WEBVTT 00:00:00.070 --> 00:00:01.770 The following content is provided 00:00:01.770 --> 00:00:04.010 under a Creative Commons license. 00:00:04.010 --> 00:00:06.860 Your support will help MIT OpenCourseWare continue 00:00:06.860 --> 00:00:10.720 to offer high quality educational resources for free. 00:00:10.720 --> 00:00:13.330 To make a donation or view additional materials 00:00:13.330 --> 00:00:17.207 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.207 --> 00:00:17.832 at ocw.mit.edu. 00:00:21.800 --> 00:00:22.500 PROFESSOR: OK. 00:00:22.500 --> 00:00:25.150 Who's going to lecture? 00:00:25.150 --> 00:00:26.642 Wow. 00:00:26.642 --> 00:00:27.290 Nice. 00:00:27.290 --> 00:00:29.500 Does lecture make sense? 00:00:29.500 --> 00:00:31.720 One, two, three-- sort of. 00:00:31.720 --> 00:00:32.759 OK. 00:00:32.759 --> 00:00:35.080 Then I'm going to have an easy job. 00:00:35.080 --> 00:00:38.020 So any pointed questions? 00:00:38.020 --> 00:00:40.510 Any specific pinpoints? 00:00:40.510 --> 00:00:43.540 We talked about DFS, but DFS and BFS are sort of related, 00:00:43.540 --> 00:00:45.642 so I'm happy to take both. 00:00:45.642 --> 00:00:47.800 AUDIENCE: The classification of edges? 00:00:47.800 --> 00:00:48.474 PROFESSOR: OK. 00:00:48.474 --> 00:00:50.140 We're going to talk about that for sure. 00:01:02.050 --> 00:01:04.459 Anything else? 00:01:04.459 --> 00:01:07.710 AUDIENCE: I thought it was funny that in undirected graphs, 00:01:07.710 --> 00:01:10.745 you could have a backward-- or something like that. 00:01:10.745 --> 00:01:12.620 PROFESSOR: You don't have to give a radiance. 00:01:12.620 --> 00:01:14.120 I will derive it. 00:01:14.120 --> 00:01:20.290 OK So what edges do we have in directed versus undirected. 00:01:24.630 --> 00:01:25.130 OK. 00:01:25.130 --> 00:01:25.629 What else? 00:01:30.370 --> 00:01:30.950 Cool. 00:01:30.950 --> 00:01:34.260 So I'm going to go through the concepts in DFS really quickly. 00:01:34.260 --> 00:01:36.160 And we're going to focus on this. 00:01:36.160 --> 00:01:38.960 Because it seems like this is where the issues are. 00:01:41.500 --> 00:01:42.310 I want this. 00:01:42.310 --> 00:01:43.235 So what's a graph? 00:01:46.262 --> 00:01:48.367 AUDIENCE: Interconnected notes. 00:01:48.367 --> 00:01:48.950 PROFESSOR: OK. 00:01:48.950 --> 00:01:49.620 Fancy names. 00:01:49.620 --> 00:01:49.940 Come on. 00:01:49.940 --> 00:01:51.315 We had this in the last workshop. 00:01:51.315 --> 00:01:54.810 AUDIENCE: A set of edges and vertices. 00:01:54.810 --> 00:01:56.100 PROFESSOR: All right. 00:01:56.100 --> 00:01:58.426 How do I draw my vertices? 00:01:58.426 --> 00:01:59.302 AUDIENCE: Dots. 00:01:59.302 --> 00:02:01.098 Circles. 00:02:01.098 --> 00:02:02.562 PROFESSOR: Dots. 00:02:02.562 --> 00:02:05.978 How do I draw my edges? 00:02:05.978 --> 00:02:06.942 AUDIENCE: Dots. 00:02:06.942 --> 00:02:07.442 Lines. 00:02:10.050 --> 00:02:10.550 Oops. 00:02:42.960 --> 00:02:44.580 PROFESSOR: Looks like a graph? 00:02:44.580 --> 00:02:47.460 What kind of graph? 00:02:47.460 --> 00:02:47.960 Directed. 00:02:47.960 --> 00:02:50.500 Because there are arrows and not straight up lines. 00:02:50.500 --> 00:02:51.400 Right? 00:02:51.400 --> 00:02:53.620 How do we store graphs in Python? 00:02:57.300 --> 00:03:00.450 Fancy name first and then implementation. 00:03:00.450 --> 00:03:01.580 AUDIENCE: [INAUDIBLE] list. 00:03:01.580 --> 00:03:02.163 PROFESSOR: OK. 00:03:09.020 --> 00:03:12.300 What's an adjacency list in Python? 00:03:12.300 --> 00:03:14.940 AUDIENCE: It just shows what notes 00:03:14.940 --> 00:03:16.807 are adjacent to other notes. 00:03:16.807 --> 00:03:17.390 PROFESSOR: OK. 00:03:17.390 --> 00:03:19.710 So what data structure do we use? 00:03:19.710 --> 00:03:22.442 What elementary python data structure do we use? 00:03:22.442 --> 00:03:23.400 AUDIENCE: A dictionary? 00:03:23.400 --> 00:03:24.420 PROFESSOR: All right. 00:03:24.420 --> 00:03:29.298 So an adjacency list is a dictionary that keys our-- 00:03:29.298 --> 00:03:30.760 AUDIENCE: [INAUDIBLE]. 00:03:30.760 --> 00:03:32.500 PROFESSOR: OK. 00:03:32.500 --> 00:03:35.520 So the keys are the vertices. 00:03:39.284 --> 00:03:41.365 So on and so forth. 00:03:41.365 --> 00:03:42.365 And what are the values? 00:03:45.508 --> 00:03:48.376 AUDIENCE: The nodes? 00:03:48.376 --> 00:03:50.097 Adjacent nodes? 00:03:50.097 --> 00:03:50.680 PROFESSOR: OK. 00:03:54.160 --> 00:03:57.840 A list of something. 00:03:57.840 --> 00:03:59.970 So after our presentation, is the object the one 00:03:59.970 --> 00:04:02.550 that has edges and the simplified 00:04:02.550 --> 00:04:04.640 one that has vertices. 00:04:04.640 --> 00:04:07.220 So suppose I want to go for the simple one. 00:04:07.220 --> 00:04:10.840 Where vertices do I have in A's list? 00:04:10.840 --> 00:04:12.420 AUDIENCE: B and G. Right? 00:04:15.940 --> 00:04:17.606 PROFESSOR: B and G. Sounds good. 00:04:20.152 --> 00:04:20.652 Right. 00:04:20.652 --> 00:04:24.466 What vertices do I have in B's list? 00:04:24.466 --> 00:04:28.920 AUDIENCE: C. 00:04:28.920 --> 00:04:32.094 PROFESSOR: Why do I not have A? 00:04:32.094 --> 00:04:36.714 AUDIENCE: Because it's an arrow, so it can't get to A. 00:04:36.714 --> 00:04:37.380 PROFESSOR: Yeah. 00:04:37.380 --> 00:04:39.625 So the nodes that I have in the list 00:04:39.625 --> 00:04:42.860 are the nodes that are reachable from that node. 00:04:42.860 --> 00:04:45.610 I have an edge from B to C, so C is reachable from B. 00:04:45.610 --> 00:04:48.510 I do not have an edge from B to A 00:04:48.510 --> 00:04:49.760 because my edges are directed. 00:04:49.760 --> 00:04:51.881 So A is not there. 00:04:51.881 --> 00:04:52.380 OK. 00:04:52.380 --> 00:04:56.480 Let's write a bit of pseudo- So although it's Python, 00:04:56.480 --> 00:05:01.480 let's say our graph is represented by a class G. Let's 00:05:01.480 --> 00:05:05.450 say in this class I have a dictionary called 00:05:05.450 --> 00:05:09.420 al, which has what we want. 00:05:09.420 --> 00:05:12.820 If I want to get all the vertices in the graph, 00:05:12.820 --> 00:05:15.300 I want to write the method that returns all the vertices. 00:05:19.040 --> 00:05:21.650 What should it look like? 00:05:21.650 --> 00:05:24.784 Just to make sure that we're all getting the data structure. 00:05:24.784 --> 00:05:25.700 AUDIENCE: [INAUDIBLE]. 00:05:25.700 --> 00:05:26.283 PROFESSOR: OK. 00:05:28.417 --> 00:05:29.750 So the keys are in the vertices. 00:05:34.840 --> 00:05:37.110 Now, given the vertex, I want its neighbors. 00:05:42.240 --> 00:05:44.278 How do I write that? 00:05:44.278 --> 00:05:47.632 AUDIENCE: You mean only the vertices that it can go to? 00:05:47.632 --> 00:05:48.298 PROFESSOR: Yeah. 00:05:51.526 --> 00:05:52.150 AUDIENCE: Yeah. 00:05:52.150 --> 00:05:54.400 Dictionary and that vertex. 00:05:54.400 --> 00:05:58.955 PROFESSOR: So return dictionary-- 00:05:58.955 --> 00:06:04.244 AUDIENCE: al bracket vertex v. 00:06:04.244 --> 00:06:05.760 PROFESSOR: Cool. 00:06:05.760 --> 00:06:06.260 OK. 00:06:06.260 --> 00:06:07.926 So this makes sense for everyone, right? 00:06:12.110 --> 00:06:13.920 Let's start writing DFS because no one 00:06:13.920 --> 00:06:15.960 said they don't understand DFS. 00:06:15.960 --> 00:06:22.320 So let's say we have a DFS procedure. 00:06:22.320 --> 00:06:25.930 And it gets a graph. 00:06:25.930 --> 00:06:27.040 And it has to do DFS. 00:06:29.840 --> 00:06:31.070 What do you need for DFS? 00:06:33.680 --> 00:06:36.240 Starting points would be good. 00:06:36.240 --> 00:06:39.060 Just like for DFS, right? 00:06:39.060 --> 00:06:41.740 So how do we do DFS? 00:06:47.030 --> 00:06:49.700 AUDIENCE: Start with one node and then we 00:06:49.700 --> 00:06:53.220 keep on following a path down by picking the first vertex. 00:06:53.220 --> 00:06:55.870 So like for A, we'd start with A. 00:06:55.870 --> 00:06:58.715 Then we'd pick the first vertex in its list. 00:06:58.715 --> 00:07:02.600 And then we pick the first vertex in that one's list, 00:07:02.600 --> 00:07:05.243 first vertex in that one's list, until we run out. 00:07:05.243 --> 00:07:05.826 PROFESSOR: OK. 00:07:05.826 --> 00:07:08.890 So basically you start today. 00:07:08.890 --> 00:07:12.220 And for every neighbor, you recursively 00:07:12.220 --> 00:07:14.500 do DFS on that neighbor. 00:07:14.500 --> 00:07:15.000 Right? 00:07:15.000 --> 00:07:17.700 So since this is recursive, let's write it 00:07:17.700 --> 00:07:20.190 in a recursive manner, nice and easy. 00:07:20.190 --> 00:07:29.090 So say DFS calls DFS visit of g and s. 00:07:29.090 --> 00:07:37.990 And let's say DFS visit takes the graph, 00:07:37.990 --> 00:07:40.020 takes the node that we're visiting, 00:07:40.020 --> 00:07:43.030 and how are we implementing this? 00:07:46.530 --> 00:07:50.270 AUDIENCE: Just return [INAUDIBLE]. 00:07:50.270 --> 00:07:50.770 No. 00:07:50.770 --> 00:07:53.197 It marks as visited. 00:07:53.197 --> 00:07:53.780 PROFESSOR: OK. 00:07:53.780 --> 00:07:56.870 Marking something as visited is important to lead this. 00:07:56.870 --> 00:08:00.910 So what's the main thing that you do in DFS? 00:08:00.910 --> 00:08:02.750 Iterate over the neighbors and recurse. 00:08:02.750 --> 00:08:03.380 Right? 00:08:03.380 --> 00:08:25.950 So for n in g neighbors of v called DFS visit of g and n. 00:08:25.950 --> 00:08:29.870 So this is the basic idea. 00:08:29.870 --> 00:08:33.140 It has some holes that we need to fill because otherwise it 00:08:33.140 --> 00:08:34.080 doesn't quite work. 00:08:34.080 --> 00:08:36.330 This is going to recurse forever, 00:08:36.330 --> 00:08:38.940 which is not very good if you have a time 00:08:38.940 --> 00:08:41.549 limit that your code needs to obey. 00:08:41.549 --> 00:08:45.529 So how do I make it not recurse forever? 00:08:45.529 --> 00:08:48.070 AUDIENCE: Every time you visit, you're about to visit a node, 00:08:48.070 --> 00:08:51.215 check if you've already visited it. 00:08:51.215 --> 00:08:52.090 PROFESSOR: All right. 00:08:52.090 --> 00:08:53.589 So we should keep track of the nodes 00:08:53.589 --> 00:08:56.690 that we've already visited and not visit them again. 00:08:56.690 --> 00:08:57.190 Right. 00:08:57.190 --> 00:09:02.332 So you start with A, go to B, go to C, 00:09:02.332 --> 00:09:04.540 you'd better not go to A again because you've already 00:09:04.540 --> 00:09:06.980 been there. 00:09:06.980 --> 00:09:08.420 So let's implement that this way. 00:09:08.420 --> 00:09:12.650 Let's make a new object called DFS result that 00:09:12.650 --> 00:09:17.560 is sort of that drawer that you have 00:09:17.560 --> 00:09:19.160 where you stick everything in where 00:09:19.160 --> 00:09:20.564 you don't have a good place. 00:09:20.564 --> 00:09:22.230 So everything I do not have a good place 00:09:22.230 --> 00:09:24.240 to put it, you just stick it in a drawer. 00:09:24.240 --> 00:09:25.512 And then you can close the drawing that be like, 00:09:25.512 --> 00:09:26.180 it's clean. 00:09:26.180 --> 00:09:27.430 Everything's clean. 00:09:27.430 --> 00:09:29.777 So let's say DFS result is that drawer. 00:09:29.777 --> 00:09:31.360 Everything that they need from now on, 00:09:31.360 --> 00:09:33.130 we're just going to stick in there. 00:09:33.130 --> 00:09:39.100 So I have an object DFS result. 00:09:39.100 --> 00:09:41.100 And we're going to figure out what we put in it. 00:09:41.100 --> 00:09:44.995 But at the beginning of DFS, I'm going to create that. 00:09:49.060 --> 00:09:50.490 So create the new object. 00:09:50.490 --> 00:09:54.110 And then I'm going to pass it onto to DFS visit. 00:09:59.740 --> 00:10:04.790 So now we have a drawer where we can put everything we want. 00:10:04.790 --> 00:10:10.610 So let's keep track of the visited information here. 00:10:10.610 --> 00:10:14.770 How would I keep track of what nodes I visited? 00:10:14.770 --> 00:10:16.150 AUDIENCE: In a list? 00:10:16.150 --> 00:10:17.910 PROFESSOR: In a list. 00:10:17.910 --> 00:10:22.530 So if I do a list, it turns out that in DFS, 00:10:22.530 --> 00:10:24.170 and VFS, for that matter, you check 00:10:24.170 --> 00:10:25.760 if you visited a node pretty often. 00:10:25.760 --> 00:10:28.130 So it would be kind of slow. 00:10:28.130 --> 00:10:29.270 So let's have a dictionary. 00:10:29.270 --> 00:10:31.103 What are we going to put in that dictionary? 00:10:33.694 --> 00:10:37.422 AUDIENCE: The node [INAUDIBLE] and listed [INAUDIBLE]. 00:10:37.422 --> 00:10:38.005 PROFESSOR: OK. 00:10:38.005 --> 00:10:43.710 So we're going to have a visited dictionary. 00:10:43.710 --> 00:10:46.040 And then the keys are going to be nodes. 00:10:46.040 --> 00:10:49.320 And let's say we're only going to put the nodes that we visit 00:10:49.320 --> 00:10:52.080 in it so we don't have to initialize it. 00:10:52.080 --> 00:10:53.580 And the values are going to be true. 00:10:53.580 --> 00:10:55.288 Because if the node is in the dictionary, 00:10:55.288 --> 00:10:59.090 it means we visited it. 00:10:59.090 --> 00:11:00.920 So for all the nodes that are in there, 00:11:00.920 --> 00:11:02.500 visited is going to be true. 00:11:06.024 --> 00:11:06.523 Sorry. 00:11:10.140 --> 00:11:10.640 OK. 00:11:10.640 --> 00:11:14.490 So where do I fill this in? 00:11:14.490 --> 00:11:15.060 Let's use it. 00:11:19.090 --> 00:11:26.470 AUDIENCE: Four n and g neighbors, set it to true. 00:11:26.470 --> 00:11:27.910 But before that-- 00:11:27.910 --> 00:11:29.920 PROFESSOR: So here? 00:11:29.920 --> 00:11:33.280 AUDIENCE: R not. 00:11:33.280 --> 00:11:36.047 Um. 00:11:36.047 --> 00:11:36.630 PROFESSOR: OK. 00:11:40.530 --> 00:11:41.430 So let's say this. 00:11:41.430 --> 00:11:44.090 Let's say these are lines 1, 2, 3. 00:11:44.090 --> 00:11:47.920 Line 1.5 or 2.5? 00:11:47.920 --> 00:11:54.840 AUDIENCE: So at n 1.5, check if v is in the dictionary. 00:11:54.840 --> 00:11:55.520 PROFESSOR: Here? 00:11:55.520 --> 00:11:57.110 OK. 00:11:57.110 --> 00:12:06.646 If v in r dot visited, then-- 00:12:09.017 --> 00:12:09.725 AUDIENCE: Return. 00:12:19.360 --> 00:12:26.750 Then 2.5, set r dot n to true. 00:12:26.750 --> 00:12:31.114 You would only do it n times-- 00:12:31.114 --> 00:12:31.780 PROFESSOR: Wait. 00:12:31.780 --> 00:12:35.983 So here you're saying say r dot visited of n equals true? 00:12:35.983 --> 00:12:37.462 AUDIENCE: Oh, no. 00:12:37.462 --> 00:12:40.913 Change the v in v visited return, 00:12:40.913 --> 00:12:42.885 change that into a loop. 00:12:42.885 --> 00:12:44.910 Put that after the loop. 00:12:44.910 --> 00:12:48.392 Put that into the loop. 00:12:48.392 --> 00:12:52.368 And it's n visited to the [INAUDIBLE]. 00:12:59.350 --> 00:13:08.804 PROFESSOR: n in r dot visited-- we probably want not n, right? 00:13:08.804 --> 00:13:09.720 AUDIENCE: [INAUDIBLE]. 00:13:14.147 --> 00:13:14.730 PROFESSOR: OK. 00:13:19.670 --> 00:13:28.637 R dot visited is true and then the DFS. 00:13:28.637 --> 00:13:31.220 So will this achieve the thing that I wanted at the beginning. 00:13:31.220 --> 00:13:33.080 You start at A, you go to B, you go to C, 00:13:33.080 --> 00:13:35.280 and then you do not go to A? 00:13:35.280 --> 00:13:37.000 Not quite. 00:13:37.000 --> 00:13:39.020 AUDIENCE: We haven't had our starting yet. 00:13:39.020 --> 00:13:39.520 Yeah. 00:13:43.414 --> 00:13:48.529 We could add V at the beginning and check at the end. 00:13:48.529 --> 00:13:52.310 Because that's only the visited, right? 00:13:52.310 --> 00:13:54.860 PROFESSOR: So I think I like Christian's suggestion. 00:13:54.860 --> 00:14:02.760 r dot visited of v is true. 00:14:02.760 --> 00:14:04.260 And then we can remove it from here. 00:14:07.704 --> 00:14:11.640 AUDIENCE: It has to be [INAUDIBLE] 00:14:11.640 --> 00:14:14.840 because the other case is going to cover 00:14:14.840 --> 00:14:16.405 for if you visited again. 00:14:16.405 --> 00:14:21.855 But the first time it's being called, it's not checking-- 00:14:21.855 --> 00:14:23.730 PROFESSOR: Well you mean the first time here? 00:14:23.730 --> 00:14:26.000 AUDIENCE: The first call that's being made, 00:14:26.000 --> 00:14:27.914 and the recursive call, it's going 00:14:27.914 --> 00:14:30.434 to be as if it's not into the visited node. 00:14:30.434 --> 00:14:31.100 PROFESSOR: Yeah. 00:14:31.100 --> 00:14:34.090 But this is at the beginning. 00:14:34.090 --> 00:14:35.130 Right? 00:14:35.130 --> 00:14:38.210 So this calls DFS visit to the starting node. 00:14:38.210 --> 00:14:38.810 Do we care? 00:14:42.340 --> 00:14:43.550 So I think this should work. 00:14:43.550 --> 00:14:46.091 It's not the only way to write it, but it's reasonably clean. 00:14:46.091 --> 00:14:50.370 It's easy to reason about, and it works. 00:14:50.370 --> 00:14:52.630 Does it make sense to everyone? 00:14:52.630 --> 00:14:53.130 OK. 00:14:53.130 --> 00:14:54.879 We're going to change things a little bit. 00:14:54.879 --> 00:14:56.690 So instead of using visited, we're 00:14:56.690 --> 00:15:00.050 going to keep track for every node of its parent. 00:15:00.050 --> 00:15:03.000 So the node that we visited from. 00:15:03.000 --> 00:15:04.750 So we're not going to use visited anymore. 00:15:04.750 --> 00:15:10.220 Instead we're going to use parent of v 00:15:10.220 --> 00:15:13.370 is going to be some other node. 00:15:17.930 --> 00:15:21.130 So what I want is for this graph, 00:15:21.130 --> 00:15:24.220 I said I'm going to go from A to B, and then from B to C. 00:15:24.220 --> 00:15:32.340 So I want the parent of B to be A, and the parent of C to be B. 00:15:32.340 --> 00:15:34.020 Make sense for everyone? 00:15:34.020 --> 00:15:36.230 So what's the parent of A? 00:15:36.230 --> 00:15:37.110 AUDIENCE: None. 00:15:37.110 --> 00:15:39.180 PROFESSOR: None's a good value, right? 00:15:39.180 --> 00:15:41.100 There's no parent. 00:15:41.100 --> 00:15:42.090 OK. 00:15:42.090 --> 00:15:44.880 So how would I modify this code to write that? 00:15:52.730 --> 00:15:59.940 AUDIENCE: Stick a line in 2.75 and establish 3b 00:15:59.940 --> 00:16:03.370 as the parent of n. 00:16:03.370 --> 00:16:08.300 PROFESSOR: So r dot parent? 00:16:08.300 --> 00:16:12.700 AUDIENCE: Yeah, of n is v. 00:16:12.700 --> 00:16:15.480 PROFESSOR: Of n is v. Good. 00:16:15.480 --> 00:16:17.200 And I don't have visited anymore, 00:16:17.200 --> 00:16:19.050 so this line is going to blow up. 00:16:19.050 --> 00:16:20.250 So what do I do instead? 00:16:23.080 --> 00:16:24.510 AUDIENCE: Oh. 00:16:24.510 --> 00:16:28.030 It doesn't have a parent? 00:16:28.030 --> 00:16:28.700 PROFESSOR: Yep. 00:16:28.700 --> 00:16:31.525 So if it doesn't have a parent, we didn't visit it. 00:16:31.525 --> 00:16:33.150 So the parents works just like visited, 00:16:33.150 --> 00:16:35.108 except the values are not going to be all true. 00:16:35.108 --> 00:16:38.015 They're going to be something a bit more useful. 00:16:38.015 --> 00:16:41.232 AUDIENCE: We can't test for none because the first one has 00:16:41.232 --> 00:16:41.899 a value of none. 00:16:41.899 --> 00:16:43.731 PROFESSOR: But they're not testing for none, 00:16:43.731 --> 00:16:45.689 we're saying is it in the dictionary or not? 00:16:45.689 --> 00:16:46.230 AUDIENCE: Oh. 00:16:46.230 --> 00:16:46.730 OK. 00:16:52.359 --> 00:16:53.650 PROFESSOR: That's a good point. 00:16:53.650 --> 00:16:54.230 You could be testing for none. 00:16:54.230 --> 00:16:55.640 That wouldn't work. 00:16:55.640 --> 00:16:58.370 So you have to write the check like this. 00:16:58.370 --> 00:17:00.954 And we're going to erase this because otherwise, it's 00:17:00.954 --> 00:17:02.120 going to throw an exception. 00:17:06.210 --> 00:17:08.955 And what else do we need? 00:17:08.955 --> 00:17:10.079 There's one missing parent. 00:17:14.534 --> 00:17:15.937 AUDIENCE: Inverse form? 00:17:15.937 --> 00:17:16.520 PROFESSOR: OK. 00:17:16.520 --> 00:17:18.696 Where do I say that? 00:17:18.696 --> 00:17:21.460 AUDIENCE: In DFS. 00:17:21.460 --> 00:17:23.060 PROFESSOR: OK. 00:17:23.060 --> 00:17:26.599 One, two, three, which line? 00:17:26.599 --> 00:17:30.380 1.5, 2.5, 3.5? 00:17:30.380 --> 00:17:31.375 AUDIENCE: 2.5. 00:17:31.375 --> 00:17:32.000 PROFESSOR: 2.5. 00:17:32.000 --> 00:17:34.476 All right, so here. 00:17:34.476 --> 00:17:36.023 What do I write? 00:17:39.470 --> 00:17:39.970 Yes? 00:17:42.538 --> 00:17:44.980 AUDIENCE: Parent of s is none. 00:17:44.980 --> 00:17:46.700 PROFESSOR: OK. 00:17:46.700 --> 00:17:54.890 So r dot parent s is none. 00:17:54.890 --> 00:17:57.820 So now this works. 00:17:57.820 --> 00:17:58.510 Right? 00:17:58.510 --> 00:17:59.650 Any questions so far? 00:18:03.470 --> 00:18:03.970 Nope. 00:18:03.970 --> 00:18:05.360 Basic DFS works. 00:18:05.360 --> 00:18:06.650 Everyone's happy with it? 00:18:11.460 --> 00:18:16.970 Let's try to track it for this graph here. 00:18:16.970 --> 00:18:19.180 And you're going to be in a better position than me 00:18:19.180 --> 00:18:21.480 because I'm not sure I can see the graph all 00:18:21.480 --> 00:18:23.990 the way from here. 00:18:23.990 --> 00:18:25.460 Almost. 00:18:25.460 --> 00:18:30.750 So let's have the parents or the parent's dictionary here. 00:18:35.560 --> 00:18:39.830 And here, let's write the call structure as it will happen. 00:18:39.830 --> 00:18:43.920 So where do I start? 00:18:43.920 --> 00:18:46.840 DFS of A, right? 00:18:46.840 --> 00:18:56.130 DFS of A. And this is going to call DFS visit A. 00:18:56.130 --> 00:18:57.130 What are A's neighbors? 00:18:59.890 --> 00:19:00.872 AUDIENCE: b and g. 00:19:03.914 --> 00:19:04.580 PROFESSOR: Cool. 00:19:04.580 --> 00:19:07.480 Is b in parents? 00:19:07.480 --> 00:19:09.802 What's in parents, at this point? 00:19:09.802 --> 00:19:12.122 AUDIENCE: Just a. 00:19:12.122 --> 00:19:14.770 Oh. 00:19:14.770 --> 00:19:16.020 PROFESSOR: A is none. 00:19:16.020 --> 00:19:16.666 Good. 00:19:16.666 --> 00:19:18.290 Excellent. 00:19:18.290 --> 00:19:21.560 So B is not in parents, so what am I going to do? 00:19:21.560 --> 00:19:23.572 And it's the parents in visited. 00:19:23.572 --> 00:19:24.530 Right? 00:19:24.530 --> 00:19:25.767 What's B's parents? 00:19:28.629 --> 00:19:31.500 AUDIENCE: A. 00:19:31.500 --> 00:19:32.380 PROFESSOR: OK. 00:19:32.380 --> 00:19:38.480 I'm going to call DFS visit of B. These children. 00:19:43.360 --> 00:19:45.940 C. Is C parents? 00:19:45.940 --> 00:19:46.440 No? 00:19:46.440 --> 00:19:47.022 What happens? 00:19:47.022 --> 00:19:47.980 Feel free to take over. 00:19:47.980 --> 00:19:49.813 When you get it, feel free to start talking, 00:19:49.813 --> 00:19:51.263 and I can just write. 00:19:51.263 --> 00:19:52.347 AUDIENCE: Of C. 00:19:52.347 --> 00:19:52.930 PROFESSOR: OK. 00:19:56.660 --> 00:19:58.737 What else? 00:19:58.737 --> 00:20:03.860 PROFESSOR: C's parent is B. So I'm 00:20:03.860 --> 00:20:06.810 in B. The only child is C. C is not in parents. 00:20:06.810 --> 00:20:07.960 C gets in parent. 00:20:07.960 --> 00:20:13.270 C's parent is B. Call DFS visit of C, what are C's children? 00:20:13.270 --> 00:20:14.770 AUDIENCE: [INAUDIBLE] and d. 00:20:17.002 --> 00:20:19.210 PROFESSOR: So we're wondering about the order, right? 00:20:19.210 --> 00:20:20.980 Let's assume that in the adjacency list, 00:20:20.980 --> 00:20:24.500 all the nodes are in alphabetical order. 00:20:24.500 --> 00:20:34.930 So I'm going to have A, D, and F. OK. 00:20:34.930 --> 00:20:37.870 Is A in parents? 00:20:37.870 --> 00:20:39.960 AUDIENCE: Yes. 00:20:39.960 --> 00:20:42.729 PROFESSOR: Is D in parents? 00:20:42.729 --> 00:20:43.270 AUDIENCE: No. 00:20:43.270 --> 00:20:45.543 PROFESSOR: So what happens? 00:20:45.543 --> 00:20:54.390 AUDIENCE: Parents visit D. D's parent is C. 00:20:54.390 --> 00:20:56.664 PROFESSOR: D's neighbors? 00:20:56.664 --> 00:21:02.400 AUDIENCE: E and F. 00:21:02.400 --> 00:21:04.800 PROFESSOR: E is in the-- no. 00:21:11.354 --> 00:21:12.270 AUDIENCE: [INAUDIBLE]. 00:21:19.140 --> 00:21:19.980 PROFESSOR: Cool. 00:21:19.980 --> 00:21:21.790 Does E have any children? 00:21:21.790 --> 00:21:22.610 AUDIENCE: No. 00:21:22.610 --> 00:21:23.760 PROFESSOR: So I'm done. 00:21:23.760 --> 00:21:27.310 I get out of it, right? 00:21:27.310 --> 00:21:30.112 Is F in parents? 00:21:30.112 --> 00:21:33.088 AUDIENCE: No. 00:21:33.088 --> 00:21:36.460 PROFESSOR: F's parent is? 00:21:36.460 --> 00:21:36.960 Cool. 00:21:43.464 --> 00:21:44.630 F doesn't have any children. 00:21:44.630 --> 00:21:45.220 Right? 00:21:45.220 --> 00:21:49.210 So I get out of this DFS visit. 00:21:49.210 --> 00:21:51.800 I'm done with F. So I get out of this visit. 00:21:51.800 --> 00:21:53.655 And I get back to this one. 00:21:53.655 --> 00:21:58.060 I'm done with D, and I'm at F. Is F in parents? 00:21:58.060 --> 00:21:59.240 OK. 00:21:59.240 --> 00:22:05.594 So what kind of edge is C to F? 00:22:05.594 --> 00:22:06.344 AUDIENCE: Forward. 00:22:08.812 --> 00:22:09.395 PROFESSOR: OK. 00:22:12.690 --> 00:22:14.569 So forward maybe tree. 00:22:14.569 --> 00:22:15.860 Let's start with the easy ones. 00:22:15.860 --> 00:22:20.790 So now we've gone forward with AB, BC, CD, DE, and DF. 00:22:20.790 --> 00:22:22.792 What kind of edges are these ones? 00:22:26.520 --> 00:22:27.020 Tree edges. 00:22:32.540 --> 00:22:34.520 Does anyone know why they're called tree edges? 00:22:38.504 --> 00:22:41.970 So if you look at the parent pointers that they have there, 00:22:41.970 --> 00:22:46.280 they're going to end up defining a tree. 00:22:46.280 --> 00:22:48.530 And the tree is, the tree shows the order 00:22:48.530 --> 00:22:51.350 in which DFS looked at the notes. 00:22:51.350 --> 00:22:56.330 So let me erase this and draw the tree for DFS 00:22:56.330 --> 00:22:57.210 that we had so far. 00:23:01.480 --> 00:23:05.450 So we started at A, then we went to B. Then we went to C, then 00:23:05.450 --> 00:23:12.200 we went to D, E, F. So this is all part of the DFS tree.You 00:23:12.200 --> 00:23:13.940 can see that for all these nodes, 00:23:13.940 --> 00:23:17.090 their parent pointers point the right way. 00:23:17.090 --> 00:23:19.010 So if you have parent pointers for every node, 00:23:19.010 --> 00:23:21.940 this is going to give you a tree. 00:23:21.940 --> 00:23:24.950 So these are tree edges because they're part of this tree. 00:23:24.950 --> 00:23:30.270 So now we have this edge from C to F. 00:23:30.270 --> 00:23:32.780 And this is what kind of edge? 00:23:36.127 --> 00:23:36.668 Forward edge. 00:23:36.668 --> 00:23:39.752 Why is it a forward edge? 00:23:39.752 --> 00:23:42.142 AUDIENCE: Somewhere up higher in the tree 00:23:42.142 --> 00:23:45.010 is trying to get somewhere lower on the tree. 00:23:45.010 --> 00:23:46.560 It's going forward in time. 00:23:46.560 --> 00:23:49.740 PROFESSOR: So it's going from a node in the tree 00:23:49.740 --> 00:23:53.142 to a node's child in the tree. 00:23:53.142 --> 00:23:53.850 Sorry, not child. 00:23:53.850 --> 00:23:54.710 Descendant. 00:23:54.710 --> 00:23:59.250 So F is strictly under C. So yeah. 00:23:59.250 --> 00:24:00.480 It takes us forward in time. 00:24:00.480 --> 00:24:02.240 It's a shortcut in the tree. 00:24:02.240 --> 00:24:05.400 So that's why it's a forward edge. 00:24:05.400 --> 00:24:07.260 So we now have two types of edges so far. 00:24:07.260 --> 00:24:09.650 Tree edges and forward edges. 00:24:09.650 --> 00:24:13.370 AUDIENCE: So tree edges [INAUDIBLE]? 00:24:13.370 --> 00:24:14.151 PROFESSOR: Yep. 00:24:14.151 --> 00:24:16.150 So the tree edges are the ones that DFS follows. 00:24:19.640 --> 00:24:21.730 All of these are tree edges. 00:24:21.730 --> 00:24:22.230 Yes. 00:24:22.230 --> 00:24:25.831 AUDIENCE: So like an edge from B to C, 00:24:25.831 --> 00:24:28.186 why can't that be a forward edge? 00:24:28.186 --> 00:24:29.128 Because technically it's not a shortcut, right? 00:24:29.128 --> 00:24:31.044 But it's saying it's pointing to the children, 00:24:31.044 --> 00:24:32.430 but it only has one child. 00:24:32.430 --> 00:24:35.810 PROFESSOR: Because that's the edge that DFS took. 00:24:35.810 --> 00:24:38.180 So the edges that we actually follow in DFS 00:24:38.180 --> 00:24:41.980 have a special name, and they're tree edges. 00:24:41.980 --> 00:24:43.680 So first off, the most important edges 00:24:43.680 --> 00:24:46.480 are the edges that make up your DFS tree. 00:24:46.480 --> 00:24:49.867 And that's why those are tree edges. 00:24:49.867 --> 00:24:51.450 We don't even look at the other types. 00:24:51.450 --> 00:24:54.310 If you have an edge and DFS followed it, it's a tree edge. 00:24:54.310 --> 00:24:56.340 Done. 00:24:56.340 --> 00:24:57.930 So these are all tree edges. 00:24:57.930 --> 00:24:59.679 We don't care about the other definitions. 00:25:03.810 --> 00:25:06.140 So don't be ashamed to ask questions about this 00:25:06.140 --> 00:25:08.220 because this is hard to understand. 00:25:08.220 --> 00:25:10.410 It's an issue we're trying to clarify here. 00:25:10.410 --> 00:25:12.370 So the point, why we're doing that example, 00:25:12.370 --> 00:25:15.150 is to figure this out. 00:25:15.150 --> 00:25:16.770 So please do ask your questions. 00:25:16.770 --> 00:25:17.520 Yes? 00:25:17.520 --> 00:25:21.036 AUDIENCE: So is it edges that are followed at any point 00:25:21.036 --> 00:25:23.358 during the search process? 00:25:23.358 --> 00:25:26.985 I guess they're edges that never get touched. 00:25:26.985 --> 00:25:27.770 Right? 00:25:27.770 --> 00:25:28.395 PROFESSOR: Yep. 00:25:28.395 --> 00:25:31.320 So when I'm at C, and I look at this edge, 00:25:31.320 --> 00:25:32.990 F is already in parents. 00:25:32.990 --> 00:25:35.622 So this edge is not going to be followed by DFS. 00:25:35.622 --> 00:25:37.080 There are some other types of edges 00:25:37.080 --> 00:25:38.163 that will not be followed. 00:25:38.163 --> 00:25:41.812 And we'll get to that by the time DFS completes. 00:25:41.812 --> 00:25:44.282 Yes. 00:25:44.282 --> 00:25:48.710 AUDIENCE: You mean that like G to C is not-- 00:25:48.710 --> 00:25:50.460 PROFESSOR: You're ruining my example, man. 00:25:50.460 --> 00:25:53.670 I'm supposed to ask you that later on. 00:25:53.670 --> 00:25:56.360 So let's let go through it and see what kind of edge it is. 00:25:56.360 --> 00:25:58.790 You're right, but let's pretend we don't know that. 00:25:58.790 --> 00:26:01.720 And we'll see as we follow the DFS. 00:26:01.720 --> 00:26:02.220 OK. 00:26:02.220 --> 00:26:04.107 So yep. 00:26:04.107 --> 00:26:05.190 That one's a forward edge. 00:26:05.190 --> 00:26:06.900 We'll deal with it later. 00:26:06.900 --> 00:26:08.380 We're here. 00:26:08.380 --> 00:26:12.510 We decided that C to F is a forward edge. 00:26:12.510 --> 00:26:13.630 F is already in parents. 00:26:13.630 --> 00:26:15.420 So we're going to return from DFS visit 00:26:15.420 --> 00:26:21.530 C. We're in DFS visit V. We're done with C. 00:26:21.530 --> 00:26:24.830 Do we have anything else to do here? 00:26:24.830 --> 00:26:31.790 So we go up to A. We're done with B. And we look at G. A 00:26:31.790 --> 00:26:35.490 to G. Is G in parents? 00:26:35.490 --> 00:26:35.990 No. 00:26:35.990 --> 00:26:40.060 So we're going to call it. 00:26:40.060 --> 00:26:52.955 So A, B. DFS visit G. What are G's neighbors? 00:26:52.955 --> 00:26:58.410 AUDIENCE: C. 00:26:58.410 --> 00:27:00.330 PROFESSOR: OK. 00:27:00.330 --> 00:27:05.090 So is C in parents? 00:27:07.776 --> 00:27:08.650 Actually I was wrong. 00:27:08.650 --> 00:27:09.780 Sorry. 00:27:09.780 --> 00:27:13.275 So C is in parents, so we're not going to visit it. 00:27:13.275 --> 00:27:15.400 So let's see how the tree looks like at this point. 00:27:20.000 --> 00:27:23.090 So it looks like this. 00:27:23.090 --> 00:27:27.995 Is C a child of G? 00:27:31.480 --> 00:27:35.270 So I'm wondering about this edge that we chose not to follow. 00:27:35.270 --> 00:27:39.400 We followed A to G so we know this is actually the edge. 00:27:39.400 --> 00:27:43.640 So from this edge, G to C, I'm wondering what kind of edge 00:27:43.640 --> 00:27:44.626 it is. 00:27:49.560 --> 00:27:52.490 So C is not G's child, right? 00:27:52.490 --> 00:27:53.770 So another forward edge. 00:27:53.770 --> 00:27:54.270 Sorry. 00:27:54.270 --> 00:27:55.728 I got confused earlier because they 00:27:55.728 --> 00:27:58.140 look the same in a drawing. 00:27:58.140 --> 00:28:02.220 So it's not a forward edge, then it's a cross edge. 00:28:06.629 --> 00:28:08.170 There's one more edge that we haven't 00:28:08.170 --> 00:28:10.770 talked about because it was too early in the search. 00:28:10.770 --> 00:28:18.000 And that edge is from C to A. So if you remember 00:28:18.000 --> 00:28:23.310 there, we were in DFS visit C. And the first thing we did 00:28:23.310 --> 00:28:26.410 was we looked at A, we said, hey it's already in parents, 00:28:26.410 --> 00:28:28.170 so we're not going to visit it. 00:28:28.170 --> 00:28:31.557 So DFS did not follow the edge C to A. 00:28:31.557 --> 00:28:32.640 That edge looks like this. 00:28:32.640 --> 00:28:35.398 What kind of edge is it? 00:28:35.398 --> 00:28:37.749 Back edge. 00:28:37.749 --> 00:28:38.374 AUDIENCE: Wait. 00:28:38.374 --> 00:28:41.040 So it has to be direct descendant in order 00:28:41.040 --> 00:28:42.946 to be a forward edge? 00:28:42.946 --> 00:28:47.810 Because they are related through A, right? 00:28:47.810 --> 00:28:49.357 C and G? 00:28:49.357 --> 00:28:51.940 PROFESSOR: So you have to look and which direction does it go? 00:28:51.940 --> 00:28:54.900 Does it go down the tree or up the tree? 00:28:54.900 --> 00:29:00.460 AUDIENCE: It's going down if you consider G to C. From G to C, 00:29:00.460 --> 00:29:02.334 so it's a cross edge. 00:29:02.334 --> 00:29:03.000 PROFESSOR: Yeah. 00:29:03.000 --> 00:29:07.139 But is C G's descendant? 00:29:07.139 --> 00:29:07.680 AUDIENCE: No. 00:29:07.680 --> 00:29:10.790 But they are related. 00:29:10.790 --> 00:29:13.090 PROFESSOR: They're related, but they're just 00:29:13.090 --> 00:29:15.621 related because they're in the tree. 00:29:15.621 --> 00:29:16.162 AUDIENCE: OK. 00:29:16.162 --> 00:29:18.820 So it's only directly descendants? 00:29:18.820 --> 00:29:20.140 PROFESSOR: Yep. 00:29:20.140 --> 00:29:22.900 So DFS puts together a tree. 00:29:22.900 --> 00:29:25.910 A forward edge is a shortcut in that tree. 00:29:25.910 --> 00:29:29.160 It lets you go forward in the VFS. 00:29:29.160 --> 00:29:32.660 A backward edge points to a parent in the tree, 00:29:32.660 --> 00:29:34.860 so it lets you go back in time. 00:29:34.860 --> 00:29:37.950 A cross edge takes you from one point to another point. 00:29:37.950 --> 00:29:40.120 And it's not a forward edge, not a backward edge. 00:29:40.120 --> 00:29:44.180 So it takes you in a whole different world. 00:29:44.180 --> 00:29:49.500 So C, A, G, has-- A, G is the sub-tree, 00:29:49.500 --> 00:29:51.880 and then this guy's this other sub-tree, 00:29:51.880 --> 00:29:54.660 and they're different little worlds. 00:29:54.660 --> 00:29:57.500 And this edge goes from one to the other. 00:30:00.161 --> 00:30:02.660 So the algorithm to decide what kind of an edge it is, let's 00:30:02.660 --> 00:30:04.380 put it together. 00:30:04.380 --> 00:30:08.800 Which edge do we look at, which type do we think about first? 00:30:08.800 --> 00:30:09.400 Tree edge. 00:30:09.400 --> 00:30:10.920 So DFS followed the edge. 00:30:10.920 --> 00:30:12.940 It's a tree edge. 00:30:12.940 --> 00:30:15.980 So we have tree edges. 00:30:15.980 --> 00:30:18.730 What do we do next? 00:30:18.730 --> 00:30:25.420 So say we have an edge from u to v. If DFS follows it, 00:30:25.420 --> 00:30:26.620 it's a tree edge. 00:30:26.620 --> 00:30:31.228 If not, then what question do I ask myself? 00:30:31.228 --> 00:30:33.713 AUDIENCE: It could be one of three edges then, 00:30:33.713 --> 00:30:34.877 back, forward, or cross. 00:30:34.877 --> 00:30:35.460 PROFESSOR: OK. 00:30:35.460 --> 00:30:37.380 So let's try to write a simple algorithm 00:30:37.380 --> 00:30:38.880 so that if I ask you this on a quiz, 00:30:38.880 --> 00:30:40.846 you can decide which is which. 00:30:40.846 --> 00:30:44.916 AUDIENCE: So I think that you look forward 00:30:44.916 --> 00:30:47.627 if u is the parent of v. 00:30:47.627 --> 00:30:48.210 PROFESSOR: OK. 00:30:48.210 --> 00:30:58.852 So if u is a parent of v-- well, let's say ascendant. 00:30:58.852 --> 00:31:02.860 It's not just the direct parent, but this 00:31:02.860 --> 00:31:04.180 is the right intuition. 00:31:04.180 --> 00:31:06.200 Then what kind of edge is it? 00:31:06.200 --> 00:31:08.760 AUDIENCE: Then it's the forward edge. 00:31:08.760 --> 00:31:12.360 PROFESSOR: So the edge is from u to v. If u is a parent of v, 00:31:12.360 --> 00:31:13.040 forward edge. 00:31:13.040 --> 00:31:13.540 Good. 00:31:17.076 --> 00:31:17.575 Otherwise? 00:31:20.960 --> 00:31:21.876 AUDIENCE: [INAUDIBLE]. 00:31:25.557 --> 00:31:26.140 PROFESSOR: OK. 00:31:35.400 --> 00:31:36.429 Then? 00:31:36.429 --> 00:31:39.363 AUDIENCE: Then backward edge. 00:31:39.363 --> 00:31:40.830 Else cross edge. 00:31:48.380 --> 00:31:50.060 PROFESSOR: Does this make sense? 00:31:50.060 --> 00:31:52.130 So these are the four types of edges. 00:31:52.130 --> 00:31:54.690 DFS takes the edge, it's a forward edge. 00:31:54.690 --> 00:31:56.560 If not, we draw the DFS tree. 00:31:56.560 --> 00:31:59.510 And we see does the edge go forward 00:31:59.510 --> 00:32:00.880 in the tree, forward edge. 00:32:00.880 --> 00:32:03.160 Does it go backward, backward edge. 00:32:03.160 --> 00:32:07.935 Does it grow between completely unrelated nodes, cross edge. 00:32:11.351 --> 00:32:11.850 OK. 00:32:11.850 --> 00:32:13.710 Now let me ask another question. 00:32:13.710 --> 00:32:16.870 Let's suppose we do this DFS thing again, 00:32:16.870 --> 00:32:19.830 but let's suppose we do it on an undirected graph. 00:32:19.830 --> 00:32:23.582 What types of edges do I have? 00:32:23.582 --> 00:32:26.954 Anyone remember off the top of their heads? 00:32:26.954 --> 00:32:27.870 Don't think so, right? 00:32:27.870 --> 00:32:29.036 So let's do it and find out. 00:32:44.740 --> 00:32:51.350 So let me see where do I get some room. 00:32:55.380 --> 00:32:55.880 Here. 00:33:02.364 --> 00:33:04.280 Let's write the tree that's we're going to go. 00:33:04.280 --> 00:33:07.310 So let's do a VFS of this quickly and write the tree. 00:33:07.310 --> 00:33:10.630 So we start at A, then what? 00:33:14.362 --> 00:33:14.862 Then? 00:33:18.280 --> 00:33:19.990 Then? 00:33:19.990 --> 00:33:25.445 D. E. 00:33:25.445 --> 00:33:33.637 AUDIENCE: And go back to D. It could go from C to A. 00:33:33.637 --> 00:33:34.220 PROFESSOR: OK. 00:33:34.220 --> 00:33:38.330 So we have C to A. What kind of edge is this? 00:33:38.330 --> 00:33:40.387 AUDIENCE: Backward edge. 00:33:40.387 --> 00:33:40.970 PROFESSOR: OK. 00:33:40.970 --> 00:33:44.550 So we still have tree edges for sure and backward edges. 00:33:47.872 --> 00:33:51.210 AUDIENCE: How do you get from C to A? 00:33:51.210 --> 00:33:53.570 PROFESSOR: This was supposed to happen way before. 00:33:53.570 --> 00:33:59.644 So we went A, B, C, and then when we were at C, oh. 00:33:59.644 --> 00:34:01.910 Sorry. 00:34:01.910 --> 00:34:04.970 My bad. 00:34:04.970 --> 00:34:07.280 So we went from A to B, B to C, and then A 00:34:07.280 --> 00:34:09.179 was the first thing in C's adjacency list. 00:34:09.179 --> 00:34:10.980 So we saw this back edge. 00:34:10.980 --> 00:34:14.204 So C to D, D to E, D to F. 00:34:14.204 --> 00:34:15.638 AUDIENCE: E to F? 00:34:15.638 --> 00:34:17.550 Or F to C? 00:34:17.550 --> 00:34:20.334 PROFESSOR: And then F to C. So what kind of edge is this? 00:34:20.334 --> 00:34:21.125 AUDIENCE: Backward. 00:34:23.690 --> 00:34:26.560 PROFESSOR: So none of these guys. 00:34:26.560 --> 00:34:32.883 And then we're all the way back at A, right? 00:34:32.883 --> 00:34:37.521 A to-- oh. 00:34:37.521 --> 00:34:38.020 Sorry. 00:34:38.020 --> 00:34:40.800 No we're not done. 00:34:40.800 --> 00:34:50.070 So C to D, D to E, D to F, then we're back at C. And C to G. 00:34:50.070 --> 00:34:53.719 AUDIENCE: There's supposed to be a line between C and F? 00:34:53.719 --> 00:34:54.860 PROFESSOR: Oh, yeah. 00:34:54.860 --> 00:34:55.480 Sorry. 00:34:55.480 --> 00:34:57.410 I guess I can't copy. 00:34:57.410 --> 00:35:01.382 One, two, three, four, five, six, seven, eight, nine. 00:35:01.382 --> 00:35:03.590 One, two, three, four, five, six, seven, eight, nine. 00:35:03.590 --> 00:35:05.880 Now it's right. 00:35:05.880 --> 00:35:07.350 Thank you. 00:35:07.350 --> 00:35:07.850 OK. 00:35:07.850 --> 00:35:09.860 So what am I missing? 00:35:09.860 --> 00:35:18.795 Edge from C to G. What kind of edge? 00:35:18.795 --> 00:35:20.194 AUDIENCE: C and G? 00:35:22.607 --> 00:35:23.190 PROFESSOR: OK. 00:35:23.190 --> 00:35:28.460 So we have tree edges and we have backward edges. 00:35:28.460 --> 00:35:29.630 What do we not have? 00:35:29.630 --> 00:35:32.520 Forward edges and cross edges, right? 00:35:32.520 --> 00:35:39.392 AUDIENCE: You need one from A to G. A and G are related. 00:35:39.392 --> 00:35:40.600 PROFESSOR: What edge is this? 00:35:43.030 --> 00:35:43.530 Cool. 00:35:43.530 --> 00:35:44.210 Thanks. 00:35:44.210 --> 00:35:47.030 AUDIENCE: You never add those, though, right? 00:35:47.030 --> 00:35:51.814 You just see that A was in the parent's list, and be like, OK. 00:35:51.814 --> 00:35:52.480 PROFESSOR: Yeah. 00:35:52.480 --> 00:35:54.570 So for all the edges that are not tree edges, 00:35:54.570 --> 00:35:57.540 DFS doesn't actually follow them. 00:35:57.540 --> 00:35:59.720 We just care about them because other algorithms 00:35:59.720 --> 00:36:00.620 care about them. 00:36:00.620 --> 00:36:03.640 They let you compute fancy things on the graph. 00:36:03.640 --> 00:36:05.640 OK. 00:36:05.640 --> 00:36:09.270 So are we good with the types? 00:36:09.270 --> 00:36:13.165 So why can't I have a forward edge? 00:36:13.165 --> 00:36:15.040 So there are two types of edges I can't have, 00:36:15.040 --> 00:36:18.110 cross edges and forward edges. 00:36:18.110 --> 00:36:20.770 Why can't I have a forward edge? 00:36:24.109 --> 00:36:26.017 On an undirected graph? 00:36:28.880 --> 00:36:30.740 So in order to have a forward edge, 00:36:30.740 --> 00:36:37.060 I would have to go C, D, F. And then not follow this edge. 00:36:37.060 --> 00:36:39.410 I mean not see this edge here. 00:36:39.410 --> 00:36:42.240 Right? 00:36:42.240 --> 00:36:43.760 If this edge would be undirected, 00:36:43.760 --> 00:36:47.520 I would say, hey, it goes from F to C. It's a background edge. 00:36:47.520 --> 00:36:50.660 So I would have seen this edge when I would have been in C. 00:36:50.660 --> 00:36:52.450 So I'm comparing this tree with this tree, 00:36:52.450 --> 00:36:54.408 and trying to figure out why they're different. 00:36:57.250 --> 00:37:00.033 How does a forward edge look like? 00:37:00.033 --> 00:37:05.480 You have a node u, then you have some more tree stuff. 00:37:05.480 --> 00:37:09.000 And from here, you got to v. And then when you're at v, 00:37:09.000 --> 00:37:11.421 you didn't see the edge. 00:37:11.421 --> 00:37:11.920 Right? 00:37:11.920 --> 00:37:13.336 Because if you would have seen it, 00:37:13.336 --> 00:37:15.330 it would have been a background edge. 00:37:15.330 --> 00:37:18.470 So you can only see the edge from u to v later on. 00:37:18.470 --> 00:37:21.160 So this is tree edges, a bunch of tree edges, 00:37:21.160 --> 00:37:23.010 and this is a forward edge. 00:37:23.010 --> 00:37:24.480 In an undirected graph, this never 00:37:24.480 --> 00:37:27.325 happens because when you're at v, 00:37:27.325 --> 00:37:29.070 you're going to see the edge. 00:37:29.070 --> 00:37:32.120 And you're going to mark it as a backward edge. 00:37:32.120 --> 00:37:33.550 So forward edges can ever happen. 00:37:37.490 --> 00:37:38.210 Yes? 00:37:38.210 --> 00:37:40.130 No? 00:37:40.130 --> 00:37:42.680 Is everyone happy? 00:37:42.680 --> 00:37:43.450 So cross edges. 00:37:43.450 --> 00:37:45.400 Why can't they happen? 00:37:45.400 --> 00:37:48.070 A cross edge can never happen because in order 00:37:48.070 --> 00:37:51.740 for a cross edge to happen, I would have to go A, B, C, 00:37:51.740 --> 00:37:56.750 visit C's children, then go up and go somewhere else. 00:37:56.750 --> 00:37:59.200 And then see this edge. 00:37:59.200 --> 00:38:02.420 But, hey, when I was at C, why didn't I see this edge? 00:38:02.420 --> 00:38:03.930 Why did I only see it later? 00:38:03.930 --> 00:38:07.140 If it's an undirected graph, I would see this edge here 00:38:07.140 --> 00:38:09.230 and it would be a forward edge. 00:38:09.230 --> 00:38:09.730 Sorry. 00:38:09.730 --> 00:38:12.580 A tree edge, because that would take it. 00:38:12.580 --> 00:38:13.170 Yes? 00:38:13.170 --> 00:38:15.169 AUDIENCE: So is it just when you're-- is it just 00:38:15.169 --> 00:38:17.850 because it's the way you traverse in DFS or would it 00:38:17.850 --> 00:38:20.920 apply generally to VFS as well? 00:38:20.920 --> 00:38:24.130 PROFESSOR: VFS doesn't have forward and backward. 00:38:24.130 --> 00:38:24.630 Yeah. 00:38:24.630 --> 00:38:27.390 VFS is completely different. 00:38:27.390 --> 00:38:31.730 So these are all DFS terms, purely DFS. 00:38:31.730 --> 00:38:32.230 OK. 00:38:32.230 --> 00:38:34.400 So no forward edges, no cross edges. 00:38:34.400 --> 00:38:36.930 And if you forget which ones you can and can't have, 00:38:36.930 --> 00:38:40.830 now you know how to reason about it quickly and remember. 00:38:40.830 --> 00:38:42.770 Yes? 00:38:42.770 --> 00:38:45.480 OK. 00:38:45.480 --> 00:38:46.440 So this is DFS. 00:38:46.440 --> 00:38:49.130 Are we all happy with DFS? 00:38:49.130 --> 00:38:51.890 Let's talk about topological sorting then. 00:38:51.890 --> 00:38:53.180 Because it's really useful. 00:38:53.180 --> 00:38:55.400 It's one of the few algorithms that is really useful 00:38:55.400 --> 00:38:58.330 and that you might have to write yourself later. 00:38:58.330 --> 00:39:01.650 So suppose these are classes. 00:39:01.650 --> 00:39:03.390 So let's get back to this oriented thing 00:39:03.390 --> 00:39:05.390 and suppose these are classes. 00:39:05.390 --> 00:39:08.670 And the edges show prerequisites. 00:39:08.670 --> 00:39:13.590 So A is a prerequisite of B. Like say A is 601 00:39:13.590 --> 00:39:14.880 and B is 6006. 00:39:14.880 --> 00:39:17.610 And you have to take 601 before you take 6006. 00:39:17.610 --> 00:39:21.690 Otherwise you will cry during programming assignments. 00:39:21.690 --> 00:39:24.190 So what we want to do is these are 00:39:24.190 --> 00:39:27.030 all the classes you need to graduate. 00:39:27.030 --> 00:39:30.027 We need to come up with an order in which you can take them 00:39:30.027 --> 00:39:32.610 so that when you take a class, you took all the prerequisites. 00:39:32.610 --> 00:39:35.790 So you don't cry while you're taking that class. 00:39:35.790 --> 00:39:37.480 How do we do that? 00:39:37.480 --> 00:39:39.435 And let's use this graph as an example. 00:39:43.167 --> 00:39:48.477 AUDIENCE: Use a directed graph that's acyclic? 00:39:48.477 --> 00:39:49.060 PROFESSOR: OK. 00:39:49.060 --> 00:39:52.520 Is this graph acyclic? 00:39:52.520 --> 00:39:54.247 AUDIENCE: No. 00:39:54.247 --> 00:39:54.830 PROFESSOR: OK. 00:39:54.830 --> 00:39:57.089 So what then? 00:39:57.089 --> 00:39:58.005 AUDIENCE: [INAUDIBLE]. 00:40:03.190 --> 00:40:06.054 PROFESSOR: So if the graph has a cycle, 00:40:06.054 --> 00:40:07.220 you can't solve the problem. 00:40:07.220 --> 00:40:09.432 It has no solution. 00:40:09.432 --> 00:40:11.140 AUDIENCE: It's like those cardboard boxes 00:40:11.140 --> 00:40:13.640 that [INAUDIBLE]. 00:40:13.640 --> 00:40:16.140 It's really annoying. 00:40:16.140 --> 00:40:21.015 PROFESSOR: I don't know the cardboard box problem. 00:40:21.015 --> 00:40:21.640 AUDIENCE: Yeah. 00:40:21.640 --> 00:40:25.510 You know those cardboard boxes with the four flaps 00:40:25.510 --> 00:40:31.110 when you close them, and they close like-- Yeah. 00:40:31.110 --> 00:40:33.900 Those are so annoying. 00:40:33.900 --> 00:40:36.462 That's just what this reminds me of. 00:40:36.462 --> 00:40:37.040 That's all. 00:40:37.040 --> 00:40:37.380 Continue. 00:40:37.380 --> 00:40:38.506 PROFESSOR: There's a cycle there, 00:40:38.506 --> 00:40:40.000 so you can't just put them in an order. 00:40:40.000 --> 00:40:40.500 Right? 00:40:40.500 --> 00:40:42.870 You have to violate the repeating variance somehow. 00:40:42.870 --> 00:40:45.950 And you do that by twisting the edges. 00:40:45.950 --> 00:40:50.160 So if we had classes A, B, C, you cannot take them in any 00:40:50.160 --> 00:40:50.660 order. 00:40:50.660 --> 00:40:51.220 Right? 00:40:51.220 --> 00:40:54.790 If you take A first, you would need C. If you take B first, 00:40:54.790 --> 00:40:58.090 you would need A. If you take C first, you would need B. 00:40:58.090 --> 00:41:00.820 So you can't start with any of them. 00:41:00.820 --> 00:41:03.340 So if you have a cycle in a graph, 00:41:03.340 --> 00:41:04.580 it's not a dependency graph. 00:41:04.580 --> 00:41:07.540 You can't compute dependencies. 00:41:07.540 --> 00:41:11.040 So topological sorts only works on cyclic graphs. 00:41:15.610 --> 00:41:19.900 So acyclic graph. 00:41:19.900 --> 00:41:21.360 And what else? 00:41:21.360 --> 00:41:25.220 So a dependency graph is a special kind of graph. 00:41:25.220 --> 00:41:28.106 And I'm looking for three fancy words. 00:41:28.106 --> 00:41:28.980 You already have two. 00:41:28.980 --> 00:41:31.635 So we have acyclic and we have graph. 00:41:31.635 --> 00:41:32.135 Directed. 00:41:36.144 --> 00:41:37.560 So you have to have a directed, so 00:41:37.560 --> 00:41:40.060 that you know which class goes first. 00:41:40.060 --> 00:41:42.582 Needs to be acyclic, otherwise there's no solution. 00:41:42.582 --> 00:41:44.040 Needs to be a graph, because that's 00:41:44.040 --> 00:41:45.081 what we're talking about. 00:41:48.600 --> 00:41:50.490 So the shortening for this is a DAG. 00:41:53.030 --> 00:41:53.530 OK. 00:41:53.530 --> 00:41:55.820 How do you compute an order? 00:41:55.820 --> 00:41:57.570 So say we remove that edge. 00:42:06.408 --> 00:42:09.767 How do we compute an order? 00:42:09.767 --> 00:42:16.134 AUDIENCE: You run DFS and then print out the reverse. 00:42:16.134 --> 00:42:17.883 PROFESSOR: And then print out the reverse? 00:42:17.883 --> 00:42:20.520 AUDIENCE: Of your final output. 00:42:20.520 --> 00:42:22.928 PROFESSOR: Which is? 00:42:22.928 --> 00:42:24.720 AUDIENCE: Your final output? 00:42:24.720 --> 00:42:29.180 You want me to do it on a graph? 00:42:29.180 --> 00:42:31.825 PROFESSOR: You can tell me how to change the codes, too. 00:42:31.825 --> 00:42:34.450 So I'm not sure what-- you said print the reverse of something, 00:42:34.450 --> 00:42:35.672 but what's that thing? 00:42:35.672 --> 00:42:37.640 AUDIENCE: So I go down a path, right, 00:42:37.640 --> 00:42:39.517 until you've hit all the nodes, then 00:42:39.517 --> 00:42:41.689 basically you start from the last node 00:42:41.689 --> 00:42:44.930 you visited and print out the reverse of that. 00:42:44.930 --> 00:42:46.350 Go back to the first node. 00:42:46.350 --> 00:42:46.933 PROFESSOR: OK. 00:42:46.933 --> 00:42:53.050 So you're saying I go A, B, C, D, E, and then 00:42:53.050 --> 00:42:55.631 I print E, D, C, B, A? 00:42:55.631 --> 00:42:56.553 AUDIENCE: No. 00:42:56.553 --> 00:42:58.532 Because that wouldn't hit all the nodes. 00:42:58.532 --> 00:42:59.115 PROFESSOR: OK. 00:43:02.763 --> 00:43:07.130 He's on the right track, by the way, so that's why we're-- 00:43:07.130 --> 00:43:10.700 AUDIENCE: You go A, B, C, D, E, F, 00:43:10.700 --> 00:43:15.860 G, and you print G, F, E, D, C, B, A. 00:43:15.860 --> 00:43:19.480 PROFESSOR: OK so we print them in the reverse. 00:43:19.480 --> 00:43:22.250 So you're printing them in the order in which you're 00:43:22.250 --> 00:43:24.760 done with them after DFS? 00:43:24.760 --> 00:43:25.260 No. 00:43:25.260 --> 00:43:26.968 You're printing them in the reverse order 00:43:26.968 --> 00:43:30.090 of the order which you visited them? 00:43:30.090 --> 00:43:33.101 So you visited A, B, C, D, E, F, G, 00:43:33.101 --> 00:43:35.920 so you're saying print G, F, E, D, C, B, A. 00:43:35.920 --> 00:43:37.889 AUDIENCE: Yeah. 00:43:37.889 --> 00:43:39.430 Because in lecture wasn't it flipped, 00:43:39.430 --> 00:43:44.805 so if an arrow points from A to B then A depends on B. 00:43:44.805 --> 00:43:47.180 And we're looking at the other way around. 00:43:50.130 --> 00:43:51.730 In lecture, based on the graph you 00:43:51.730 --> 00:43:55.572 gave, if the arrow goes from A to B, then A depends on B. 00:43:55.572 --> 00:43:58.906 PROFESSOR: If the arrow goes from A to B, then what? 00:43:58.906 --> 00:44:02.045 AUDIENCE: If you're saying A is a prerequisite for B and G, 00:44:02.045 --> 00:44:05.450 then we just sweep from left to right and say, 00:44:05.450 --> 00:44:06.610 I have to do this. 00:44:06.610 --> 00:44:08.309 And then those two. 00:44:08.309 --> 00:44:08.892 PROFESSOR: No. 00:44:08.892 --> 00:44:11.808 They have to-- 00:44:11.808 --> 00:44:15.585 AUDIENCE: It's the order that-- because when you're doing DFS, 00:44:15.585 --> 00:44:16.440 you recurse. 00:44:16.440 --> 00:44:19.364 It's the order that they finished the recurse. 00:44:19.364 --> 00:44:20.030 PROFESSOR: Yeah. 00:44:20.030 --> 00:44:21.250 You have the right answer. 00:44:21.250 --> 00:44:23.440 I'm just trying to build out the annotation for it. 00:44:23.440 --> 00:44:27.510 I'm pretty sure this is the right way. 00:44:27.510 --> 00:44:30.761 I can look at this, but I can't promise you 00:44:30.761 --> 00:44:32.635 that this is the right way because I coded it 00:44:32.635 --> 00:44:34.526 and it works. 00:44:34.526 --> 00:44:36.740 So no. 00:44:36.740 --> 00:44:39.990 This is the right way. 00:44:39.990 --> 00:44:40.490 Yes. 00:44:40.490 --> 00:44:42.156 This is the right way to represent them. 00:44:42.156 --> 00:44:50.880 So forward edge means B depends on A. 00:44:50.880 --> 00:44:51.380 OK. 00:44:51.380 --> 00:44:54.139 So let's figure out how we do this. 00:44:54.139 --> 00:44:55.930 Let's do this in pseudo-code and then build 00:44:55.930 --> 00:44:57.340 the intuition for it. 00:44:57.340 --> 00:45:01.200 So you said reverse of the finishing times. 00:45:01.200 --> 00:45:04.960 So let's build a list that has all the nodes 00:45:04.960 --> 00:45:07.190 in the order of their finishing times. 00:45:07.190 --> 00:45:14.852 So let's build a list finished that is empty at first. 00:45:14.852 --> 00:45:16.810 Then I'm going to add all the nodes in the list 00:45:16.810 --> 00:45:18.080 as I'm done with them. 00:45:18.080 --> 00:45:21.420 And then I'll reverse the list. 00:45:21.420 --> 00:45:23.452 Where do I add nodes to the list? 00:45:28.272 --> 00:45:29.718 AUDIENCE: In the check. 00:45:33.650 --> 00:45:38.366 Line 2.75. 00:45:38.366 --> 00:45:39.241 PROFESSOR: Not quite. 00:45:42.400 --> 00:45:46.814 So line 4. 00:45:46.814 --> 00:45:50.180 So I'm adding them in the order in which I'm done with them. 00:45:50.180 --> 00:45:53.410 So when I'm about to leave a node completely, 00:45:53.410 --> 00:45:54.576 I'll add it to that list. 00:46:02.680 --> 00:46:03.730 Which node? 00:46:03.730 --> 00:46:05.615 What's the name? 00:46:05.615 --> 00:46:08.390 AUDIENCE: V. 00:46:08.390 --> 00:46:09.420 PROFESSOR: OK. 00:46:09.420 --> 00:46:12.858 And then where can I reverse the list? 00:46:12.858 --> 00:46:16.050 AUDIENCE: Line 4 of the first-- 00:46:16.050 --> 00:46:17.300 PROFESSOR: OK. 00:46:17.300 --> 00:46:20.110 How would I do this? 00:46:20.110 --> 00:46:29.130 r dot finished dot reverse. 00:46:29.130 --> 00:46:31.203 Like this? 00:46:31.203 --> 00:46:33.090 AUDIENCE: I think that's OK in Python. 00:46:33.090 --> 00:46:35.460 PROFESSOR: I think so, too. 00:46:35.460 --> 00:46:39.610 So this will give me a topological sort. 00:46:39.610 --> 00:46:43.650 Let's figure out why this works intuitively. 00:46:43.650 --> 00:46:45.820 Yes? 00:46:45.820 --> 00:46:50.330 So while we're building the topological sort, 00:46:50.330 --> 00:46:54.480 while we're building the inverse of the final list. 00:46:54.480 --> 00:46:56.250 So the first thing that we put in the list 00:46:56.250 --> 00:46:59.060 is the last class we're going to take. 00:46:59.060 --> 00:47:02.040 So as I go forward in this graph, 00:47:02.040 --> 00:47:09.940 my VFS is going to go A, B, C, D, E. There's nothing after E. 00:47:09.940 --> 00:47:11.920 And it's done with E. 00:47:11.920 --> 00:47:15.710 So this means there is no class that depends on E. Otherwise, 00:47:15.710 --> 00:47:18.410 DFS would keep recursing. 00:47:18.410 --> 00:47:20.690 So E is the last class I take. 00:47:20.690 --> 00:47:23.730 If I take this last, there's definitely no dependencies 00:47:23.730 --> 00:47:25.450 that I'm violating. 00:47:25.450 --> 00:47:26.260 All right? 00:47:26.260 --> 00:47:35.310 So it's safe to start with E. 00:47:35.310 --> 00:47:42.470 Now I'm out of E, I'm back to D. I go from D to F, I print that. 00:47:42.470 --> 00:47:46.550 Let's not worry about this for a little bit. 00:47:46.550 --> 00:47:52.410 And let's go back to D. When DFS comes 00:47:52.410 --> 00:47:55.630 back, it's going to print D. So I 00:47:55.630 --> 00:47:59.470 know that by the time DFS is out, 00:47:59.470 --> 00:48:04.270 I printed all the classes that depend on D. Right? 00:48:04.270 --> 00:48:06.510 So when I'm at D, whenever I have forward 00:48:06.510 --> 00:48:08.450 edges that I haven't visited yet, 00:48:08.450 --> 00:48:11.250 I will call DFS visit on them. 00:48:11.250 --> 00:48:16.170 DFS visit returns before I can get out of D. 00:48:16.170 --> 00:48:20.040 So all the edges that depend on D have been printed. 00:48:20.040 --> 00:48:20.540 Sorry. 00:48:20.540 --> 00:48:23.950 All the nodes that depend on D have been printed. 00:48:23.950 --> 00:48:27.000 So when I get out of D, I know that all the nodes that 00:48:27.000 --> 00:48:29.300 depend on D have been printed, so it's 00:48:29.300 --> 00:48:33.080 safe to print D. This is the intuition 00:48:33.080 --> 00:48:36.450 behind topological sort. 00:48:36.450 --> 00:48:41.580 So you can build sort of an induction proof based on this. 00:48:41.580 --> 00:48:46.590 So whenever I'm here, I assume that all the nodes 00:48:46.590 --> 00:48:49.670 that I have forward edges to are somewhere in my results. 00:48:49.670 --> 00:48:51.770 So I can include my nodes. 00:48:51.770 --> 00:48:56.170 So this means that whenever you put a node here, 00:48:56.170 --> 00:49:00.020 all the nodes that have forward edges to this node 00:49:00.020 --> 00:49:03.590 have already been output. 00:49:03.590 --> 00:49:06.020 So this means that no dependency relationships 00:49:06.020 --> 00:49:08.350 are going to be violated. 00:49:08.350 --> 00:49:08.850 OK. 00:49:08.850 --> 00:49:11.230 Let's keep building this and see the result. 00:49:11.230 --> 00:49:14.410 So D, E, F, we get out of B. We get out 00:49:14.410 --> 00:49:18.960 of C. What do we do here? 00:49:18.960 --> 00:49:19.800 Print C, right? 00:49:25.260 --> 00:49:27.126 What classes depend on C? 00:49:27.126 --> 00:49:33.630 D, E, and F. I had a tree edge to D. 00:49:33.630 --> 00:49:37.400 I had DFS visit here, so I know that all the classes that 00:49:37.400 --> 00:49:41.670 depend on C because they depend on D have been output. 00:49:41.670 --> 00:49:44.780 And then I have a forward edge on F. 00:49:44.780 --> 00:49:48.280 So I didn't recurse from C to F, but I 00:49:48.280 --> 00:49:50.291 know that has been covered somewhere. 00:49:50.291 --> 00:49:50.790 Right? 00:49:50.790 --> 00:49:54.220 Forward edge means that I've already seen it in DFS, 00:49:54.220 --> 00:49:55.930 and that I've already returned from it. 00:49:55.930 --> 00:49:57.760 So it has already been printed. 00:50:02.800 --> 00:50:05.670 So I'm going to write this. 00:50:05.670 --> 00:50:14.390 Now I get out of C. I get out of B. I go into G. I get out of B, 00:50:14.390 --> 00:50:20.820 and I go into A. And I go out of A, and I print it. 00:50:20.820 --> 00:50:21.320 OK. 00:50:21.320 --> 00:50:24.540 So I have tree edges that I can handle. 00:50:29.020 --> 00:50:30.260 So these are all tree edges. 00:50:33.930 --> 00:50:40.310 And the reason that topological sorting works on tree edges 00:50:40.310 --> 00:50:46.480 is that I call DFS visit on the tree edge, 00:50:46.480 --> 00:50:49.120 and I know it returns by the time I return. 00:50:49.120 --> 00:50:51.310 So I know that whatever's underneath 00:50:51.310 --> 00:50:53.670 that tree has already been printed. 00:50:53.670 --> 00:50:58.450 I have forward edges that will just take me forward 00:50:58.450 --> 00:50:59.930 in the DFS. 00:50:59.930 --> 00:51:03.490 So I know that by the time I return from a node, 00:51:03.490 --> 00:51:05.447 I've already returned from all the nodes 00:51:05.447 --> 00:51:07.280 that I have forward edges to because they're 00:51:07.280 --> 00:51:08.190 lower in the tree. 00:51:08.190 --> 00:51:08.690 Right? 00:51:08.690 --> 00:51:10.360 Forward edges work like this. 00:51:10.360 --> 00:51:11.900 So by the time I'm out of C, I've 00:51:11.900 --> 00:51:21.240 definitely printed F. Now I have this cross edge from G to C. 00:51:21.240 --> 00:51:27.210 A cross edge means that there is no direct relationship here, 00:51:27.210 --> 00:51:32.250 but I've already visited C. If there's no direct relationship, 00:51:32.250 --> 00:51:34.760 it means that for sure I'm done visiting C 00:51:34.760 --> 00:51:39.575 and I've returned so that I can get to G. 00:51:39.575 --> 00:51:44.410 So there's some common parent between C and G. 00:51:44.410 --> 00:51:46.850 I'm already done with C, and I've returned to that parent. 00:51:46.850 --> 00:51:49.450 And then I went to G. 00:51:49.450 --> 00:51:53.840 So all the nodes that are pointed to my cross edges 00:51:53.840 --> 00:51:59.671 have also been printed in topological sort. 00:51:59.671 --> 00:52:00.170 OK. 00:52:00.170 --> 00:52:01.560 Now what about back edges? 00:52:01.560 --> 00:52:02.915 What if I had a back edge? 00:52:05.420 --> 00:52:08.510 What if I had this back edge between C and A? 00:52:08.510 --> 00:52:09.410 What happens then? 00:52:09.410 --> 00:52:15.590 If I had a back edge, then that would break topological sort. 00:52:15.590 --> 00:52:16.090 Right? 00:52:16.090 --> 00:52:18.580 Because this is saying that hey, you 00:52:18.580 --> 00:52:22.000 should print A before you print C. 00:52:22.000 --> 00:52:24.110 But I know that I'm going to come out of C way 00:52:24.110 --> 00:52:27.040 before I have a chance to come out of A. 00:52:27.040 --> 00:52:30.470 So if I have a back edge, topological sort doesn't work. 00:52:30.470 --> 00:52:33.740 When do I have a back edge? 00:52:33.740 --> 00:52:35.200 When I have a cycle. 00:52:35.200 --> 00:52:37.650 So this is why I don't care about back edges. 00:52:37.650 --> 00:52:41.300 Back edges [INAUDIBLE] cycles. 00:52:41.300 --> 00:52:44.470 So back edge means I have a path forward. 00:52:44.470 --> 00:52:47.120 And I have a path backward. 00:52:47.120 --> 00:52:48.060 That does a cycle. 00:52:53.820 --> 00:52:54.320 OK. 00:52:54.320 --> 00:52:57.890 There's an awful lot of silence here. 00:52:57.890 --> 00:52:59.290 Does everything makes sense or? 00:53:02.911 --> 00:53:03.410 OK. 00:53:03.410 --> 00:53:03.993 Any questions? 00:53:06.691 --> 00:53:07.190 Nope? 00:53:07.190 --> 00:53:08.050 Everyone's happy? 00:53:10.961 --> 00:53:12.960 So you don't need to reason about this formally. 00:53:12.960 --> 00:53:16.750 Just remember the intuition that the reason we're printing them 00:53:16.750 --> 00:53:20.320 in this order is the first thing you 00:53:20.320 --> 00:53:23.040 print is the last class you're going to take. 00:53:23.040 --> 00:53:27.620 Because for sure there are no dependencies left on it, 00:53:27.620 --> 00:53:29.380 otherwise DFS would keep recursing. 00:53:29.380 --> 00:53:32.070 And then there's that recursive structure that makes this work. 00:53:35.800 --> 00:53:36.300 OK. 00:53:36.300 --> 00:53:37.784 Cool.