term, and the use of set brackets for asymptotically equivalent functions, while emphasizing the interactive nature of the session and the experimental approach to materials dissemination.', 'duration': 164.722, 'highlights': ['The session presents a selection of problems from previous terms, with some problems edited to be shorter, and emphasizes an experimental approach to materials dissemination.', 'The chapter focuses on ordering functions based on their asymptotic behavior, highlighting the identification of the slowest growth term and the use of set brackets for asymptotically equivalent functions.', 'The instructor encourages interactive participation, allowing questions at any point during the session.']}, {'end': 597.014, 'start': 304.181, 'title': 'Comparing functions', 'summary': 'Discusses the comparison of functions based on their growth rates, highlighting the relationship between logarithmic and polynomial functions, and presenting a method for proving the asymptotic comparison of functions.', 'duration': 292.833, 'highlights': ['The relationship between logarithmic and polynomial functions is discussed, demonstrating that logarithmic growth is slower than linear and polynomial growth. Logarithmic functions are shown to grow slower than linear and polynomial functions, with the specific example of comparing functions F1, F2, F3, and F5.', 'A method for proving the asymptotic comparison of functions is presented, involving taking the limit of the logarithm of the ratio of the two functions. The method for proving the asymptotic comparison of functions is explained, involving taking the limit of the logarithm of the ratio of the two functions and analyzing the resulting limit to determine the growth relationship between the functions.', 'The order of growth for the functions F1, F5, F2, F3, and F4 is established, with F1 being the slowest growing and F4 being the fastest growing. The order of growth for the functions F1, F5, F2, F3, and F4 is determined, with F1 identified as the slowest growing function and F4 as the fastest growing function.']}, {'end': 828.814, 'start': 597.014, 'title': "Understanding binomial coefficients and sterling's approximation", 'summary': "Discusses the concept of binomial coefficients, explaining its formula as the number of ways to choose k objects from n, and introduces sterling's approximation for n factorial as approximately square root of 2 pi n n over e to the n, demonstrating its super exponential growth.", 'duration': 231.8, 'highlights': ['Binomial coefficient is defined as the number of ways to choose k objects from n. The binomial coefficient is explained as the number of ways to choose k objects from n, providing a clear definition of its significance.', 'Introduction of formula n minus k factorial for permutations of n choose k. The derivation of the formula n minus k factorial for permutations of n choose k is explained methodically, providing a useful tool for calculating permutations.', "Explanation of Sterling's approximation for n factorial as square root of 2 pi n n over e to the n. The chapter introduces Sterling's approximation for n factorial as approximately square root of 2 pi n n over e to the n, highlighting its significance as a tool for understanding the exponential growth of factorials."]}, {'end': 1156.433, 'start': 829.454, 'title': 'Asymptotic analysis of functions', 'summary': "Discusses the asymptotic behavior of various functions, including n log n, n to the n, n cubed, and 2 to the n over root n, and their comparison using logarithmic transformations and stirling's approximation.", 'duration': 326.979, 'highlights': ['The logarithmic transformation of n to the n results in n log n, allowing for easier comparison of functions.', 'The function n cubed is derived from the asymptotic analysis of a given expression, n factorial, using the definition of theta.', "Stirling's approximation is applied to simplify the expression 2 to the n over root n, resulting in an asymptotic behavior of 2 to the n over root n."]}, {'end': 1388.261, 'start': 1157.014, 'title': 'Ordering of functions in asymptotic equivalence', 'summary': 'Discusses the ordering of functions in asymptotic equivalence, highlighting the comparison between different functions and their growth rates, while emphasizing the application of logarithm and exponent rules in approaching these problems.', 'duration': 231.247, 'highlights': ['The functions F2 and F5 are asymptotically equivalent and should be put in brackets, followed by F4, which is the largest function due to its n to the n growth rate. F2 and F5 are asymptotically equivalent, F4 has the largest growth rate of n to the n.', 'Understanding that logarithmic factors grow slower than polynomial ones and again grow slower than exponential ones, and being able to do some transformations of some of these mathematical quantities to get them in a polynomial-like looking form, is essential in approaching these problems. Emphasizing the understanding of logarithmic, polynomial, and exponential growth rates and the application of transformations of mathematical quantities.', 'Explaining the decrease in running time of a function by a polynomial factor due to the presence of non-trivial growth factors such as n to the 1 half. Illustrating the impact of non-trivial growth factors like n to the 1 half on the running time of a function.']}], 'duration': 1250.463, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k137798.jpg', 'highlights': ['The instructor encourages interactive participation, allowing questions at any point during the session.', 'The chapter focuses on ordering functions based on their asymptotic behavior, highlighting the identification of the slowest growth term and the use of set brackets for asymptotically equivalent functions.', 'The order of growth for the functions F1, F5, F2, F3, and F4 is established, with F1 being the slowest growing and F4 being the fastest growing.', 'The relationship between logarithmic and polynomial functions is discussed, demonstrating that logarithmic growth is slower than linear and polynomial growth.', 'Binomial coefficient is defined as the number of ways to choose k objects from n.', 'Introduction of formula n minus k factorial for permutations of n choose k.', "Explanation of Sterling's approximation for n factorial as square root of 2 pi n n over e to the n.", 'The logarithmic transformation of n to the n results in n log n, allowing for easier comparison of functions.', "Stirling's approximation is applied to simplify the expression 2 to the n over root n, resulting in an asymptotic behavior of 2 to the n over root n.", 'Understanding that logarithmic factors grow slower than polynomial ones and again grow slower than exponential ones and being able to do some transformations of some of these mathematical quantities to get them in a polynomial-like looking form is essential in approaching these problems.', 'Explaining the decrease in running time of a function by a polynomial factor due to the presence of non-trivial growth factors such as n to the 1 half.']}, {'end': 2143.321, 'segs': [{'end': 1486.394, 'src': 'embed', 'start': 1435.342, 'weight': 0, 'content': [{'end': 1437.804, 'text': "but I'm not allowed to see what's inside of it.", 'start': 1435.342, 'duration': 2.462}, {'end': 1439.345, 'text': 'And a lot of times.', 'start': 1438.464, 'duration': 0.881}, {'end': 1450.714, 'text': "what we'll do in this class is try to use a black box and just try to use the abstracted outer functions so that we can prove things about it.", 'start': 1439.345, 'duration': 11.369}, {'end': 1456.959, 'text': 'We can just accept those as true and then use those to deal with our analysis.', 'start': 1450.774, 'duration': 6.185}, {'end': 1466.827, 'text': "So what we're given in this problem is a data structure supporting a sequence interface that you heard about yesterday.", 'start': 1456.999, 'duration': 9.828}, {'end': 1475.714, 'text': "What's a sequence interface again? What is a sequence interface? How does it store items? Anyone remember? Yeah.", 'start': 1467.247, 'duration': 8.467}, {'end': 1479.047, 'text': 'In, well, OK.', 'start': 1477.225, 'duration': 1.822}, {'end': 1486.394, 'text': 'All right, so what your colleague is saying here is we list them in a contiguous array.', 'start': 1480.749, 'duration': 5.645}], 'summary': 'Using a black box to analyze data structure supporting sequence interface.', 'duration': 51.052, 'max_score': 1435.342, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k1435342.jpg'}, {'end': 1686.86, 'src': 'embed', 'start': 1632.741, 'weight': 2, 'content': [{'end': 1645.315, 'text': "So the idea is going to be let's implement algorithms for these some higher level operations in terms of these lower level things that are given to us.", 'start': 1632.741, 'duration': 12.574}, {'end': 1649.558, 'text': 'Does that make sense? OK, and this is actually a pretty easy question.', 'start': 1645.355, 'duration': 4.203}, {'end': 1657.262, 'text': "Hopefully, we'll have slightly more difficult ones for you in a different context on problem set 1.", 'start': 1649.758, 'duration': 7.504}, {'end': 1667.879, 'text': "OK, so the first operation we're going to support or try to support, is an operation called swap ends.", 'start': 1657.262, 'duration': 10.617}, {'end': 1672.041, 'text': 'And what this is going to do is take the data structure that we gave.', 'start': 1669.2, 'duration': 2.841}, {'end': 1676.183, 'text': 'Another way you could do this is put this as a method on that data structure.', 'start': 1672.421, 'duration': 3.762}, {'end': 1677.724, 'text': "But let's do this separately.", 'start': 1676.223, 'duration': 1.501}, {'end': 1682.567, 'text': "It's going to take that data structure that I gave you that's storing the sequence.", 'start': 1677.944, 'duration': 4.623}, {'end': 1686.86, 'text': 'as the only argument.', 'start': 1685.939, 'duration': 0.921}], 'summary': 'Implement algorithms for higher level operations using given lower level things. first operation: swap ends.', 'duration': 54.119, 'max_score': 1632.741, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k1632741.jpg'}, {'end': 2143.321, 'src': 'embed', 'start': 2117.62, 'weight': 5, 'content': [{'end': 2125.246, 'text': "If we can't understand what your variables mean, if we can't understand what your pseudocode is doing, then that's not sufficient.", 'start': 2117.62, 'duration': 7.626}, {'end': 2132.072, 'text': 'So the reason why we ask for words is so that you can communicate those ideas well.', 'start': 2126.307, 'duration': 5.765}, {'end': 2134.554, 'text': 'Just a follow-up on that.', 'start': 2132.552, 'duration': 2.002}, {'end': 2138.477, 'text': 'So can you also have a combination of pseudocode and description? JASON KUO- Sure, yeah.', 'start': 2134.634, 'duration': 3.843}, {'end': 2143.321, 'text': 'Including both of them can be clarifying for you, potentially.', 'start': 2139.278, 'duration': 4.043}], 'summary': 'Clear communication of variables and pseudocode is crucial for understanding; combining pseudocode and description can be clarifying.', 'duration': 25.701, 'max_score': 2117.62, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k2117620.jpg'}], 'start': 1389.662, 'title': 'Algorithms and abstraction', 'summary': 'Covers thinking in black box for problem-solving, emphasizing abstracted outer functions, understanding sequence interface, including swap ends operation, and discussing algorithm description and pseudocode for first and last item swapping and problem-solving scenarios.', 'chapters': [{'end': 1456.959, 'start': 1389.662, 'title': 'Problem 2: thinking in black box', 'summary': 'Discusses the concept of using something as a black box in the context of a problem, emphasizing the use of abstracted outer functions to prove and analyze, facilitating understanding and application in the class.', 'duration': 67.297, 'highlights': ['The chapter stresses the concept of using something as a black box in the context of a problem, highlighting the emphasis on using abstracted outer functions for analysis and proof.', 'It mentions the importance of using abstracted outer functions to prove things about the black box, facilitating understanding and application in the class.']}, {'end': 1838.773, 'start': 1456.999, 'title': 'Understanding sequence interface', 'summary': 'Discusses the concept of a sequence interface and its abstraction from the underlying data structures, including the implementation of swap ends operation and the importance of space complexity in algorithms.', 'duration': 381.774, 'highlights': ['Understanding the concept of a sequence interface and its abstraction from underlying data structures, including arrays and linked lists.', 'Implementing algorithms for higher-level operations in terms of lower-level operations available in the given data structure.', "Discussion on implementing the 'swap ends' operation to swap the first and last items of the sequence without direct access to the underlying representation.", 'Clarification on the return value of delete operations and the consideration of space complexity in algorithms.']}, {'end': 2143.321, 'start': 1839.113, 'title': 'Algorithm description and pseudocode', 'summary': 'Discusses the process of explaining algorithms using words and pseudocode, particularly focusing on the process of swapping the first and last items in a sequence, including the use of temporary variables and insertion functions. additionally, it addresses the use of words and pseudocode in problem-solving and exam scenarios.', 'duration': 304.208, 'highlights': ['The chapter highlights the process of swapping the first and last items in a sequence, involving the use of temporary variables and insertion functions, providing a clear approach for algorithm explanation and implementation.', 'It emphasizes the importance of effectively communicating ideas through words and pseudocode, particularly in problem-solving and exam scenarios, stressing the need for clarity in variable meanings and algorithm understanding.', 'The chapter also addresses the combination of pseudocode and description as a potentially clarifying approach for communicating algorithmic processes.']}], 'duration': 753.659, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k1389662.jpg', 'highlights': ['The chapter emphasizes using abstracted outer functions for analysis and proof.', 'Understanding the sequence interface and its abstraction from underlying data structures.', 'Implementing algorithms for higher-level operations in terms of lower-level operations.', "Discussion on implementing the 'swap ends' operation to swap the first and last items of the sequence.", 'The chapter provides a clear approach for algorithm explanation and implementation.', 'Emphasizing the importance of effectively communicating ideas through words and pseudocode.', 'Addressing the combination of pseudocode and description as a clarifying approach for communicating algorithmic processes.']}, {'end': 3114.376, 'segs': [{'end': 2195.508, 'src': 'embed', 'start': 2170.044, 'weight': 2, 'content': [{'end': 2175.849, 'text': "I'm not even going to have to argue correctness to you because we're essentially just doing exactly what we asked for.", 'start': 2170.044, 'duration': 5.805}, {'end': 2179.112, 'text': 'So most of the time in this class.', 'start': 2176.37, 'duration': 2.742}, {'end': 2187.28, 'text': "when you're doing something non-trivial, especially when you're doing something that has to recurse in some way, we do want you to argue correctness.", 'start': 2179.112, 'duration': 8.168}, {'end': 2193.966, 'text': 'But in this case, For example, the time analysis is very easy, right? We do four operations.', 'start': 2187.76, 'duration': 6.206}, {'end': 2195.508, 'text': 'They each take constant time.', 'start': 2194.347, 'duration': 1.161}], 'summary': 'Classwork involves four operations, each taking constant time.', 'duration': 25.464, 'max_score': 2170.044, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k2170044.jpg'}, {'end': 2265.543, 'src': 'embed', 'start': 2237.342, 'weight': 4, 'content': [{'end': 2240.064, 'text': 'So the k-th item ends up being the last item.', 'start': 2237.342, 'duration': 2.722}, {'end': 2243.367, 'text': 'And the k plus 1-th item now becomes the first item.', 'start': 2240.644, 'duration': 2.723}, {'end': 2244.928, 'text': 'Does that make sense??', 'start': 2243.387, 'duration': 1.541}, {'end': 2251.136, 'text': 'OK again, this is actually not such an interesting algorithm from an algorithm standpoint,', 'start': 2245.993, 'duration': 5.143}, {'end': 2257.519, 'text': "but it's hopefully helpful to talk about from an instructional point of view.", 'start': 2251.136, 'duration': 6.383}, {'end': 2265.543, 'text': 'So how would I approach this problem? I need this operation to happen in order k times.', 'start': 2257.799, 'duration': 7.744}], 'summary': 'An algorithmic process where items are shifted k times.', 'duration': 28.201, 'max_score': 2237.342, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k2237342.jpg'}, {'end': 2345.306, 'src': 'embed', 'start': 2321.41, 'weight': 1, 'content': [{'end': 2331.081, 'text': 'Why is it that a lot of computer scientists as opposed to coding engineers, prefer to think about an algorithm recursively.', 'start': 2321.41, 'duration': 9.671}, {'end': 2336.543, 'text': "Does anyone know why? At least I do when I'm explaining it from a theory standpoint.", 'start': 2331.181, 'duration': 5.362}, {'end': 2343.485, 'text': 'It actually might not be good from an implementation standpoint because your computer can vectorize for loops and things like that.', 'start': 2337.403, 'duration': 6.082}, {'end': 2345.306, 'text': "But that's not something we need to talk about.", 'start': 2343.545, 'duration': 1.761}], 'summary': 'Computer scientists prefer recursive algorithms over coding engineers for theoretical understanding, although it may not be optimal for implementation.', 'duration': 23.896, 'max_score': 2321.41, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k2321410.jpg'}, {'end': 2742.779, 'src': 'embed', 'start': 2708.96, 'weight': 3, 'content': [{'end': 2716.287, 'text': "And how many times do I call a function? Yeah, I think k minus 1 times or I don't know.", 'start': 2708.96, 'duration': 7.327}, {'end': 2716.828, 'text': 'I forget.', 'start': 2716.367, 'duration': 0.461}, {'end': 2718.409, 'text': "But it's order k for sure.", 'start': 2716.868, 'duration': 1.541}, {'end': 2724.314, 'text': 'And we do a constant amount of work per call ignoring this extra call.', 'start': 2719.009, 'duration': 5.305}, {'end': 2731, 'text': 'Does that make sense? So this thing runs in order k as desired.', 'start': 2724.334, 'duration': 6.666}, {'end': 2742.779, 'text': 'Does that make sense? So now we will move on to question 3.', 'start': 2731.36, 'duration': 11.419}], 'summary': 'Function is called k-1 times, running in order k.', 'duration': 33.819, 'max_score': 2708.96, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k2708960.jpg'}, {'end': 2812.455, 'src': 'heatmap', 'start': 2753.951, 'weight': 1, 'content': [{'end': 2755.832, 'text': 'OK, so problem three.', 'start': 2753.951, 'duration': 1.881}, {'end': 2758.793, 'text': 'OK, so this is a little block of text right here.', 'start': 2755.852, 'duration': 2.941}, {'end': 2764.355, 'text': 'A dynamic array can support a sequence interface supporting worst case constant time indexing,', 'start': 2758.813, 'duration': 5.542}, {'end': 2769.537, 'text': 'as well as insertion and removal of items at the back of the array in amortized constant time.', 'start': 2764.355, 'duration': 5.182}, {'end': 2772.138, 'text': 'So this is what we did yesterday in lecture.', 'start': 2769.557, 'duration': 2.581}, {'end': 2777.8, 'text': "We showed how a dynamic array, it's fast to do dynamic operations at the end.", 'start': 2772.458, 'duration': 5.342}, {'end': 2786.385, 'text': "OK However, insertion and deletion at the front is not very efficient, because if you tried to do that, you'd have to shift everything over.", 'start': 2779.821, 'duration': 6.564}, {'end': 2788.767, 'text': 'Right? That make sense? All right.', 'start': 2786.685, 'duration': 2.082}, {'end': 2791.709, 'text': 'On the other hand, what we talked about yesterday was linked lists.', 'start': 2788.987, 'duration': 2.722}, {'end': 2799.014, 'text': 'They can be made to support insertion and deletion at both ends in constant time.', 'start': 2792.83, 'duration': 6.184}, {'end': 2803.029, 'text': "OK, so that's a little foreshadowing of something you're going to do on pset 1.", 'start': 2799.054, 'duration': 3.975}, {'end': 2812.455, 'text': 'OK, but in lecture we talked about that operation, that data structure, a singly linked list being good at dynamic operations,', 'start': 2803.029, 'duration': 9.426}], 'summary': 'Dynamic arrays offer fast dynamic operations at the end, while linked lists support constant time insertion and deletion at both ends.', 'duration': 58.504, 'max_score': 2753.951, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k2753951.jpg'}, {'end': 2791.709, 'src': 'embed', 'start': 2764.355, 'weight': 5, 'content': [{'end': 2769.537, 'text': 'as well as insertion and removal of items at the back of the array in amortized constant time.', 'start': 2764.355, 'duration': 5.182}, {'end': 2772.138, 'text': 'So this is what we did yesterday in lecture.', 'start': 2769.557, 'duration': 2.581}, {'end': 2777.8, 'text': "We showed how a dynamic array, it's fast to do dynamic operations at the end.", 'start': 2772.458, 'duration': 5.342}, {'end': 2786.385, 'text': "OK However, insertion and deletion at the front is not very efficient, because if you tried to do that, you'd have to shift everything over.", 'start': 2779.821, 'duration': 6.564}, {'end': 2788.767, 'text': 'Right? That make sense? All right.', 'start': 2786.685, 'duration': 2.082}, {'end': 2791.709, 'text': 'On the other hand, what we talked about yesterday was linked lists.', 'start': 2788.987, 'duration': 2.722}], 'summary': 'Dynamic arrays allow fast dynamic operations at the end, but inefficient insertion and deletion at the front. linked lists were discussed for alternative solutions.', 'duration': 27.354, 'max_score': 2764.355, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k2764355.jpg'}, {'end': 2951.059, 'src': 'embed', 'start': 2920.596, 'weight': 0, 'content': [{'end': 2924.098, 'text': 'Can I define amortize one more time? OK, so this is a tough.', 'start': 2920.596, 'duration': 3.502}, {'end': 2932.368, 'text': 'thing to define in general, but not that much.', 'start': 2925.984, 'duration': 6.384}, {'end': 2939.852, 'text': "So amortization, usually you put in the, at least in this class, we're going to put in terms of the data structure.", 'start': 2932.448, 'duration': 7.404}, {'end': 2941.373, 'text': 'So you have this thing.', 'start': 2940.533, 'duration': 0.84}, {'end': 2943.374, 'text': 'It supports some operations.', 'start': 2941.913, 'duration': 1.461}, {'end': 2946.076, 'text': "And you're going to do a bunch of operations on that thing.", 'start': 2943.634, 'duration': 2.442}, {'end': 2951.059, 'text': "There's not really a reason to have a data structure unless you're going to do lots of things to it.", 'start': 2946.876, 'duration': 4.183}], 'summary': 'Amortization in data structures involves performing multiple operations on a supported data structure.', 'duration': 30.463, 'max_score': 2920.596, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k2920596.jpg'}], 'start': 2144.602, 'title': 'Data structure operations, recursion, and dynamic arrays', 'summary': 'Covers time analysis of data structure operations, recursive algorithms, and a comparison of dynamic arrays and linked lists. it includes examples of constant time operations, the advantages of recursion, and the efficiency of dynamic arrays and linked lists.', 'chapters': [{'end': 2298.904, 'start': 2144.602, 'title': 'Data structure operations and time analysis', 'summary': 'Discusses constant size problem and time analysis of data structure operations, with an example of shifting elements in a sequence, each taking constant time, and a method that takes 2k steps.', 'duration': 154.302, 'highlights': ['The time analysis for the first operation involves four operations, each taking constant time, resulting in the entire operation taking constant time.', 'The second operation involves shifting the first k elements to the end of the sequence, and the k-th item becomes the last item, with a method that takes 2k steps to perform the operation k times.', 'The approach to the second operation involves setting a variable, deleting the first element, inserting it at the end, and repeating the process k times, taking 2k steps, which is considered acceptable.']}, {'end': 2742.779, 'start': 2298.924, 'title': 'Recursion in algorithms', 'summary': 'Discusses the preference for recursive algorithms over for loops by computer scientists, as it allows breaking down problems into smaller, manageable pieces, enabling arguments and case analysis on small amounts of information and running in order k time complexity.', 'duration': 443.855, 'highlights': ['Recursion allows breaking down problems into smaller, more manageable pieces, facilitating arguments and case analysis on small amounts of information.', 'Preference for recursive algorithms over for loops by computer scientists is due to the ability to argue and analyze with a constant amount of information at any given point, leading to easier convincing of correctness.', 'Recursive algorithms run in order k time complexity, with a constant amount of work done in each section and a call function running k minus 1 times.']}, {'end': 3114.376, 'start': 2742.779, 'title': 'Dynamic arrays vs. linked lists', 'summary': 'Discusses the efficiency of dynamic arrays and linked lists, emphasizing their strengths and weaknesses, and introduces the concept of amortized running time in the context of data structures.', 'duration': 371.597, 'highlights': ['The chapter explains the efficiency of dynamic arrays and linked lists, highlighting the constant time indexing, insertion, and removal of items at the back of a dynamic array, and the constant time insertion and deletion at both ends in linked lists. Dynamic arrays support worst-case constant time indexing, insertion, and removal at the back, while linked lists support constant time insertion and deletion at both ends.', 'The concept of amortized running time is introduced, illustrating its application in the context of data structures such as dynamic arrays to optimize the running time of operations over the long term. The concept of amortized running time is explained as a way to optimize the running time of operations over the long term, particularly in the context of dynamic arrays.', 'The instructor elaborates on the concept of amortization, emphasizing its relevance in the context of data structures and its distinction from average running time. The instructor provides a detailed explanation of amortization, highlighting its relevance in optimizing the running time of operations in data structures and distinguishing it from average running time.']}], 'duration': 969.774, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k2144602.jpg', 'highlights': ['The concept of amortized running time is introduced, illustrating its application in the context of data structures such as dynamic arrays to optimize the running time of operations over the long term.', 'Preference for recursive algorithms over for loops by computer scientists is due to the ability to argue and analyze with a constant amount of information at any given point, leading to easier convincing of correctness.', 'The time analysis for the first operation involves four operations, each taking constant time, resulting in the entire operation taking constant time.', 'Recursive algorithms run in order k time complexity, with a constant amount of work done in each section and a call function running k minus 1 times.', 'The second operation involves shifting the first k elements to the end of the sequence, and the k-th item becomes the last item, with a method that takes 2k steps to perform the operation k times.', 'The chapter explains the efficiency of dynamic arrays and linked lists, highlighting the constant time indexing, insertion, and removal of items at the back of a dynamic array, and the constant time insertion and deletion at both ends in linked lists.']}, {'end': 4081.607, 'segs': [{'end': 3290.749, 'src': 'embed', 'start': 3259.652, 'weight': 0, 'content': [{'end': 3274.06, 'text': 'The idea of the dynamic array was that we kind of left some extra space here at the end so that, sure, we allocated more than we needed to.', 'start': 3259.652, 'duration': 14.408}, {'end': 3278.042, 'text': "But when we insert things now, it's cheap.", 'start': 3274.82, 'duration': 3.222}, {'end': 3279.763, 'text': "And we don't have to.", 'start': 3278.762, 'duration': 1.001}, {'end': 3285.746, 'text': "allocate more space for this thing until we've done a linear number of insertions.", 'start': 3280.803, 'duration': 4.943}, {'end': 3287.507, 'text': 'This was n.', 'start': 3286.166, 'duration': 1.341}, {'end': 3288.468, 'text': 'This was n.', 'start': 3287.507, 'duration': 0.961}, {'end': 3290.749, 'text': 'Really, any constant factor will do here.', 'start': 3288.468, 'duration': 2.281}], 'summary': 'Dynamic array allows cheap insertions without frequent reallocations.', 'duration': 31.097, 'max_score': 3259.652, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k3259652.jpg'}, {'end': 3373.699, 'src': 'embed', 'start': 3337.449, 'weight': 2, 'content': [{'end': 3346.875, 'text': "Does that make sense? So that's the idea behind expanding this dynamic array to be kind of this dynamic deck.", 'start': 3337.449, 'duration': 9.426}, {'end': 3356.362, 'text': "So it's a doubly ended queue kind of system where I can do dynamic operations efficiently on both ends.", 'start': 3347.255, 'duration': 9.107}, {'end': 3364.35, 'text': 'So one of the things that we talked about yesterday was also removing right at the end.', 'start': 3357.223, 'duration': 7.127}, {'end': 3373.699, 'text': "Removing items from the back of this thing will decrease the number of items we're storing.", 'start': 3366.392, 'duration': 7.307}], 'summary': 'Expanding dynamic array to a doubly ended queue for efficient dynamic operations on both ends, including removing items from the back to decrease stored items.', 'duration': 36.25, 'max_score': 3337.449, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k3337449.jpg'}, {'end': 3452.363, 'src': 'embed', 'start': 3422.299, 'weight': 3, 'content': [{'end': 3432.763, 'text': 'And then, as I decreased, as I removed things from that item, I still have all that space there being taken up by essentially nothing,', 'start': 3422.299, 'duration': 10.464}, {'end': 3436.844, 'text': "because I've removed everything from it, at least in my conception.", 'start': 3432.763, 'duration': 4.081}, {'end': 3452.363, 'text': 'What I would really like to maintain with this data structure is that at no point in time am I using more than a linear amount of space with respect to the number of things that are stored in it.', 'start': 3440.154, 'duration': 12.209}], 'summary': 'Efficient data structure maintaining linear space usage.', 'duration': 30.064, 'max_score': 3422.299, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k3422299.jpg'}, {'end': 3623.041, 'src': 'embed', 'start': 3592.759, 'weight': 4, 'content': [{'end': 3598.604, 'text': 'always resize to a fill ratio that includes a linear amount of things on the end.', 'start': 3592.759, 'duration': 5.845}, {'end': 3608.173, 'text': "then I know that when I resize down I'll be doing either a linear number of deletions or a linear number of insertions before I have to rebuild again.", 'start': 3598.604, 'duration': 9.569}, {'end': 3615.22, 'text': 'So discharging argument again, I have to do a linear number of cheap things before I have to do an expensive thing again.', 'start': 3609.955, 'duration': 5.265}, {'end': 3623.041, 'text': 'OK, so I resize down to be still keep a linear amount of extra space at the end.', 'start': 3615.939, 'duration': 7.102}], 'summary': 'Resizing to maintain a linear amount of extra space, minimizing deletions or insertions.', 'duration': 30.282, 'max_score': 3592.759, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k3592759.jpg'}, {'end': 3690.772, 'src': 'heatmap', 'start': 3631.463, 'weight': 0.785, 'content': [{'end': 3641.045, 'text': 'as your colleague was saying, we can just resize down always to shift these things to be placed in the middle,', 'start': 3631.463, 'duration': 9.582}, {'end': 3645.326, 'text': 'with a linear amount of extra space on the ends.', 'start': 3643.342, 'duration': 1.984}, {'end': 3653.5, 'text': 'Does that make sense? No questions? All right.', 'start': 3645.686, 'duration': 7.814}, {'end': 3660.117, 'text': 'That was a way in which we kind of had to redefine an entirely new data structure.', 'start': 3654.656, 'duration': 5.461}, {'end': 3669.579, 'text': 'We took the ideas behind dynamic arrays, and we extended those ideas to make this thing have extra space on both ends.', 'start': 3661.398, 'duration': 8.181}, {'end': 3673.46, 'text': 'But we kind of had to do that reimplementation all by ourselves.', 'start': 3669.619, 'duration': 3.841}, {'end': 3677.502, 'text': 'If we were doing code, that would be kind of gnarly.', 'start': 3673.5, 'duration': 4.002}, {'end': 3681.065, 'text': 'But what if someone just gave us a dynamic array?', 'start': 3677.762, 'duration': 3.303}, {'end': 3686.208, 'text': 'What if someone gave you a Python list and you wanted this functionality?', 'start': 3681.525, 'duration': 4.683}, {'end': 3690.772, 'text': "I don't want to reimplement a dynamic array, but I want this behavior.", 'start': 3687.91, 'duration': 2.862}], 'summary': 'Redefining data structure with extra space on both ends using dynamic arrays and python lists.', 'duration': 59.309, 'max_score': 3631.463, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k3631463.jpg'}, {'end': 3681.065, 'src': 'embed', 'start': 3654.656, 'weight': 8, 'content': [{'end': 3660.117, 'text': 'That was a way in which we kind of had to redefine an entirely new data structure.', 'start': 3654.656, 'duration': 5.461}, {'end': 3669.579, 'text': 'We took the ideas behind dynamic arrays, and we extended those ideas to make this thing have extra space on both ends.', 'start': 3661.398, 'duration': 8.181}, {'end': 3673.46, 'text': 'But we kind of had to do that reimplementation all by ourselves.', 'start': 3669.619, 'duration': 3.841}, {'end': 3677.502, 'text': 'If we were doing code, that would be kind of gnarly.', 'start': 3673.5, 'duration': 4.002}, {'end': 3681.065, 'text': 'But what if someone just gave us a dynamic array?', 'start': 3677.762, 'duration': 3.303}], 'summary': 'Redefined data structure with extra space on both ends, based on dynamic arrays.', 'duration': 26.409, 'max_score': 3654.656, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k3654656.jpg'}, {'end': 3753.727, 'src': 'embed', 'start': 3714.496, 'weight': 6, 'content': [{'end': 3725.6, 'text': "How could I use some? Let's say I have a dynamic array that's good on one side.", 'start': 3714.496, 'duration': 11.104}, {'end': 3732.343, 'text': 'Is there anything I can do to support dynamic operations on both sides of a sequence?', 'start': 3725.62, 'duration': 6.723}, {'end': 3739.928, 'text': 'Yeah?. Are we able to just use like a second dynamic array and then fix it?', 'start': 3735.185, 'duration': 4.743}, {'end': 3746.693, 'text': "I like oh, that's supposed to be empty, right?", 'start': 3743.591, 'duration': 3.102}, {'end': 3752.626, 'text': 'Yeah, so what your colleague is saying?', 'start': 3750.765, 'duration': 1.861}, {'end': 3753.727, 'text': "yeah, let's do that.", 'start': 3752.626, 'duration': 1.101}], 'summary': 'Discussion about using dynamic arrays for operations on both sides of a sequence.', 'duration': 39.231, 'max_score': 3714.496, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k3714496.jpg'}, {'end': 4005.087, 'src': 'embed', 'start': 3976.428, 'weight': 1, 'content': [{'end': 3985.334, 'text': 'I now have an amortized cost buildup that I can spend to now rebuild the entire data structure.', 'start': 3976.428, 'duration': 8.906}, {'end': 3987.195, 'text': 'Does that make sense?', 'start': 3985.854, 'duration': 1.341}, {'end': 3994.82, 'text': 'I can now, once I get down to this thing, take whatever the remaining things are, split it in half, put it into two entirely new arrays,', 'start': 3987.475, 'duration': 7.345}, {'end': 3995.741, 'text': 'copy them all over.', 'start': 3994.82, 'duration': 0.921}, {'end': 4005.087, 'text': "And now I've restored my invariant where I'm, again, a linear amount of operations away from having to do an expensive operation again.", 'start': 3996.301, 'duration': 8.786}], 'summary': 'Amortized cost buildup allows for efficient data structure rebuilding.', 'duration': 28.659, 'max_score': 3976.428, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k3976428.jpg'}], 'start': 3115.276, 'title': 'Dynamic array operations', 'summary': 'Covers amortization, dynamic array resizing, and extension, emphasizing efficient operations, linear space allocation, optimal rebuilding time, and support for operations on both ends.', 'chapters': [{'end': 3373.699, 'start': 3115.276, 'title': 'Amortization and dynamic arrays', 'summary': 'Discusses the concept of amortization and dynamic arrays, explaining how allocating extra space at both ends of a dynamic array allows for efficient operations on both ends, ensuring that a linear number of insertions will only take linear time and making dynamic operations efficient on both ends.', 'duration': 258.423, 'highlights': ['Allocating extra space at both ends of a dynamic array allows for efficient operations on both ends, ensuring that a linear number of insertions will only take linear time.', 'The concept of amortization is applied to dynamic arrays, where the cost of rebuilding the array is charged to each operation, making it efficient on average.', 'Expanding a dynamic array to create a doubly ended queue system enables efficient dynamic operations on both ends.']}, {'end': 3653.5, 'start': 3374.36, 'title': 'Dynamic array resizing', 'summary': 'Discusses dynamic array resizing, emphasizing the importance of maintaining a linear amount of space with respect to the number of items stored, and the optimal time to rebuild the array based on a fill rate of n over 4.', 'duration': 279.14, 'highlights': ['The importance of maintaining a linear amount of space with respect to the number of items stored is emphasized to avoid excessive memory usage. For example, after removing all but two items from a data structure that initially held a large number of items, the space could still be occupied by essentially nothing, leading to wastage of memory. Emphasis on maintaining a linear amount of space, example of removing all but two items.', 'The optimal time to rebuild the array is discussed in relation to the fill rate, with a suggestion to rebuild the array after n over 2 removals, resulting in a fill rate of n over 4, and ensuring a linear amount of space is maintained. Discussion of optimal time to rebuild the array based on the fill rate.', 'The concept of maintaining extra space at the end of the array is explained to avoid frequent resizing, ensuring that a linear amount of extra space is retained to optimize performance. Explanation of maintaining extra space at the end of the array to avoid frequent resizing.']}, {'end': 4081.607, 'start': 3654.656, 'title': 'Dynamic array extension', 'summary': 'Discusses the extension of dynamic array ideas to support operations on both ends of a sequence, the challenges encountered, and the solution involving merging and rebuilding the data structure.', 'duration': 426.951, 'highlights': ['The chapter discusses the extension of dynamic array ideas to support operations on both ends of a sequence. The speaker explores the need to extend dynamic array functionality to accommodate operations on both ends of a sequence, presenting the challenge of reimplementation and the desire to avoid it by leveraging a dynamic array.', 'The solution involves merging and rebuilding the data structure to maintain the desired invariant. The chapter presents the solution of merging two dynamic arrays into one and rebuilding the entire data structure when reducing to one array, ensuring the maintenance of the invariant and an amortized cost buildup for future expensive operations.', 'Challenges encountered in implementing the extended dynamic array functionality are discussed. The speaker highlights the challenges of maintaining invariance, handling empty lists, and the need for a linear amount of operations to rebuild the data structure, providing insights into the complexities involved in implementing the extended dynamic array functionality.']}], 'duration': 966.331, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k3115276.jpg', 'highlights': ['Allocating extra space at both ends of a dynamic array allows for efficient operations on both ends, ensuring linear time for insertions.', 'The concept of amortization is applied to dynamic arrays, making it efficient on average by charging the cost of rebuilding to each operation.', 'Expanding a dynamic array to create a doubly ended queue system enables efficient dynamic operations on both ends.', 'The importance of maintaining a linear amount of space with respect to the number of items stored is emphasized to avoid excessive memory usage.', 'The optimal time to rebuild the array is discussed in relation to the fill rate, ensuring a linear amount of space is maintained.', 'The concept of maintaining extra space at the end of the array is explained to avoid frequent resizing and optimize performance.', 'The chapter discusses the extension of dynamic array ideas to support operations on both ends of a sequence.', 'The solution involves merging and rebuilding the data structure to maintain the desired invariant and amortized cost buildup for future expensive operations.', 'Challenges encountered in implementing the extended dynamic array functionality are discussed, providing insights into the complexities involved.']}, {'end': 4537.746, 'segs': [{'end': 4213.229, 'src': 'heatmap', 'start': 4153.421, 'weight': 0.913, 'content': [{'end': 4164.953, 'text': 'And The question is asking if we give you a linked list that has two n nodes.', 'start': 4153.421, 'duration': 11.532}, {'end': 4173.854, 'text': 'I want you to take the last n nodes and reverse their order and do this to the data structure.', 'start': 4164.953, 'duration': 8.901}, {'end': 4175.796, 'text': "You're not going to return a new data structure.", 'start': 4173.875, 'duration': 1.921}, {'end': 4177.475, 'text': "You're going to do this.", 'start': 4176.676, 'duration': 0.799}, {'end': 4180.096, 'text': "You're going to modify the existing nodes.", 'start': 4178.036, 'duration': 2.06}, {'end': 4182.877, 'text': 'And actually, here goes back to your question.', 'start': 4180.136, 'duration': 2.741}, {'end': 4189.871, 'text': 'What are we limited to in how we approach this problem?', 'start': 4186.567, 'duration': 3.304}, {'end': 4198.8, 'text': 'What this problem says is your algorithm should not make any new linked list nodes or instantiate any new non-constant sized data structures.', 'start': 4190.291, 'duration': 8.509}, {'end': 4206.946, 'text': "So it's not like I can read through this whole thing, find out where the n plus 1 node is.", 'start': 4199.401, 'duration': 7.545}, {'end': 4213.229, 'text': 'read out all of those names, store them in an array somewhere and then rewrite them back out.', 'start': 4206.946, 'duration': 6.283}], 'summary': 'Reverse the order of the last n nodes in a linked list without using new nodes or non-constant sized data structures.', 'duration': 59.808, 'max_score': 4153.421, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k4153421.jpg'}, {'end': 4206.946, 'src': 'embed', 'start': 4178.036, 'weight': 0, 'content': [{'end': 4180.096, 'text': "You're going to modify the existing nodes.", 'start': 4178.036, 'duration': 2.06}, {'end': 4182.877, 'text': 'And actually, here goes back to your question.', 'start': 4180.136, 'duration': 2.741}, {'end': 4189.871, 'text': 'What are we limited to in how we approach this problem?', 'start': 4186.567, 'duration': 3.304}, {'end': 4198.8, 'text': 'What this problem says is your algorithm should not make any new linked list nodes or instantiate any new non-constant sized data structures.', 'start': 4190.291, 'duration': 8.509}, {'end': 4206.946, 'text': "So it's not like I can read through this whole thing, find out where the n plus 1 node is.", 'start': 4199.401, 'duration': 7.545}], 'summary': 'Algorithm should not create new nodes or non-constant sized data structures.', 'duration': 28.91, 'max_score': 4178.036, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k4178036.jpg'}, {'end': 4425.955, 'src': 'embed', 'start': 4302.021, 'weight': 2, 'content': [{'end': 4310.447, 'text': 'a lot of times it makes sense to develop an outline or a game plan of constituent parts that you might want to approach this problem with.', 'start': 4302.021, 'duration': 8.426}, {'end': 4316.911, 'text': 'So the first thing that your colleague over here was saying was, at some point, we need to find out where the middle of this thing is.', 'start': 4311.087, 'duration': 5.824}, {'end': 4335.222, 'text': 'Does that make sense? So maybe the first thing we want to do to approach this problem is 1 find nth node.', 'start': 4317.171, 'duration': 18.051}, {'end': 4337.803, 'text': "That's the end of the first set of children.", 'start': 4335.442, 'duration': 2.361}, {'end': 4348.63, 'text': 'Then I have a second thing that I want to do.', 'start': 4338.444, 'duration': 10.186}, {'end': 4350.15, 'text': "What's the next thing I have to do??", 'start': 4349.07, 'duration': 1.08}, {'end': 4355.113, 'text': 'I have to kind of reverse the pointers of everything after the nth note, right?', 'start': 4350.17, 'duration': 4.943}, {'end': 4365.277, 'text': 'OK, so second thing, reverse, I guess, next pointers.', 'start': 4356.093, 'duration': 9.184}, {'end': 4382.748, 'text': 'of everything after the n-th node, the nodes n plus 1 to 2n.', 'start': 4375.782, 'duration': 6.966}, {'end': 4393.657, 'text': "Does that make sense? And after I've reversed all of those things, what do I have? I have a first block.", 'start': 4382.848, 'duration': 10.809}, {'end': 4394.678, 'text': 'This points like that.', 'start': 4393.677, 'duration': 1.001}, {'end': 4398.481, 'text': "And now we've got this thing.", 'start': 4395.418, 'duration': 3.063}, {'end': 4404.726, 'text': "And we've reversed all the pointers like this.", 'start': 4399.562, 'duration': 5.164}, {'end': 4410.225, 'text': "So that's after step 2.", 'start': 4407.503, 'duration': 2.722}, {'end': 4422.193, 'text': "Is that what we want? Yeah? So step 3 would be nullify the function of the nth node, because that's now the new end.", 'start': 4410.225, 'duration': 11.968}, {'end': 4425.955, 'text': 'JASON KUO- So this is my new end.', 'start': 4422.233, 'duration': 3.722}], 'summary': 'Develop a plan to find the middle node, reverse pointers, and nullify the nth node.', 'duration': 123.934, 'max_score': 4302.021, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k4302021.jpg'}], 'start': 4082.148, 'title': 'Reversing linked lists', 'summary': 'Discusses reversing the last half of a singly linked list with restrictions on new node creation, and constructing an algorithm for reversing a linked list involving finding the middle node, reversing next pointers, and nullifying the function of the nth node, to maintain a fair learning approach for students.', 'chapters': [{'end': 4226.894, 'start': 4082.148, 'title': 'Reverse last half of singly linked list', 'summary': 'Discusses reversing the last half of a singly linked list with restrictions on creating new nodes and non-constant sized data structures, aiming to modify the existing nodes and maintain a fair approach for students, to achieve a more balanced learning environment.', 'duration': 144.746, 'highlights': ['The problem entails reversing the order of the last n nodes in a linked list with limitations on creating new nodes and non-constant sized data structures, emphasizing the need to modify the existing nodes.', 'Students aim to create a fair approach by reversing the last half of the line, promoting a balanced learning environment.', 'The singly linked list consists of a size and a head, where the head is a pointer to a node containing the name of the child and the next pointer to the subsequent node.', 'The algorithm is constrained from creating new linked list nodes or non-constant sized data structures, necessitating the manipulation of existing nodes without generating additional data structures or nodes.']}, {'end': 4537.746, 'start': 4227.134, 'title': 'Reverse linked list algorithm', 'summary': 'Discusses the approach to construct an algorithm for reversing a linked list, involving finding the middle node, reversing next pointers, and nullifying the function of the nth node.', 'duration': 310.612, 'highlights': ['The first step is to find the nth node in the linked list. This step involves identifying the middle of the linked list, forming a crucial part of the algorithm construction.', 'The second step is to reverse the next pointers of everything after the nth node. This involves reversing the pointers of the nodes from n+1 to 2n, contributing to the process of reversing the linked list.', 'The third step is to nullify the function of the nth node, as it becomes the new end of the linked list. This step involves nullifying the function of the nth node, setting the stage for the successful reversal of the linked list.']}], 'duration': 455.598, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k4082148.jpg', 'highlights': ['The problem entails reversing the order of the last n nodes in a linked list with limitations on creating new nodes and non-constant sized data structures, emphasizing the need to modify the existing nodes.', 'The algorithm is constrained from creating new linked list nodes or non-constant sized data structures, necessitating the manipulation of existing nodes without generating additional data structures or nodes.', 'The first step is to find the nth node in the linked list. This step involves identifying the middle of the linked list, forming a crucial part of the algorithm construction.', 'The second step is to reverse the next pointers of everything after the nth node. This involves reversing the pointers of the nodes from n+1 to 2n, contributing to the process of reversing the linked list.', 'The third step is to nullify the function of the nth node, as it becomes the new end of the linked list. This step involves nullifying the function of the nth node, setting the stage for the successful reversal of the linked list.']}, {'end': 5190.079, 'segs': [{'end': 4579.463, 'src': 'embed', 'start': 4539.554, 'weight': 4, 'content': [{'end': 4547.941, 'text': 'So basically, the last step here is clean up ends.', 'start': 4539.554, 'duration': 8.387}, {'end': 4556.788, 'text': "And in a LaTeX write up, you'd want to specify what are the things that you're relinking.", 'start': 4548.741, 'duration': 8.047}, {'end': 4561.752, 'text': 'But this was a coding question, and so we actually gave you code to work with.', 'start': 4557.609, 'duration': 4.143}, {'end': 4569.038, 'text': "So I'm going to see whether I can live code this for you in front of you.", 'start': 4562.973, 'duration': 6.065}, {'end': 4579.463, 'text': 'OK, so here was our code submission site from last term.', 'start': 4573.82, 'duration': 5.643}], 'summary': 'Demonstrating live coding with a provided code submission site.', 'duration': 39.909, 'max_score': 4539.554, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k4539554.jpg'}, {'end': 4639.419, 'src': 'embed', 'start': 4605.738, 'weight': 0, 'content': [{'end': 4609.841, 'text': 'And then we have two more code questions, a test file and a reorder students file.', 'start': 4605.738, 'duration': 4.103}, {'end': 4613.544, 'text': 'So reorder students looks something like this.', 'start': 4610.642, 'duration': 2.902}, {'end': 4621.028, 'text': "It has a template of the code that we're going to want you to write with inputs and outputs, and you're putting your code here.", 'start': 4615.084, 'duration': 5.944}, {'end': 4623.91, 'text': "And this function doesn't need to return anything.", 'start': 4622.069, 'duration': 1.841}, {'end': 4630.653, 'text': "And then we also give you this linked list implementation, which is what's in your recitation handout.", 'start': 4624.65, 'duration': 6.003}, {'end': 4639.419, 'text': "I'm actually going to ignore most of this stuff, really just that this thing contains an item in next in your node.", 'start': 4631.194, 'duration': 8.225}], 'summary': 'Code questions include test file and reorder students file, with linked list implementation.', 'duration': 33.681, 'max_score': 4605.738, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k4605738.jpg'}, {'end': 4708.831, 'src': 'embed', 'start': 4682.91, 'weight': 2, 'content': [{'end': 4690.317, 'text': 'What we can see is when I do this, here are my test cases.', 'start': 4682.91, 'duration': 7.407}, {'end': 4692.178, 'text': "Here's a linked list.", 'start': 4690.717, 'duration': 1.461}, {'end': 4695.521, 'text': "And what's happening is it's just spitting out the same linked list.", 'start': 4692.979, 'duration': 2.542}, {'end': 4696.802, 'text': "I haven't done anything to it.", 'start': 4695.621, 'duration': 1.181}, {'end': 4700.667, 'text': 'Right? All right, so we need to do something to it.', 'start': 4697.526, 'duration': 3.141}, {'end': 4705.109, 'text': "How are we going to do that? All right, so let's implement this function.", 'start': 4701.168, 'duration': 3.941}, {'end': 4708.831, 'text': "And I'm going to get rid of this stuff because get rid of that.", 'start': 4705.129, 'duration': 3.702}], 'summary': 'Discussion on implementing a function to modify a linked list.', 'duration': 25.921, 'max_score': 4682.91, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k4682910.jpg'}, {'end': 4840.936, 'src': 'embed', 'start': 4811.234, 'weight': 3, 'content': [{'end': 4817.697, 'text': "And so in my mind, I'm going to want to use the same kind of notation here so that I can understand my code.", 'start': 4811.234, 'duration': 6.463}, {'end': 4820.599, 'text': 'So B is going to be the next thing.', 'start': 4817.997, 'duration': 2.602}, {'end': 4828.82, 'text': "And now in this process, as I'm going to flip things around, what I'm going to do is I'm going to keep track of three nodes.", 'start': 4823.513, 'duration': 5.307}, {'end': 4834.467, 'text': "I'm going to keep track of x, which is the node that I'm going to be relinking.", 'start': 4829.481, 'duration': 4.986}, {'end': 4840.936, 'text': 'And what else do I need to keep track of? Destination.', 'start': 4834.487, 'duration': 6.449}], 'summary': 'Using notation to understand code, tracking 3 nodes: x, b, destination.', 'duration': 29.702, 'max_score': 4811.234, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k4811234.jpg'}, {'end': 4926.52, 'src': 'embed', 'start': 4896.51, 'weight': 1, 'content': [{'end': 4897.291, 'text': 'Either way is fine.', 'start': 4896.51, 'duration': 0.781}, {'end': 4900.073, 'text': 'And then I want to go through a loop.', 'start': 4897.991, 'duration': 2.082}, {'end': 4901.594, 'text': "I'm going to be doing it a loop way.", 'start': 4900.213, 'duration': 1.381}, {'end': 4903.295, 'text': 'You can do it a recursive way if you want.', 'start': 4901.654, 'duration': 1.641}, {'end': 4907.638, 'text': "Here's a loop way in which I'm just going to loop through.", 'start': 4904.596, 'duration': 3.042}, {'end': 4908.238, 'text': 'how many times?', 'start': 4907.638, 'duration': 0.6}, {'end': 4912.081, 'text': 'How many pointers am I going to relink as I go down this thing??', 'start': 4908.498, 'duration': 3.583}, {'end': 4917.368, 'text': 'I need to relink the pointers of all of these guys.', 'start': 4914.585, 'duration': 2.783}, {'end': 4921.914, 'text': 'How many are there? How many? n.', 'start': 4917.408, 'duration': 4.506}, {'end': 4922.675, 'text': 'There are n of them.', 'start': 4921.914, 'duration': 0.761}, {'end': 4926.52, 'text': "So 4, I don't care about the loop variable here either.", 'start': 4923.055, 'duration': 3.465}], 'summary': 'Discussing loop and recursion methods for relinking n pointers.', 'duration': 30.01, 'max_score': 4896.51, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k4896510.jpg'}], 'start': 4539.554, 'title': 'Code cleaning and linked list reordering', 'summary': 'Covers cleaning up code and live coding in latex, and implementing linked list reordering with 100% success.', 'chapters': [{'end': 4623.91, 'start': 4539.554, 'title': 'Cleaning up code and live coding in latex', 'summary': 'Covers the process of cleaning up code and live coding in latex, involving the use of code submissions and linking files, while also providing a live demonstration of code organization and structure.', 'duration': 84.356, 'highlights': ['The chapter covers the process of cleaning up code and live coding in LaTeX, involving the use of code submissions and linking files, while also providing a live demonstration of code organization and structure.', 'The speaker demonstrates the process of cleaning up code and live coding in LaTeX using a template from the previous term, including the presence of a linked list sequence and two additional code questions.', 'The speaker mentions providing a version of a linked list sequence and two more code questions, a test file and a reorder students file.']}, {'end': 5190.079, 'start': 4624.65, 'title': 'Linked list reordering', 'summary': 'Discusses the implementation of a function to reorder students in a linked list, iterating through the process and successfully passing the test cases, achieving 100% success.', 'duration': 565.429, 'highlights': ['The function is implemented to reorder students in a linked list, successfully passing the test cases with 100% accuracy.', 'The process involves finding the nth node, relinking the pointers, and verifying the correctness of the reordering.', 'The implementation involves iterating through the linked list, using loop structures to manipulate the nodes and achieve the desired reordering.', 'The importance of maintaining the correct pointers and tracking the nodes throughout the process is emphasized to ensure the accuracy of the reordering.']}], 'duration': 650.525, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IPSaG9RRc-k/pics/IPSaG9RRc-k4539554.jpg', 'highlights': ['The function is implemented to reorder students in a linked list, successfully passing the test cases with 100% accuracy.', 'The process involves finding the nth node, relinking the pointers, and verifying the correctness of the reordering.', 'The implementation involves iterating through the linked list, using loop structures to manipulate the nodes and achieve the desired reordering.', 'The importance of maintaining the correct pointers and tracking the nodes throughout the process is emphasized to ensure the accuracy of the reordering.', 'The chapter covers the process of cleaning up code and live coding in LaTeX, involving the use of code submissions and linking files, while also providing a live demonstration of code organization and structure.']}], 'highlights': ['The chapter introduces the first-ever 6.006 problem session on Fridays, aiming to provide recorded problem-solving methods to complement the foundational material presented in lectures.', "The goal of these problem sessions is to show how to approach working on problems similar to those in the problem sets, aiming to enhance students' problem-solving skills.", 'The problem sets are applications of the fundamental material, and there is usually a different feel between the problems and the underlying foundational material.', 'The sessions aim to provide a safe space for students to interact with the material in a smaller environment, fostering interaction and one-on-one questions.', 'The instructor encourages interactive participation, allowing questions at any point during the session.', 'The chapter focuses on ordering functions based on their asymptotic behavior, highlighting the identification of the slowest growth term and the use of set brackets for asymptotically equivalent functions.', 'The order of growth for the functions F1, F5, F2, F3, and F4 is established, with F1 being the slowest growing and F4 being the fastest growing.', 'The relationship between logarithmic and polynomial functions is discussed, demonstrating that logarithmic growth is slower than linear and polynomial growth.', 'Binomial coefficient is defined as the number of ways to choose k objects from n.', 'Introduction of formula n minus k factorial for permutations of n choose k.', 'The logarithmic transformation of n to the n results in n log n, allowing for easier comparison of functions.', "Stirling's approximation is applied to simplify the expression 2 to the n over root n, resulting in an asymptotic behavior of 2 to the n over root n.", 'Understanding that logarithmic factors grow slower than polynomial ones and again grow slower than exponential ones and being able to do some transformations of some of these mathematical quantities to get them in a polynomial-like looking form is essential in approaching these problems.', 'The concept of amortized running time is introduced, illustrating its application in the context of data structures such as dynamic arrays to optimize the running time of operations over the long term.', 'Preference for recursive algorithms over for loops by computer scientists is due to the ability to argue and analyze with a constant amount of information at any given point, leading to easier convincing of correctness.', 'The time analysis for the first operation involves four operations, each taking constant time, resulting in the entire operation taking constant time.', 'Recursive algorithms run in order k time complexity, with a constant amount of work done in each section and a call function running k minus 1 times.', 'The second operation involves shifting the first k elements to the end of the sequence, and the k-th item becomes the last item, with a method that takes 2k steps to perform the operation k times.', 'Allocating extra space at both ends of a dynamic array allows for efficient operations on both ends, ensuring linear time for insertions.', 'The concept of amortization is applied to dynamic arrays, making it efficient on average by charging the cost of rebuilding to each operation.', 'Expanding a dynamic array to create a doubly ended queue system enables efficient dynamic operations on both ends.', 'The importance of maintaining a linear amount of space with respect to the number of items stored is emphasized to avoid excessive memory usage.', 'The optimal time to rebuild the array is discussed in relation to the fill rate, ensuring a linear amount of space is maintained.', 'The concept of maintaining extra space at the end of the array is explained to avoid frequent resizing and optimize performance.', 'The problem entails reversing the order of the last n nodes in a linked list with limitations on creating new nodes and non-constant sized data structures, emphasizing the need to modify the existing nodes.', 'The algorithm is constrained from creating new linked list nodes or non-constant sized data structures, necessitating the manipulation of existing nodes without generating additional data structures or nodes.', 'The first step is to find the nth node in the linked list. This step involves identifying the middle of the linked list, forming a crucial part of the algorithm construction.', 'The second step is to reverse the next pointers of everything after the nth node. This involves reversing the pointers of the nodes from n+1 to 2n, contributing to the process of reversing the linked list.', 'The third step is to nullify the function of the nth node, as it becomes the new end of the linked list. This step involves nullifying the function of the nth node, setting the stage for the successful reversal of the linked list.', 'The function is implemented to reorder students in a linked list, successfully passing the test cases with 100% accuracy.', 'The process involves finding the nth node, relinking the pointers, and verifying the correctness of the reordering.', 'The implementation involves iterating through the linked list, using loop structures to manipulate the nodes and achieve the desired reordering.', 'The importance of maintaining the correct pointers and tracking the nodes throughout the process is emphasized to ensure the accuracy of the reordering.', 'The chapter covers the process of cleaning up code and live coding in LaTeX, involving the use of code submissions and linking files, while also providing a live demonstration of code organization and structure.']}