WEBVTT 00:00:00.070 --> 00:00:02.500 The following content is provided under a Creative 00:00:02.500 --> 00:00:04.019 Commons license. 00:00:04.019 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.730 continue to offer high quality educational resources for free. 00:00:10.730 --> 00:00:13.340 To make a donation or view additional materials 00:00:13.340 --> 00:00:17.236 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.236 --> 00:00:17.861 at ocw.mit.edu. 00:00:22.776 --> 00:00:24.150 PROFESSOR: So this is a 2-3 tree. 00:00:24.150 --> 00:00:27.340 So as you can see, every node has-- so the 2-3 is either 00:00:27.340 --> 00:00:28.840 two children or three children. 00:00:28.840 --> 00:00:31.350 Every node can have either one key or two keys. 00:00:31.350 --> 00:00:34.665 And the correlation is that every-- 00:00:34.665 --> 00:00:37.610 so if there are n keys in a node, it has n plus 1 children. 00:00:37.610 --> 00:00:41.740 So the way that works is similar to binary search trees. 00:00:41.740 --> 00:00:46.350 So if you have value here, the two children surrounding it-- 00:00:46.350 --> 00:00:48.850 so this side is less, this side is more. 00:00:48.850 --> 00:00:53.010 So it's essentially sort of going in order reversal, left 00:00:53.010 --> 00:00:54.830 child, root, right child. 00:00:54.830 --> 00:00:56.440 So [INAUDIBLE], it's ordered. 00:00:56.440 --> 00:01:04.010 So generally a B tree will have some nodes. 00:01:04.010 --> 00:01:10.610 So let's say n and n plus 1 children. 00:01:16.920 --> 00:01:20.130 And if you take anything in the middle, 00:01:20.130 --> 00:01:23.260 look at the two children, all the keys in this sub-tree 00:01:23.260 --> 00:01:26.360 are smaller than the key here, and all the keys 00:01:26.360 --> 00:01:28.680 in this sub-tree are larger than the key here. 00:01:28.680 --> 00:01:30.320 So that's the general node. 00:01:30.320 --> 00:01:34.570 So before we go into more details 00:01:34.570 --> 00:01:36.520 of the properties and everything, 00:01:36.520 --> 00:01:39.330 the question is why use B-trees. 00:01:39.330 --> 00:01:42.380 So if we do a quick depth analysis, 00:01:42.380 --> 00:01:44.760 we can see that the depth is to log n rate. 00:01:44.760 --> 00:01:48.630 Is that clear to everyone sort of, why the depth is log n? 00:01:48.630 --> 00:01:52.025 Because you have branching just like in binary search trees. 00:01:52.025 --> 00:01:53.860 In fact, you have more branching. 00:01:53.860 --> 00:01:56.020 But in any case, depth is to log n. 00:01:56.020 --> 00:02:00.260 But why use B-trees over binary search trees? 00:02:00.260 --> 00:02:04.920 Anyone have a reason why you would 00:02:04.920 --> 00:02:07.500 prefer to use B-trees or not? 00:02:07.500 --> 00:02:11.370 So all the operations are still log n. 00:02:11.370 --> 00:02:11.910 Any guesses? 00:02:16.140 --> 00:02:17.080 None. 00:02:17.080 --> 00:02:18.500 OK. 00:02:18.500 --> 00:02:22.240 Well, OK, the reason is memory hierarchy. 00:02:22.240 --> 00:02:24.340 So normally in [INAUDIBLE], we just 00:02:24.340 --> 00:02:27.320 assume that the computer has access to memory, 00:02:27.320 --> 00:02:30.150 and you can just pick up things from disk and constant time 00:02:30.150 --> 00:02:32.890 and do your operations with it, and you don't worry 00:02:32.890 --> 00:02:34.294 about caches and everything. 00:02:34.294 --> 00:02:35.710 But that's not how computers work. 00:02:35.710 --> 00:02:37.692 So in a computer, you have-- so those of you 00:02:37.692 --> 00:02:39.400 who have taken some computer architecture 00:02:39.400 --> 00:02:42.730 class [INAUDIBLE] or something, you will know that hierarchy. 00:02:42.730 --> 00:02:46.570 So there's a CPU-- so let's draw it somewhere. 00:02:46.570 --> 00:02:48.020 So you have your CPU. 00:02:48.020 --> 00:02:50.570 And [INAUDIBLE] CPU, you have some registers. 00:02:50.570 --> 00:02:52.930 You have your caches, L1, L2, L3, whatever. 00:02:52.930 --> 00:02:54.080 You have your RAM. 00:02:54.080 --> 00:02:55.610 You have disk after that. 00:02:55.610 --> 00:02:57.860 So disk [? loads. ?] Then you have your, I don't know, 00:02:57.860 --> 00:02:59.090 your cloud, whatever. 00:02:59.090 --> 00:03:01.130 So each level, your memory size grows 00:03:01.130 --> 00:03:04.430 and your access time grows as well. 00:03:04.430 --> 00:03:07.650 So in the basic memory hierarchy model, 00:03:07.650 --> 00:03:11.930 we have just two levels of hierarchy, let's say. 00:03:11.930 --> 00:03:16.370 So you have cache connected by a high bandwidth channel 00:03:16.370 --> 00:03:19.370 to the CPU, and you have a low bandwidth channel to disk. 00:03:23.040 --> 00:03:25.754 So the difference is-- so essentially 00:03:25.754 --> 00:03:27.920 you can consider that cache just has infinite speed. 00:03:27.920 --> 00:03:29.790 Cache, just like, whatever you can take it. 00:03:29.790 --> 00:03:32.790 You don't have any cost for bringing in stuff from cache. 00:03:32.790 --> 00:03:34.680 But it's finite size. 00:03:34.680 --> 00:03:37.560 So the way cache works is it has a bunch of words, which 00:03:37.560 --> 00:03:38.740 is a finite number of words. 00:03:38.740 --> 00:03:44.915 So each word has size B, and let's say you have m words. 00:03:47.610 --> 00:03:51.340 However, hard disk is just, let's say, infinite memory, 00:03:51.340 --> 00:03:56.130 but it has some cost associated to accessing things. 00:03:56.130 --> 00:03:58.580 Also when you access things from hard disk, 00:03:58.580 --> 00:03:59.630 you copy them into cache. 00:03:59.630 --> 00:04:01.990 When you copy a block of size b, you take it up 00:04:01.990 --> 00:04:04.800 from the hard disk, and you take a block, 00:04:04.800 --> 00:04:05.920 and you put it into cache. 00:04:05.920 --> 00:04:08.510 And you have to get rid of something because it's fine. 00:04:08.510 --> 00:04:11.460 So what you want to do is you want to utilize 00:04:11.460 --> 00:04:14.010 that b block efficiently. 00:04:14.010 --> 00:04:16.459 You just want to bring a b block every time 00:04:16.459 --> 00:04:18.170 you want to access a new node. 00:04:18.170 --> 00:04:21.170 In a binary search tree, normal operations are what? 00:04:21.170 --> 00:04:25.520 You start in the root and go to a node. 00:04:25.520 --> 00:04:27.880 But that's not very easily correlated with this. 00:04:27.880 --> 00:04:28.380 Right? 00:04:28.380 --> 00:04:30.779 So if you want to utilize an entire block, 00:04:30.779 --> 00:04:33.320 you would want something like a block which sort of goes down 00:04:33.320 --> 00:04:35.709 the tree. 00:04:35.709 --> 00:04:37.500 But that's not how binary trees are stored. 00:04:37.500 --> 00:04:41.450 Binary trees are stored this way. 00:04:41.450 --> 00:04:43.660 So that's the nice thing about B-trees. 00:04:43.660 --> 00:04:45.452 So this is just a 2-3 tree. 00:04:45.452 --> 00:04:46.660 This is not a general B-tree. 00:04:46.660 --> 00:04:48.840 A general B-tree will have a bunch of nodes, 00:04:48.840 --> 00:04:50.690 and we'll come to that number. 00:04:50.690 --> 00:04:53.610 But generally you want to make that number of nodes something 00:04:53.610 --> 00:04:57.380 like the cache-- what is it? 00:04:57.380 --> 00:04:58.910 The word size in the cache. 00:04:58.910 --> 00:05:02.860 So once you do that, you can get an entire node from disk, 00:05:02.860 --> 00:05:05.950 like work on that, and then get another [INAUDIBLE], 00:05:05.950 --> 00:05:07.629 so your height is reduced. 00:05:07.629 --> 00:05:09.420 And you can do your operation much quicker, 00:05:09.420 --> 00:05:12.940 because you're not accessing disk every time you're 00:05:12.940 --> 00:05:15.060 going down a level. 00:05:15.060 --> 00:05:15.560 Sorry. 00:05:15.560 --> 00:05:17.930 You are accessing disk every time you go down a level, 00:05:17.930 --> 00:05:20.490 but you're utilizing the whole block 00:05:20.490 --> 00:05:22.226 when you're accessing disk. 00:05:22.226 --> 00:05:23.100 Good? 00:05:23.100 --> 00:05:25.180 Sort of make sense? 00:05:25.180 --> 00:05:27.370 OK. 00:05:27.370 --> 00:05:31.460 So let's write down the specifications for B-trees now. 00:05:34.352 --> 00:05:35.316 All right. 00:05:40.620 --> 00:05:43.460 So number of children. 00:05:50.090 --> 00:05:52.510 So first of all, a B-tree has something 00:05:52.510 --> 00:05:54.960 called a branching factor. 00:05:54.960 --> 00:05:58.567 So in the 2-3 tree, the branching factor is two. 00:05:58.567 --> 00:06:00.650 So what that means is simply that it just balances 00:06:00.650 --> 00:06:01.608 the number of children. 00:06:01.608 --> 00:06:04.460 So the number of children has to be greater than or equal to 2. 00:06:04.460 --> 00:06:05.690 Other than the root node. 00:06:05.690 --> 00:06:07.970 The root node can have less than B children. 00:06:07.970 --> 00:06:09.200 It's fine. 00:06:09.200 --> 00:06:13.670 Also it's upper bounded by 2B [? plus ?] 2B. 00:06:13.670 --> 00:06:15.760 Notice that this is a strict upper bound. 00:06:15.760 --> 00:06:22.140 So you can have at most 2B minus 1 children from a node. 00:06:22.140 --> 00:06:29.790 Also remember that the number of keys, the number of keys 00:06:29.790 --> 00:06:32.640 is just 1 less than the number of children. 00:06:32.640 --> 00:06:40.730 Therefore, these inequalities are just reduced by 1. 00:06:40.730 --> 00:06:45.260 So you have minus 1 and you have 2B minus 1. 00:06:45.260 --> 00:06:49.760 So the number of keys can be between minus 1 and 2B minus 2. 00:06:49.760 --> 00:06:52.270 The rationale for that will become clear-- yeah? 00:06:52.270 --> 00:06:54.345 AUDIENCE: Is B the height of the tree? 00:06:54.345 --> 00:06:56.567 PROFESSOR: No, B is the branching. 00:06:56.567 --> 00:06:57.650 B is the branching factor. 00:06:57.650 --> 00:06:59.497 So that is the number of children. 00:06:59.497 --> 00:07:00.830 It's not the number of children. 00:07:00.830 --> 00:07:02.650 It's a bound of the number of children. 00:07:02.650 --> 00:07:07.050 So like in the 2-3 tree, B is equal to 2, 00:07:07.050 --> 00:07:08.840 and this is a 2-3 tree. 00:07:13.300 --> 00:07:18.950 So the 2 refers to-- you can have either two children 00:07:18.950 --> 00:07:21.660 or you can have three children. 00:07:21.660 --> 00:07:24.520 And so the upper bound on children is 2B minus 1. 00:07:27.110 --> 00:07:29.240 2B minus 1 is equal to 3. 00:07:29.240 --> 00:07:32.180 So you can have two or three children. 00:07:32.180 --> 00:07:34.830 And correspondingly, you can have either one or two 00:07:34.830 --> 00:07:37.251 keys in a node. 00:07:37.251 --> 00:07:37.750 Make sense? 00:07:37.750 --> 00:07:38.374 AUDIENCE: Yeah. 00:07:38.374 --> 00:07:39.340 PROFESSOR: Cool. 00:07:39.340 --> 00:07:42.200 OK So coming back to this. 00:07:42.200 --> 00:07:44.630 So the root does not have a lower bound. 00:07:44.630 --> 00:07:47.160 The root can have one child in any tree. 00:07:47.160 --> 00:07:50.355 So you have a B equal to 5 tree, the root 00:07:50.355 --> 00:07:52.050 can still have one child-- sorry. 00:07:52.050 --> 00:07:55.480 Not one child, one key element, two children. 00:07:55.480 --> 00:07:56.900 All right. 00:07:56.900 --> 00:07:57.760 It's good. 00:07:57.760 --> 00:08:00.370 Also it's completely balanced. 00:08:00.370 --> 00:08:04.310 So all the leaves are the same depth. 00:08:19.330 --> 00:08:21.780 So you can see it here, right? 00:08:21.780 --> 00:08:24.350 So you can't have a dangling node here. 00:08:24.350 --> 00:08:25.880 This is not allowed. 00:08:25.880 --> 00:08:27.047 You have to have a leaf. 00:08:27.047 --> 00:08:28.630 You have to have something going down, 00:08:28.630 --> 00:08:31.400 and everything ends at the same level. 00:08:31.400 --> 00:08:31.900 All right. 00:08:31.900 --> 00:08:33.110 So that's the thing. 00:08:33.110 --> 00:08:36.520 So also the leaves obviously don't have children, 00:08:36.520 --> 00:08:43.130 so this condition is violated by the leaf. 00:08:43.130 --> 00:08:46.380 So that's the basic structure of a B-tree. 00:08:49.101 --> 00:08:51.100 So the first operation we'll consider on B-trees 00:08:51.100 --> 00:08:52.560 is searching. 00:08:52.560 --> 00:08:55.160 So that should be relatively straightforward. 00:08:55.160 --> 00:08:58.110 So remember how searching is done in the binary search tree. 00:08:58.110 --> 00:09:01.900 You bring in a value x compared to the key. 00:09:01.900 --> 00:09:04.875 Let's say x is less than K, you go down this path. 00:09:04.875 --> 00:09:08.120 Let's say x is greater than K, you go down this path. 00:09:08.120 --> 00:09:09.570 So similarly in a B-tree. 00:09:09.570 --> 00:09:12.010 So let's say we bring in a value. 00:09:12.010 --> 00:09:15.770 Let's say you are looking for 20. 00:09:15.770 --> 00:09:18.120 So you bring in 20 compared to this. 00:09:18.120 --> 00:09:21.190 20 is less than 30, so you go down here. 00:09:21.190 --> 00:09:23.852 Now you have two values. 00:09:23.852 --> 00:09:25.060 So where does 20 fit in here? 00:09:25.060 --> 00:09:25.851 Not here. 00:09:25.851 --> 00:09:26.350 Not here. 00:09:26.350 --> 00:09:26.951 It fits here. 00:09:26.951 --> 00:09:27.450 OK. 00:09:27.450 --> 00:09:29.110 Go down this tree. 00:09:29.110 --> 00:09:30.980 You find 20, that's it. 00:09:30.980 --> 00:09:37.410 So in general, you bring in a key K, you look at this node, 00:09:37.410 --> 00:09:39.110 and you go through all the values. 00:09:39.110 --> 00:09:43.590 So something I forgot to mention, which should be clear. 00:09:43.590 --> 00:09:47.250 All the keys in a node, they're sorted, one after the other. 00:09:47.250 --> 00:09:49.350 So your values go like this. 00:09:49.350 --> 00:09:52.140 So they're increasing in this way. 00:09:52.140 --> 00:09:54.835 Make sense? 00:09:54.835 --> 00:09:56.440 So you bring in a key. 00:09:56.440 --> 00:09:59.900 Look at all the keys in the node you're looking at, 00:09:59.900 --> 00:10:02.230 pick the place where K fits in, unless it's already 00:10:02.230 --> 00:10:02.730 in the node. 00:10:02.730 --> 00:10:03.438 Then you're done. 00:10:03.438 --> 00:10:04.660 You've found it. 00:10:04.660 --> 00:10:08.850 Otherwise, let's say K fits in between these two guys. 00:10:08.850 --> 00:10:13.250 So you go down this child and continue. 00:10:13.250 --> 00:10:15.940 So searching with log n, similar to BSTs. 00:10:20.294 --> 00:10:21.835 So searching is not very interesting. 00:10:38.480 --> 00:10:40.225 So next is insertion. 00:10:44.440 --> 00:10:46.860 So insertion is a little more interesting than searching. 00:10:46.860 --> 00:10:48.360 So what you do in insertion is you-- 00:10:48.360 --> 00:11:09.020 [SIDE CONVERSATION] 00:11:09.020 --> 00:11:12.219 PROFESSOR: So before we resume, does anyone have any questions 00:11:12.219 --> 00:11:13.510 about the structure of B-trees. 00:11:13.510 --> 00:11:16.860 We rushed through that quite fast. 00:11:16.860 --> 00:11:18.655 About how B-trees are structured, 00:11:18.655 --> 00:11:20.165 everyone good with that? 00:11:20.165 --> 00:11:25.294 OK, also any questions about searching in a B-tree or a BST? 00:11:25.294 --> 00:11:25.794 Go ahead. 00:11:25.794 --> 00:11:27.770 AUDIENCE: Just a random question. 00:11:27.770 --> 00:11:31.722 So the 38 there, it can only have two children. 00:11:31.722 --> 00:11:33.210 PROFESSOR: Yep. 00:11:33.210 --> 00:11:36.270 So one value, two children. 00:11:36.270 --> 00:11:39.760 So you have some node in the B-tree, 00:11:39.760 --> 00:11:41.420 and whatever is below it is split 00:11:41.420 --> 00:11:43.350 into parts by the elements. 00:11:43.350 --> 00:11:45.100 So if you have n elements, it splits it up 00:11:45.100 --> 00:11:46.275 into n plus 1 segments. 00:11:49.830 --> 00:11:53.087 AUDIENCE: You said that the root didn't have to follow the root. 00:11:53.087 --> 00:11:53.670 PROFESSOR: No. 00:11:53.670 --> 00:11:54.589 AUDIENCE: Why is that? 00:11:54.589 --> 00:11:57.130 PROFESSOR: Well, you'll see when we do insertion and deletion 00:11:57.130 --> 00:11:58.005 why that's necessary. 00:11:58.005 --> 00:12:02.350 But essentially you can consider that it's an invariant. 00:12:02.350 --> 00:12:04.770 And all we have to do is preserve that invariant. 00:12:04.770 --> 00:12:08.230 So the root, it has to still have less than two-- 00:12:08.230 --> 00:12:10.140 it still has to have the upper bound. 00:12:10.140 --> 00:12:14.569 But it doesn't need to have a lower bound. 00:12:14.569 --> 00:12:17.490 AUDIENCE: How do you choose B? 00:12:17.490 --> 00:12:20.120 PROFESSOR: Well, the whole [INAUDIBLE] cache size, 00:12:20.120 --> 00:12:21.220 so something with that. 00:12:21.220 --> 00:12:23.850 So you probably want 2B to be about your cache size 00:12:23.850 --> 00:12:25.760 so you can get the whole block in one go. 00:12:25.760 --> 00:12:28.050 I've never implemented a B-tree, so I 00:12:28.050 --> 00:12:29.150 don't know how it's actually done in practice. 00:12:29.150 --> 00:12:31.399 But that is the reason, so I'm assuming it's something 00:12:31.399 --> 00:12:33.730 to do with the cache length. 00:12:33.730 --> 00:12:38.102 AUDIENCE: Is the 14, is it a child of both 10 and 17? 00:12:38.102 --> 00:12:39.935 PROFESSOR: Well, it's not a child of either. 00:12:39.935 --> 00:12:41.260 It's a child of this node. 00:12:41.260 --> 00:12:43.570 So this node has two elements, so it's 00:12:43.570 --> 00:12:46.380 being divided-- dividing the interval up into three parts. 00:12:46.380 --> 00:12:49.800 So it's in between 10 and 17 is the point here. 00:12:49.800 --> 00:12:54.600 AUDIENCE: So then this node has five children? 00:12:54.600 --> 00:12:55.857 PROFESSOR: Sorry? 00:12:55.857 --> 00:12:56.940 No, it has three children. 00:12:56.940 --> 00:13:00.010 So don't think of every key as a node. 00:13:00.010 --> 00:13:04.000 Think of the whole unit as a node. 00:13:04.000 --> 00:13:06.115 So it's not necessarily-- in a binary search tree, 00:13:06.115 --> 00:13:08.134 you have one element, but here every node 00:13:08.134 --> 00:13:09.050 has multiple elements. 00:13:09.050 --> 00:13:11.624 That's the point of it. 00:13:11.624 --> 00:13:14.840 Anyone else? 00:13:14.840 --> 00:13:17.650 OK, let's start with searching. 00:13:17.650 --> 00:13:22.236 So let's leave this here. 00:13:30.210 --> 00:13:32.770 Well, you have the formulas up there, so that's good. 00:13:42.570 --> 00:13:43.160 Insertion. 00:13:43.160 --> 00:13:44.030 Let's start with insertion. 00:13:44.030 --> 00:13:45.155 We already did searching. 00:13:52.390 --> 00:13:54.444 So insertion is you bring in a new key K, 00:13:54.444 --> 00:13:56.110 and you want to insert it into the tree. 00:13:56.110 --> 00:13:57.120 So what's the problem that could happen? 00:13:57.120 --> 00:13:59.500 You can find the location where you want to insert it, just 00:13:59.500 --> 00:14:00.125 like searching. 00:14:00.125 --> 00:14:03.390 You just go down the tree and find where it should be placed. 00:14:03.390 --> 00:14:05.280 But once you do place it, you have a problem. 00:14:05.280 --> 00:14:06.113 What is the problem? 00:14:06.113 --> 00:14:10.570 The problem is that one of your nodes will become overfull. 00:14:10.570 --> 00:14:11.070 Whatever. 00:14:11.070 --> 00:14:13.840 It'll overflow, and that's not what you want. 00:14:13.840 --> 00:14:16.760 So you want some way so you can manage this. 00:14:16.760 --> 00:14:17.940 How do you manage this? 00:14:17.940 --> 00:14:25.190 So I have this lovely prop here, which I hope to demonstrate. 00:14:25.190 --> 00:14:25.690 OK. 00:14:29.130 --> 00:14:32.770 So here we have B equal to 4. 00:14:32.770 --> 00:14:39.110 So let's first figure out the number of keys. 00:14:39.110 --> 00:14:42.230 So what is the minimum number of keys, anyone for B equal to 4? 00:14:42.230 --> 00:14:42.959 AUDIENCE: Three. 00:14:42.959 --> 00:14:44.125 PROFESSOR: Three, precisely. 00:14:44.125 --> 00:14:49.688 So what is the maximum number of keys? 00:14:49.688 --> 00:14:51.556 AUDIENCE: Six. 00:14:51.556 --> 00:14:53.770 PROFESSOR: 4 into 2 minus 3, yeah. 00:14:53.770 --> 00:14:55.000 Correct. 00:14:55.000 --> 00:14:56.170 3, 4. 00:14:56.170 --> 00:14:58.500 It's not seven, there's a strictly less than sign 00:14:58.500 --> 00:15:00.090 somewhere. 00:15:00.090 --> 00:15:01.070 Yes. 00:15:01.070 --> 00:15:05.198 And you'll see why it's not seven in a minute. 00:15:05.198 --> 00:15:06.050 [LAUGHTER] 00:15:06.050 --> 00:15:08.150 Oh. 00:15:08.150 --> 00:15:09.070 Hypocritical of me. 00:15:11.640 --> 00:15:12.140 All right. 00:15:12.140 --> 00:15:15.180 So as you can see, 1, 2, 3, 4, 5, 6, 7. 00:15:15.180 --> 00:15:18.110 So some insertion happened. 00:15:18.110 --> 00:15:19.250 Is the writing clear? 00:15:19.250 --> 00:15:22.580 Can everyone read the numbers? 00:15:22.580 --> 00:15:23.750 49 looks a little skewed. 00:15:23.750 --> 00:15:26.160 Anyway, essentially these are all sorted. 00:15:26.160 --> 00:15:27.290 This is the parent node. 00:15:27.290 --> 00:15:28.650 Doesn't matter what's over here. 00:15:28.650 --> 00:15:33.100 All that matters is 8, 56, and whatever's in between. 00:15:38.900 --> 00:15:41.580 So what we do when we have an overfull node 00:15:41.580 --> 00:15:43.790 is something that's called a split operation. 00:15:43.790 --> 00:15:45.729 So split. 00:15:45.729 --> 00:15:47.270 And there's something which is called 00:15:47.270 --> 00:15:48.670 a merge, which we'll come to later when 00:15:48.670 --> 00:15:49.680 we're doing deletion. 00:15:49.680 --> 00:15:52.720 But a split is-- very intuitively, 00:15:52.720 --> 00:15:55.486 it splits the node into two parts. 00:15:55.486 --> 00:15:57.610 So what it does is when you have an overfull node-- 00:15:57.610 --> 00:15:59.700 so the number of elements here is what? 00:15:59.700 --> 00:16:03.220 2B minus 1, which is just 1 over the max. 00:16:03.220 --> 00:16:06.210 So what do you do is you take the middle element 00:16:06.210 --> 00:16:09.115 and remove it. 00:16:09.115 --> 00:16:11.800 and now you split the node into two parts. 00:16:11.800 --> 00:16:14.730 Observe that there are three here and three here, 00:16:14.730 --> 00:16:16.111 which is perfect. 00:16:16.111 --> 00:16:17.860 And now what you do with the middle node-- 00:16:17.860 --> 00:16:19.774 so now you're actually disrupting 00:16:19.774 --> 00:16:21.440 the structure of the tree, because there 00:16:21.440 --> 00:16:23.062 was one pointer going in. 00:16:23.062 --> 00:16:23.895 There was one child. 00:16:23.895 --> 00:16:25.660 And now you have two children. 00:16:25.660 --> 00:16:27.940 So somehow you need to adjust the parent node, 00:16:27.940 --> 00:16:30.090 because the parent node had only one child. 00:16:30.090 --> 00:16:33.240 Well, at least there are other children off to the side. 00:16:33.240 --> 00:16:36.900 But here it had only one child, and now it's split apart. 00:16:36.900 --> 00:16:38.580 So you do something very simple. 00:16:38.580 --> 00:16:42.675 You just insert this guy in here. 00:16:42.675 --> 00:16:44.300 And then you say, oh, this points here, 00:16:44.300 --> 00:16:47.010 and this points here. 00:16:47.010 --> 00:16:47.610 Make sense? 00:16:50.310 --> 00:16:51.810 I'm going to get rid of these two. 00:16:59.850 --> 00:17:02.120 And you can even convince yourself 00:17:02.120 --> 00:17:04.910 that this preserves all the nice properties. 00:17:04.910 --> 00:17:09.310 So your children have nicely fallen back 00:17:09.310 --> 00:17:11.589 into their interval. 00:17:11.589 --> 00:17:13.627 Your sequence is completely correct, 00:17:13.627 --> 00:17:15.460 because this was the middle element of this. 00:17:15.460 --> 00:17:18.329 So this divides this interval properly. 00:17:18.329 --> 00:17:21.970 This is also between 8 and 56, because this was in this node. 00:17:21.970 --> 00:17:23.259 So all the properties. 00:17:23.259 --> 00:17:25.050 But there's one property that is a problem. 00:17:25.050 --> 00:17:28.890 So you have just increased the size of the parent node by 1. 00:17:28.890 --> 00:17:32.750 So now it's possible that the parent node has overflowed. 00:17:32.750 --> 00:17:33.680 So what do you do? 00:17:33.680 --> 00:17:35.800 You split it again. 00:17:35.800 --> 00:17:36.950 And split it again. 00:17:36.950 --> 00:17:39.544 And if at any point, you're fine, 00:17:39.544 --> 00:17:41.710 you look at the parent node and go, OK, that's fine. 00:17:41.710 --> 00:17:43.250 That's in the range. 00:17:43.250 --> 00:17:45.280 But every time it overflows, you can keep going. 00:17:45.280 --> 00:17:46.210 And how many times can you do this? 00:17:46.210 --> 00:17:48.180 You can do this all the way up to the root. 00:17:48.180 --> 00:17:51.710 And when you reach the root, either it's fine 00:17:51.710 --> 00:17:52.930 or the root is too big. 00:17:52.930 --> 00:17:54.840 It's reached 2B minus 1. 00:17:54.840 --> 00:17:56.680 And then you split the root, and you 00:17:56.680 --> 00:17:58.720 get one single [INAUDIBLE] up there. 00:17:58.720 --> 00:18:00.220 So that, in answer to your question, 00:18:00.220 --> 00:18:02.780 that is why you need that property in some sense. 00:18:02.780 --> 00:18:06.750 Not a very convincing argument, but sort of. 00:18:06.750 --> 00:18:08.720 So let's actually do an insertion 00:18:08.720 --> 00:18:11.540 in this tree we have here. 00:18:11.540 --> 00:18:13.600 So we are going to insert 16. 00:18:20.250 --> 00:18:23.130 So 16 comes in here. 00:18:23.130 --> 00:18:25.450 It's less than 30, it goes to the left. 00:18:25.450 --> 00:18:28.390 It's between 10 and 17, it goes in the middle. 00:18:28.390 --> 00:18:29.280 16. 00:18:29.280 --> 00:18:33.480 And it's greater than 14, so we add 16 here. 00:18:54.150 --> 00:18:54.920 All right. 00:18:54.920 --> 00:18:56.040 That seems good. 00:18:56.040 --> 00:18:57.570 All the properties fine. 00:18:57.570 --> 00:18:59.660 This still has two elements, which is the maximum, 00:18:59.660 --> 00:19:00.810 but it's good. 00:19:00.810 --> 00:19:03.324 It doesn't overflow. 00:19:03.324 --> 00:19:04.490 Let's insert something else. 00:19:04.490 --> 00:19:05.830 Let's insert 2. 00:19:08.600 --> 00:19:11.860 So 2 goes to 30, goes down, goes down. 00:19:11.860 --> 00:19:16.670 And we have a problem, because 2 has overflowed this node. 00:19:16.670 --> 00:19:18.590 So we split. 00:19:18.590 --> 00:19:21.367 And the way we split is we take the middle element. 00:19:21.367 --> 00:19:22.450 So we split the node here. 00:19:25.000 --> 00:19:31.680 And 3 goes up to the parent, so 3 goes here. 00:19:31.680 --> 00:19:38.015 And all good, except for the parent has overflowed. 00:19:38.015 --> 00:19:39.390 So what do we do with the parent? 00:19:39.390 --> 00:19:40.895 We split the parent again. 00:19:40.895 --> 00:19:44.870 And this time, it's right down the middle, the 10 goes up. 00:19:44.870 --> 00:19:48.520 So OK, let's get rid of this. 00:19:48.520 --> 00:19:53.090 So now that we split the parent, the 10 goes up here. 00:19:53.090 --> 00:19:54.490 And you're good. 00:19:54.490 --> 00:19:58.510 It's a bit cluttered, so let me reposition the 17. 00:20:07.940 --> 00:20:10.550 Did those two operations make sense? 00:20:10.550 --> 00:20:12.876 Questions? 00:20:12.876 --> 00:20:17.870 AUDIENCE: If your node size [INAUDIBLE] number of-- 00:20:17.870 --> 00:20:20.040 PROFESSOR: So just pick the-- first of all-- OK. 00:20:22.970 --> 00:20:25.387 If the way we're doing it-- when your node is overflowing, 00:20:25.387 --> 00:20:27.344 it's returning only one thing at a time, right? 00:20:27.344 --> 00:20:28.000 AUDIENCE: Yeah. 00:20:28.000 --> 00:20:29.749 PROFESSOR: So if your node is overflowing, 00:20:29.749 --> 00:20:32.587 it'll be 2t minus 1, which is an odd number always. 00:20:32.587 --> 00:20:34.670 There might be a case where you get an even number 00:20:34.670 --> 00:20:35.753 if you do something weird. 00:20:35.753 --> 00:20:38.200 Maybe you have a-- there are different ways to do B-trees. 00:20:38.200 --> 00:20:41.710 But if it does, you can probably pick the one, either of them, 00:20:41.710 --> 00:20:43.240 and then [INAUDIBLE]. 00:20:43.240 --> 00:20:44.240 I'm not sure about that. 00:20:44.240 --> 00:20:45.400 I'll look into it. 00:20:45.400 --> 00:20:48.910 But in general, if you're doing it this way, it's always odd. 00:20:48.910 --> 00:20:51.274 So you don't have to worry about that. 00:20:51.274 --> 00:20:53.908 Anything else? 00:20:53.908 --> 00:20:56.908 AUDIENCE: If we did reach all the way to the root 00:20:56.908 --> 00:20:58.150 and then went one more up-- 00:20:58.150 --> 00:20:59.030 PROFESSOR: So what you would do is-- 00:20:59.030 --> 00:21:00.446 AUDIENCE: That root would have-- 00:21:00.446 --> 00:21:03.070 PROFESSOR: That root would have two children, one element 00:21:03.070 --> 00:21:05.410 and two children, which is fine because we didn't put 00:21:05.410 --> 00:21:06.866 that restriction on the root. 00:21:06.866 --> 00:21:07.795 That's good. 00:21:07.795 --> 00:21:08.670 How we doing on time? 00:21:08.670 --> 00:21:11.480 OK, we have some time. 00:21:11.480 --> 00:21:14.966 Let's jump into deletion, unless anyone else has questions. 00:21:14.966 --> 00:21:16.840 AUDIENCE: [INAUDIBLE] any point? 00:21:16.840 --> 00:21:18.120 PROFESSOR: So-- oh, yeah. 00:21:18.120 --> 00:21:19.520 That's a good-- thank you. 00:21:19.520 --> 00:21:21.850 So you are going down to the leaves 00:21:21.850 --> 00:21:24.015 at most-- at most of the leaf ones, 00:21:24.015 --> 00:21:25.490 and you're going back up one. 00:21:25.490 --> 00:21:27.880 So it's like log n plus log n, and you're good. 00:21:31.610 --> 00:21:32.710 Let's do deletion. 00:21:43.150 --> 00:21:46.210 So deletion is more complicated. 00:21:46.210 --> 00:21:48.520 So the reason, it'll be clear. 00:21:48.520 --> 00:21:51.790 So the problem in deletion will be remove a node 00:21:51.790 --> 00:21:53.490 and a node is now underfull. 00:21:53.490 --> 00:21:57.400 So it has less than B minus 1 keys in it suddenly. 00:21:57.400 --> 00:22:00.600 So let's turn this around. 00:22:04.950 --> 00:22:07.980 So again B equal to 4. 00:22:07.980 --> 00:22:09.900 This node is a problem. 00:22:09.900 --> 00:22:11.610 Only two things in it. 00:22:11.610 --> 00:22:13.930 So what do we do? 00:22:13.930 --> 00:22:17.980 So before we go into that, let's make this assumption 00:22:17.980 --> 00:22:22.010 that-- there are two steps to deletion. 00:22:22.010 --> 00:22:24.950 The first step is making the deletion at a leaf. 00:22:24.950 --> 00:22:26.049 How do you do that? 00:22:26.049 --> 00:22:28.340 So the way you make a deletion at a leaf is, let's say, 00:22:28.340 --> 00:22:30.100 you have a key. 00:22:30.100 --> 00:22:34.470 You come down in your B-tree, and you add a node. 00:22:34.470 --> 00:22:38.017 Oh, this key needs to be deleted. 00:22:38.017 --> 00:22:38.850 But it's not a leaf. 00:22:38.850 --> 00:22:40.470 So what do you do? 00:22:40.470 --> 00:22:46.230 So essentially what you do is you look at these two subtrees. 00:22:46.230 --> 00:22:47.780 So it might have only one subtree. 00:22:47.780 --> 00:22:49.530 If it's at the end, it will have only one. 00:22:49.530 --> 00:22:51.360 Actually, no, that's not true. 00:22:51.360 --> 00:22:51.940 Ignore that. 00:22:51.940 --> 00:22:54.680 If it's not a leaf, it has two subtrees. 00:22:54.680 --> 00:22:57.130 So either take the rightmost element 00:22:57.130 --> 00:22:58.625 in this subtree, which is a leaf, 00:22:58.625 --> 00:23:00.250 because you can always keep going down, 00:23:00.250 --> 00:23:02.333 right, right, right, right till you get to a leaf, 00:23:02.333 --> 00:23:04.810 or the leftmost element in this subtree. 00:23:04.810 --> 00:23:09.310 So that is just the next element after this guy. 00:23:09.310 --> 00:23:13.700 So you delete this, and you bring this up to here. 00:23:13.700 --> 00:23:16.910 We'll do an example of this, and it'll be clearer. 00:23:16.910 --> 00:23:19.650 So you take either the rightmost element in the left subtree 00:23:19.650 --> 00:23:21.550 or the leftmost element in the right subtree 00:23:21.550 --> 00:23:22.620 and bring it up here. 00:23:22.620 --> 00:23:25.860 So you sort of like move the deletion to the leaf. 00:23:25.860 --> 00:23:27.310 And now it's easier to deal with. 00:23:27.310 --> 00:23:28.310 So we will come to that. 00:23:28.310 --> 00:23:32.990 Also just note that this is not what is done in the recitation. 00:23:32.990 --> 00:23:34.580 This algorithm for deletion, I think, 00:23:34.580 --> 00:23:35.880 is not done in the recitation notes. 00:23:35.880 --> 00:23:38.546 This is a different thing, which I'll send out a link for later. 00:23:38.546 --> 00:23:40.540 But I believe it works, because I got it 00:23:40.540 --> 00:23:44.500 from the [INAUDIBLE] reference. 00:23:44.500 --> 00:23:49.160 So once you move to the leaf-- so now let's look at this. 00:23:49.160 --> 00:23:51.510 So this is a node that is underfull. 00:23:51.510 --> 00:23:54.050 And you want to fix it. 00:23:54.050 --> 00:23:55.040 So how do you fix it? 00:23:55.040 --> 00:23:58.750 So what do is you look at its siblings. 00:23:58.750 --> 00:24:00.280 So in this case, it has one sibling. 00:24:00.280 --> 00:24:01.770 It can have up to two siblings. 00:24:01.770 --> 00:24:04.240 It can have left or right. 00:24:04.240 --> 00:24:06.390 So what you do is you look at a sibling. 00:24:06.390 --> 00:24:10.360 And this sibling is actually 1 over the minimum. 00:24:10.360 --> 00:24:13.460 And if it's 1 over the minimum, then it's really easy. 00:24:13.460 --> 00:24:15.840 All you have to do is take the leftmost thing here-- 00:24:15.840 --> 00:24:17.340 or if it's the sibling on this side, 00:24:17.340 --> 00:24:21.040 take the rightmost thing here. 00:24:21.040 --> 00:24:22.730 And look at its parent. 00:24:22.730 --> 00:24:30.330 So you bring the parent down, and you move the sibling up. 00:24:30.330 --> 00:24:31.220 And there we go. 00:24:31.220 --> 00:24:35.950 So you basically are rotating the thing into place. 00:24:35.950 --> 00:24:39.420 So you move the parent down into the underfull node, 00:24:39.420 --> 00:24:43.329 and you replace the parent by the leftmost thing here. 00:24:43.329 --> 00:24:45.120 Everyone see why that preserves everything? 00:24:50.150 --> 00:24:52.380 And the child is also shifted. 00:24:52.380 --> 00:24:53.490 Make sure you see that. 00:24:53.490 --> 00:24:57.190 So the child which was in this subtree is now in this subtree. 00:24:59.760 --> 00:25:01.770 But then you can have the situation 00:25:01.770 --> 00:25:04.620 where you don't have a nice sibling 00:25:04.620 --> 00:25:06.410 to take care of your problems. 00:25:06.410 --> 00:25:10.490 So in this scenario, the sibling is barely full. 00:25:10.490 --> 00:25:12.980 It has three things, and it can't donate anything to you. 00:25:12.980 --> 00:25:15.440 So what do you do in that case? 00:25:15.440 --> 00:25:18.170 So then you do something which is a parallel of the split 00:25:18.170 --> 00:25:18.730 operation. 00:25:18.730 --> 00:25:20.570 You do a merge. 00:25:20.570 --> 00:25:21.430 So what do you have? 00:25:21.430 --> 00:25:30.490 So here you have B minus 2, and here you have B minus 1. 00:25:30.490 --> 00:25:33.555 And you get 2B minus 3. 00:25:33.555 --> 00:25:34.930 Well, you've got another element. 00:25:34.930 --> 00:25:36.530 You also take the parent. 00:25:36.530 --> 00:25:37.040 So how do you do the merge. 00:25:37.040 --> 00:25:38.706 I just want to show you the merge first. 00:25:38.706 --> 00:25:41.891 So the way you do it is you move the parent down, 00:25:41.891 --> 00:25:42.890 and you merge these two. 00:25:46.220 --> 00:25:48.380 Seems OK? 00:25:48.380 --> 00:25:52.010 So you move the parent node down and merge these two. 00:25:52.010 --> 00:25:53.995 And, well, now this comes together, 00:25:53.995 --> 00:25:55.480 and this points into the new node. 00:25:58.400 --> 00:26:02.530 Sort of clear what's going on? 00:26:02.530 --> 00:26:05.112 Questions? 00:26:05.112 --> 00:26:06.088 Yes? 00:26:06.088 --> 00:26:09.815 AUDIENCE: So now the parent is underfull? 00:26:09.815 --> 00:26:11.690 PROFESSOR: Well, so you have-- yeah, exactly. 00:26:11.690 --> 00:26:13.990 So you have decreased the size of the parent, 00:26:13.990 --> 00:26:15.070 so it might be underfull. 00:26:15.070 --> 00:26:17.190 So you propagate. 00:26:17.190 --> 00:26:18.220 Anything else? 00:26:18.220 --> 00:26:20.630 AUDIENCE: So are these all different techniques 00:26:20.630 --> 00:26:21.774 for doing that? 00:26:21.774 --> 00:26:23.190 PROFESSOR: So there are two cases. 00:26:23.190 --> 00:26:27.110 So either you have a sibling which has extra nodes to donate 00:26:27.110 --> 00:26:29.036 to you or you don't. 00:26:29.036 --> 00:26:30.660 If you don't, then you have to do this. 00:26:30.660 --> 00:26:33.335 AUDIENCE: But what about that case? 00:26:33.335 --> 00:26:34.720 Or is that just like-- 00:26:34.720 --> 00:26:36.875 PROFESSOR: No, that is moving it down to the leaf. 00:26:36.875 --> 00:26:38.708 Once you move the deletion down to the leaf, 00:26:38.708 --> 00:26:40.300 so here we have something now. 00:26:40.300 --> 00:26:44.100 And now you move it all the way back up. 00:26:44.100 --> 00:26:45.720 So there are two cases. 00:26:45.720 --> 00:26:46.650 Let's do an example. 00:26:46.650 --> 00:26:48.850 That'll make it clearer. 00:26:48.850 --> 00:26:50.366 How are we doing on time? 00:26:50.366 --> 00:26:51.764 Five minutes, all right. 00:26:54.280 --> 00:26:58.435 So we are going to delete 38. 00:26:58.435 --> 00:27:00.840 38 is gone. 00:27:00.840 --> 00:27:02.840 But we want to move it down to the leaf. 00:27:02.840 --> 00:27:04.580 So let's take an element. 00:27:04.580 --> 00:27:07.810 Let's say we take 41. 00:27:07.810 --> 00:27:13.560 So we take 41 and move it up here. 00:27:13.560 --> 00:27:17.482 41 is the leftmost thing in the right subtree. 00:27:17.482 --> 00:27:19.440 So this vacancy doesn't really affect anything, 00:27:19.440 --> 00:27:22.381 because this node still has the right number of things, 00:27:22.381 --> 00:27:24.630 because it's still got one thing in it, which is good. 00:27:24.630 --> 00:27:25.920 So you're fine. 00:27:25.920 --> 00:27:29.080 This is now just 48. 00:27:29.080 --> 00:27:32.850 Let's say we now delete 41. 00:27:32.850 --> 00:27:36.540 So 41 is gone. 00:27:36.540 --> 00:27:41.480 So now that 41 is gone, what do you 00:27:41.480 --> 00:27:42.850 replace this blank spot with? 00:27:47.840 --> 00:27:49.390 Either this or this, right? 00:27:49.390 --> 00:27:50.520 Doesn't matter. 00:27:50.520 --> 00:27:53.900 So let's just do this one for consistency. 00:27:53.900 --> 00:27:55.420 So you have 48 here. 00:27:55.420 --> 00:27:59.230 And now you a problem because you have a blank box. 00:27:59.230 --> 00:28:03.590 So can you rotate? 00:28:03.590 --> 00:28:04.460 Yes, no? 00:28:04.460 --> 00:28:05.870 No, right? 00:28:05.870 --> 00:28:09.860 Because sibling is barely full. 00:28:09.860 --> 00:28:11.000 So what can you do? 00:28:11.000 --> 00:28:12.016 So you merge. 00:28:12.016 --> 00:28:12.890 And how do you merge? 00:28:12.890 --> 00:28:17.067 You move the 48 down, and you combine everything. 00:28:17.067 --> 00:28:18.650 So this is kind of hard to understand, 00:28:18.650 --> 00:28:20.940 but this is like a zero-element node. 00:28:23.960 --> 00:28:26.120 So when you merge, you have 32, 48, and nothing, 00:28:26.120 --> 00:28:28.070 so it's just 32 and 48. 00:28:28.070 --> 00:28:38.770 So what you do is-- so this seems weird, 00:28:38.770 --> 00:28:40.910 but this is just another empty node. 00:28:40.910 --> 00:28:43.075 You just propagated the emptiness upwards. 00:28:46.520 --> 00:28:49.860 Now you take this empty node, and you look for its siblings. 00:28:49.860 --> 00:28:54.330 Again, its sibling is-- well, it's barely full. 00:28:54.330 --> 00:28:55.330 So what do you do now? 00:28:55.330 --> 00:28:57.790 You bring the 30 down, and you merge this. 00:28:57.790 --> 00:28:58.950 So let's do that. 00:29:06.080 --> 00:29:10.765 30 comes down, and there we go. 00:29:13.410 --> 00:29:14.245 Looks fine? 00:29:14.245 --> 00:29:15.940 Does that tree look good? 00:29:15.940 --> 00:29:17.190 Questions about the operation? 00:29:20.400 --> 00:29:27.050 I'm sure it was not clear, but-- anything? 00:29:27.050 --> 00:29:28.370 Make sense? 00:29:28.370 --> 00:29:32.730 OK, let's do a deletion where we can actually do a rotation. 00:29:32.730 --> 00:29:36.200 So let's go ahead and delete 20. 00:29:36.200 --> 00:29:38.205 So you do your searching, go down the tree. 00:29:38.205 --> 00:29:40.910 You find the 20 under here. 00:29:40.910 --> 00:29:43.300 So now, OK. 00:29:43.300 --> 00:29:45.620 So you're left with just-- actually never mind. 00:29:45.620 --> 00:29:46.570 We'll do another one. 00:29:46.570 --> 00:29:47.770 So this doesn't do anything. 00:29:47.770 --> 00:29:51.020 You lost the 20, and you're left with the 24 this time. 00:29:51.020 --> 00:29:53.359 So now you delete the 24. 00:29:53.359 --> 00:29:54.900 So now that you've got rid of the 24, 00:29:54.900 --> 00:29:56.150 you have a blank box here now. 00:29:58.380 --> 00:30:00.310 But its sibling is not barely full. 00:30:00.310 --> 00:30:02.750 It has something to donate. 00:30:02.750 --> 00:30:06.864 So anyone, which elements are going to rotate? 00:30:06.864 --> 00:30:07.840 AUDIENCE: 17 and 16. 00:30:07.840 --> 00:30:09.403 PROFESSOR: 16 and 17, right. 00:30:09.403 --> 00:30:10.370 Cool. 00:30:10.370 --> 00:30:16.060 So 16 goes up, 17 goes down. 00:30:16.060 --> 00:30:17.260 And you're done. 00:30:17.260 --> 00:30:18.260 You're consistent again. 00:30:20.850 --> 00:30:21.880 So that was deletion. 00:30:21.880 --> 00:30:24.180 Those are the two cases for deletion. 00:30:24.180 --> 00:30:26.752 Does that make sense? 00:30:26.752 --> 00:30:29.220 Anyone? 00:30:29.220 --> 00:30:31.020 Any questions? 00:30:31.020 --> 00:30:32.290 OK. 00:30:32.290 --> 00:30:36.420 So that's all the topics we were supposed to cover today. 00:30:36.420 --> 00:30:39.810 Any questions about any of the operations, 00:30:39.810 --> 00:30:43.520 any of the other topics, lecture, anything?