WEBVTT 00:00:00.070 --> 00:00:02.500 The following content is provided under a Creative 00:00:02.500 --> 00:00:04.010 Commons license. 00:00:04.010 --> 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 --> 00:00:13.330 To make a donation or view additional materials 00:00:13.330 --> 00:00:17.215 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.215 --> 00:00:17.840 at ocw.mit.edu. 00:00:21.460 --> 00:00:24.280 PROFESSOR: Good morning everyone, let's get started. 00:00:24.280 --> 00:00:29.650 So part two of our two lecture sequence on network flow. 00:00:29.650 --> 00:00:33.910 So all the pain involved in the concepts 00:00:33.910 --> 00:00:41.390 and notation from Tuesday turns into algorithmic fun today. 00:00:41.390 --> 00:00:44.530 But we will do a little bit of a review just 00:00:44.530 --> 00:00:46.620 to make sure we're all on the same page 00:00:46.620 --> 00:00:48.640 with respect to all of these notions 00:00:48.640 --> 00:00:54.020 that we talked about on flow networks. 00:00:54.020 --> 00:00:58.920 So we call that a flow network is simply 00:00:58.920 --> 00:01:03.490 a directed graph, GVE, and each edge 00:01:03.490 --> 00:01:09.570 is going to have two numbers associated with it. 00:01:09.570 --> 00:01:12.580 If you see something like 1:3, this 00:01:12.580 --> 00:01:15.755 means that this is the flow, and that's the capacity. 00:01:19.600 --> 00:01:22.100 We have a basic constraint dissociated 00:01:22.100 --> 00:01:26.140 with the flow not exceeding the capacity on any given edge. 00:01:26.140 --> 00:01:28.850 And we also have the laws of conservation 00:01:28.850 --> 00:01:32.060 that say that other than the sink in the source, which 00:01:32.060 --> 00:01:35.970 are the two distinguished vertices in G V E, 00:01:35.970 --> 00:01:38.260 you're going to have a situation where 00:01:38.260 --> 00:01:43.070 for any intermediate vertex, all of the flow entering the vertex 00:01:43.070 --> 00:01:46.020 has to leave the vertex. 00:01:46.020 --> 00:01:48.910 And with those two constraints you 00:01:48.910 --> 00:01:53.960 need to find the max flow in your flow network. 00:01:53.960 --> 00:02:01.970 The flow value corresponding to this potential max 00:02:01.970 --> 00:02:10.470 flow, or some flow, is F equaling f of s of V. 00:02:10.470 --> 00:02:14.100 And so s is the source. 00:02:14.100 --> 00:02:17.390 V corresponds to all of the vertices. 00:02:17.390 --> 00:02:21.120 And you're looking at pushing flow from the source vertex 00:02:21.120 --> 00:02:30.280 to any of the vertices, and thus the V includes t, as well. 00:02:30.280 --> 00:02:33.370 And that this is essentially a flow value, 00:02:33.370 --> 00:02:37.780 and we showed-- this is implicit summation notation, 00:02:37.780 --> 00:02:41.980 we showed, using a little bit of algebra, 00:02:41.980 --> 00:02:48.240 that this equals f of V t. 00:02:51.350 --> 00:02:56.665 And that's going to hold because of flow conservation primarily. 00:02:59.190 --> 00:03:04.860 So that was not particularly surprising. 00:03:04.860 --> 00:03:07.470 Something that was a little more surprising, 00:03:07.470 --> 00:03:09.560 or is a little more surprising, is 00:03:09.560 --> 00:03:15.070 that you can have an arbitrary cut in the flow network, 00:03:15.070 --> 00:03:17.940 and a cut corresponds to any partition. 00:03:22.860 --> 00:03:32.130 S, T, such that s belongs to S, the source belongs to S, 00:03:32.130 --> 00:03:38.920 and the sink belongs to T, and that's essentially a cut. 00:03:38.920 --> 00:03:42.460 And so obviously there could be exponentially many cuts 00:03:42.460 --> 00:03:44.950 in a given flow network. 00:03:44.950 --> 00:03:47.200 But the amazing thing is that you 00:03:47.200 --> 00:03:50.870 can show a lemma that says that, regardless 00:03:50.870 --> 00:03:57.270 of what the actual flow values are in any of these edges-- 00:03:57.270 --> 00:04:01.250 across any cut you're going to see this flow. 00:04:01.250 --> 00:04:17.510 And this is true for any cut S, T. The corollary to this lemma 00:04:17.510 --> 00:04:22.890 is that the flow is going to be less than 00:04:22.890 --> 00:04:29.120 or equal to the capacity of any cut. 00:04:34.720 --> 00:04:37.495 And that simply comes from the edge constraints. 00:04:37.495 --> 00:04:39.870 You know that the flow value corresponding 00:04:39.870 --> 00:04:42.050 to any particular edge has to be less 00:04:42.050 --> 00:04:43.610 than the capacity of that edge. 00:04:43.610 --> 00:04:47.090 So you obey that constraint and you just 00:04:47.090 --> 00:04:52.070 look at the edges that go from S to T, sum up those capacities, 00:04:52.070 --> 00:05:00.670 and your maximum flow or any flow is bounded, [INAUDIBLE], 00:05:00.670 --> 00:05:01.890 by that quantity. 00:05:01.890 --> 00:05:05.840 And that's our c S T. 00:05:05.840 --> 00:05:12.280 So you now have a relationship between the minimum capacity 00:05:12.280 --> 00:05:15.750 cut and the max flow. 00:05:15.750 --> 00:05:20.570 We didn't actually exploit that in the sense 00:05:20.570 --> 00:05:27.630 of the algorithm itself, the operation of the algorithm. 00:05:27.630 --> 00:05:29.520 But we're going to actually exploit 00:05:29.520 --> 00:05:33.570 the notion of a cut when we prove this algorithm that I got 00:05:33.570 --> 00:05:38.900 to late on Tuesday's lecture that corresponds 00:05:38.900 --> 00:05:40.570 to the Ford-Fulkerson algorithm. 00:05:40.570 --> 00:05:42.052 I showed you the execution of it. 00:05:42.052 --> 00:05:43.510 In order to prove that it's correct 00:05:43.510 --> 00:05:47.450 we're going to need this notion of that 00:05:47.450 --> 00:05:52.340 bound on the flow value given any cut in the network. 00:05:52.340 --> 00:05:55.070 So the max flow related to min-cut. 00:05:55.070 --> 00:05:57.440 But before we get to that. 00:05:57.440 --> 00:06:00.780 Just to catch you up on the algorithm 00:06:00.780 --> 00:06:04.010 that we actually got to, the Ford-Fulkerson algorithm, that 00:06:04.010 --> 00:06:06.870 was based on the notion of a residual graph. 00:06:06.870 --> 00:06:11.740 This is now a new graph, G f, which is V, E f, 00:06:11.740 --> 00:06:13.520 so it depends on the flow. 00:06:13.520 --> 00:06:18.460 You computed based on G, based on the given flow f. 00:06:18.460 --> 00:06:28.850 And this graph has edges that have strictly positive residual 00:06:28.850 --> 00:06:37.050 capacities which are defined as the C f u, 00:06:37.050 --> 00:06:44.810 v equals C u,v minus f u, v. And this needs to be strictly 00:06:44.810 --> 00:06:50.400 greater than 0 in order for the edge to exist in G of f. 00:06:50.400 --> 00:06:53.250 So if this is zero then you don't have an edge between u 00:06:53.250 --> 00:06:55.680 and v in G of f. 00:06:55.680 --> 00:07:01.300 If it's one or greater assuming integral flow values, you will. 00:07:01.300 --> 00:07:06.985 And in the residual graph-- two more quick definitions. 00:07:09.530 --> 00:07:13.370 One of which you saw last time, and another 00:07:13.370 --> 00:07:19.230 that I just verbalized, is the notion of an augmenting path. 00:07:19.230 --> 00:07:24.530 And this is any path from s to t in G f. 00:07:24.530 --> 00:07:27.880 So if there's a path from S to T in G f 00:07:27.880 --> 00:07:30.340 you do not have a maximum flow. 00:07:30.340 --> 00:07:31.774 There is an augmenting path. 00:07:31.774 --> 00:07:34.190 You're going to be able to increase the flow corresponding 00:07:34.190 --> 00:07:37.740 to the f value, that gave you your residual network, G 00:07:37.740 --> 00:07:39.060 S of f. 00:07:39.060 --> 00:07:42.000 And how much can you increase this flow by? 00:07:42.000 --> 00:07:43.950 How much can you increase f by? 00:07:43.950 --> 00:07:52.590 That is termed the residual capacity. 00:07:52.590 --> 00:07:59.040 And the residual capacity of an augmenting path 00:07:59.040 --> 00:08:03.700 is simply C f of P, that's what we're 00:08:03.700 --> 00:08:07.310 going to call it, C f of P, equals 00:08:07.310 --> 00:08:14.380 the minimum value of the residual capacities 00:08:14.380 --> 00:08:16.620 along the path. 00:08:16.620 --> 00:08:25.750 So this would be C f u,v, and that's pretty much it. 00:08:25.750 --> 00:08:30.040 And some examples should make this clearer. 00:08:30.040 --> 00:08:35.280 If we look at what the actual algorithm is, and go 00:08:35.280 --> 00:08:37.640 through a simpler example than we did last time, 00:08:37.640 --> 00:08:42.480 but it'll give you a sense of how augmentations 00:08:42.480 --> 00:08:43.990 are going to change the flow. 00:08:43.990 --> 00:08:48.230 How augmenting paths appear and eventually disappear 00:08:48.230 --> 00:08:53.590 as the flow increases and reaches the maximum flow. 00:08:53.590 --> 00:08:56.270 So this simply corresponds to, as I 00:08:56.270 --> 00:08:59.380 said, walking through the augmenting path, 00:08:59.380 --> 00:09:01.610 looking at each of the residual capacities, 00:09:01.610 --> 00:09:04.260 and picking the Min value. 00:09:04.260 --> 00:09:10.330 So the Ford-Fulkerson algorithm, which 00:09:10.330 --> 00:09:17.390 is the first algorithm for max flow continues to be used, 00:09:17.390 --> 00:09:33.100 sets the flow on each of the edges to be zero. 00:09:33.100 --> 00:09:35.550 So you just set everything to zero. 00:09:35.550 --> 00:09:38.810 Clearly satisfies flow conservation, 00:09:38.810 --> 00:09:41.360 satisfies all the edge capacities. 00:09:41.360 --> 00:09:45.090 And you have literally three lines of code 00:09:45.090 --> 00:09:59.200 here, while an augmenting path in G f exists. 00:09:59.200 --> 00:10:02.922 So there's a lot going on here, you have to compute G of f, 00:10:02.922 --> 00:10:06.960 I mean it's three lines of code, but the sub 00:10:06.960 --> 00:10:09.880 routines calls, which are fairly complicated, 00:10:09.880 --> 00:10:12.690 you have to compute g of f, given f, 00:10:12.690 --> 00:10:14.760 and initially it's all zeros. 00:10:14.760 --> 00:10:16.700 And then eventually it's going to change, 00:10:16.700 --> 00:10:20.610 or immediately after you discover an augmenting path. 00:10:20.610 --> 00:10:23.040 And now you have discovered an augmenting path. 00:10:23.040 --> 00:10:26.940 So you have to now do either breadth-first search 00:10:26.940 --> 00:10:29.740 or depth-first search, in order to discover whether this 00:10:29.740 --> 00:10:31.610 is a path from s to t because that's 00:10:31.610 --> 00:10:33.500 the definition of an augmenting path. 00:10:33.500 --> 00:10:45.510 And then the last line of code says, augment f by C f of p. 00:10:45.510 --> 00:10:47.900 And again this is fairly involved. 00:10:47.900 --> 00:10:52.210 You have to take the path in G of f. 00:10:52.210 --> 00:11:00.680 You're going to increase the values of the edges that 00:11:00.680 --> 00:11:08.100 correspond to G of f translating those edges back to G. 00:11:08.100 --> 00:11:11.920 And there's possibly an inversion in direction 00:11:11.920 --> 00:11:15.070 that is associated with that translation. 00:11:15.070 --> 00:11:18.970 And so if you take a really straightforward example. 00:11:18.970 --> 00:11:25.511 Let's say if you have G is s, and this one vertex here, a, 00:11:25.511 --> 00:11:27.310 and then you have t. 00:11:27.310 --> 00:11:31.210 And right now I have 1:2 and 1:4. 00:11:34.250 --> 00:11:36.940 I now am going to compute G of f. 00:11:36.940 --> 00:11:40.395 So G of f is going to be-- I'm just going to write down s, 00:11:40.395 --> 00:11:44.599 a, and t because the vertices are all the same. 00:11:44.599 --> 00:11:46.015 The edges are obviously different. 00:11:48.710 --> 00:11:52.110 Am I going to have an edge from s to a? 00:11:52.110 --> 00:11:53.700 Yes. 00:11:53.700 --> 00:11:56.850 And what number should I put on that edge? 00:11:56.850 --> 00:12:00.400 1, and that's because it's 2 minus 1, 00:12:00.400 --> 00:12:01.845 not because it's just plain 1. 00:12:04.450 --> 00:12:10.750 And this edge here is going to have a residual capacity of 1, 00:12:10.750 --> 00:12:14.830 and that just comes directly from that 1. 00:12:14.830 --> 00:12:17.530 And what that means is, and this is important, 00:12:17.530 --> 00:12:22.340 what that means is that, in effect, this residual capacity 00:12:22.340 --> 00:12:26.900 says in this direction you still have a capacity of 1. 00:12:26.900 --> 00:12:28.360 Which means that in this direction 00:12:28.360 --> 00:12:30.110 you could reduce by one. 00:12:30.110 --> 00:12:32.790 OK, that's the important thing to remember. 00:12:32.790 --> 00:12:36.710 And over here I'm going to have-- 00:12:36.710 --> 00:12:40.100 this way I'm going to have 3 and coming back 00:12:40.100 --> 00:12:42.410 I'm going to have 1. 00:12:42.410 --> 00:12:46.260 All right, so straightforward example but I 00:12:46.260 --> 00:12:50.840 think evocative in the sense that it applies the concepts 00:12:50.840 --> 00:12:55.110 that we've seen so far other than cuts. 00:12:55.110 --> 00:12:56.370 And we'll get to that. 00:12:56.370 --> 00:13:00.230 And what happens here, is there a path from s to t? 00:13:00.230 --> 00:13:01.380 Absolutely. 00:13:01.380 --> 00:13:05.210 And what is the residual capacity 00:13:05.210 --> 00:13:06.755 of the path from s to t? 00:13:09.460 --> 00:13:12.130 It's one because you've got a 1 here, and you got a 3 here, 00:13:12.130 --> 00:13:14.340 you got to take the Min, and it's 1. 00:13:14.340 --> 00:13:17.860 Right, I guess that's the number of choice today, one. 00:13:17.860 --> 00:13:21.110 But once you do that and you find that, you say, OK, 00:13:21.110 --> 00:13:22.915 what I'm going to do is-- and this 00:13:22.915 --> 00:13:24.540 is going to be pretty straightforward-- 00:13:24.540 --> 00:13:27.280 I'm going to look at these edges, 00:13:27.280 --> 00:13:31.110 and I know that I have c of t being a one 00:13:31.110 --> 00:13:32.680 corresponding to this value. 00:13:32.680 --> 00:13:35.800 And I'm going to go add one to those corresponding edges 00:13:35.800 --> 00:13:40.642 because these are the edges that actually shows in terms 00:13:40.642 --> 00:13:41.600 of the augmenting path. 00:13:41.600 --> 00:13:44.960 So the augmenting path is that. 00:13:44.960 --> 00:13:55.660 And so this turns into s to a to t, and that becomes 2:2. 00:13:55.660 --> 00:13:59.420 This becomes 2:4 and that's it. 00:14:02.850 --> 00:14:05.540 So I now have G with the new f. 00:14:05.540 --> 00:14:10.880 So I can now think about what is G of f1 correspond to. 00:14:10.880 --> 00:14:15.810 The G is separate from f, so in some sense 00:14:15.810 --> 00:14:20.320 the G stayed the same, but the f changed, as you saw here. 00:14:20.320 --> 00:14:25.078 And now G f1 looks like s, a, t. 00:14:27.980 --> 00:14:29.380 What do I do between s and a? 00:14:32.160 --> 00:14:34.312 Well, I only have something from a to s. 00:14:34.312 --> 00:14:36.500 I don't have anything from s to a 00:14:36.500 --> 00:14:39.250 because the residual capacity in this direction, thanks 00:14:39.250 --> 00:14:42.880 to the fact that the flow has saturated the edge capacity, 00:14:42.880 --> 00:14:44.340 is actually zero. 00:14:44.340 --> 00:14:46.440 But I do have an edge this way. 00:14:46.440 --> 00:14:49.220 And what is the value for that edge? 00:14:49.220 --> 00:14:50.750 It's 2. 00:14:50.750 --> 00:14:53.480 And over here, what are the numbers 00:14:53.480 --> 00:14:55.530 for the top and the bottom? 00:14:55.530 --> 00:14:57.610 2 and 2. 00:14:57.610 --> 00:15:00.770 And now I look at G f1, and I've gone 00:15:00.770 --> 00:15:03.710 through one iteration of this pseudo code, 00:15:03.710 --> 00:15:07.290 and I now try to find an augmenting path in G f1. 00:15:07.290 --> 00:15:10.010 And is there a path from s to t? 00:15:10.010 --> 00:15:12.590 No, which means I'm done. 00:15:12.590 --> 00:15:14.740 This is the max flow. 00:15:14.740 --> 00:15:15.440 You believe me? 00:15:19.270 --> 00:15:21.310 For this example, you believe me. 00:15:21.310 --> 00:15:26.230 You believe me, like, for every example? 00:15:26.230 --> 00:15:30.260 No, absolutely not because you haven't done a proof! 00:15:30.260 --> 00:15:33.191 We haven't done a proof! 00:15:33.191 --> 00:15:34.690 That's why you shouldn't believe me. 00:15:37.310 --> 00:15:39.410 So we got to do a proof. 00:15:39.410 --> 00:15:43.740 So how is it that we know, given all of this, 00:15:43.740 --> 00:15:49.320 it's not because I told you so, that when we've converged. 00:15:49.320 --> 00:15:55.380 When we get out of this while loop, that we have a max flow. 00:15:55.380 --> 00:15:56.130 Looks pretty good. 00:15:56.130 --> 00:15:59.430 Worked for that example and it's going 00:15:59.430 --> 00:16:03.010 to work for every example, but that's not a proof. 00:16:03.010 --> 00:16:06.330 So now we have to sort of take in all of the notions 00:16:06.330 --> 00:16:09.650 that we've talked about here, including primarily the notion 00:16:09.650 --> 00:16:10.870 of cuts. 00:16:10.870 --> 00:16:13.510 And the proof, the max flow, min-cut theorem, 00:16:13.510 --> 00:16:18.790 which is going to show this key result that we require, 00:16:18.790 --> 00:16:22.240 which is that when we terminate in the Ford-Fulkerson 00:16:22.240 --> 00:16:26.420 algorithm, we're going to have a max flow. 00:16:26.420 --> 00:16:29.245 And that's the reason why it's a maximum flow algorithm. 00:16:29.245 --> 00:16:30.620 If you don't have that proof, you 00:16:30.620 --> 00:16:31.953 don't have a max flow algorithm. 00:16:34.320 --> 00:16:37.560 So hopefully all of that is clear. 00:16:37.560 --> 00:16:40.020 Pipe up if you have questions. 00:16:40.020 --> 00:16:44.360 And let's write out the max flow min-cut theorem, 00:16:44.360 --> 00:16:51.550 which I mentioned of it last time 00:16:51.550 --> 00:16:55.810 but never really got to even stating, 00:16:55.810 --> 00:16:59.840 but today we're going to state it and prove it. 00:17:09.540 --> 00:17:11.960 So this is an interesting theorem. 00:17:11.960 --> 00:17:14.569 I mean it's a legendary theorem. 00:17:14.569 --> 00:17:21.750 And it's not your usual theorem in the sense 00:17:21.750 --> 00:17:26.670 that it makes a simple statement. 00:17:26.670 --> 00:17:29.430 It actually makes three statements. 00:17:29.430 --> 00:17:32.100 It says the following are equivalent, 00:17:32.100 --> 00:17:36.470 and it's got three things which are the following. 00:17:39.140 --> 00:17:52.080 F equals c S T for some cut, S,T. 00:17:52.080 --> 00:17:57.810 This says that some cut is saturated. 00:17:57.810 --> 00:18:00.130 Right, so that's just the statement. 00:18:00.130 --> 00:18:07.650 The second statement, which is equivalent to the first one, 00:18:07.650 --> 00:18:10.410 says that f is a maximum flow. 00:18:14.990 --> 00:18:17.960 So f being a maximum flow means there's 00:18:17.960 --> 00:18:19.710 some cut that's saturated. 00:18:19.710 --> 00:18:22.520 If there's some cut that's saturated for a given flow, 00:18:22.520 --> 00:18:24.360 you have a max flow. 00:18:24.360 --> 00:18:26.900 And then the last one, which is important 00:18:26.900 --> 00:18:31.020 for our Ford-Fulkerson algorithm, 00:18:31.020 --> 00:18:40.525 is that f admits no augmenting paths. 00:18:43.860 --> 00:18:50.780 And so if I want to show that the Ford-Fulkerson algorithm 00:18:50.780 --> 00:18:54.670 terminates with a max flow, tell me, 00:18:54.670 --> 00:18:59.990 based on implications-- like i implies j, 00:18:59.990 --> 00:19:04.000 for numbers i and j, what it is that I need to prove. 00:19:04.000 --> 00:19:06.960 We're going to prove a bunch of other things along the way 00:19:06.960 --> 00:19:12.020 but crucially what is the i implies j that I want 00:19:12.020 --> 00:19:15.920 if I want to claim that the Ford-Fulkerson algorithm 00:19:15.920 --> 00:19:17.650 terminates with the max flow. 00:19:22.040 --> 00:19:23.315 Yeah, you over there. 00:19:23.315 --> 00:19:25.690 AUDIENCE: 3 implies 2. 00:19:25.690 --> 00:19:29.010 PROFESSOR: 3 implies 2. 00:19:29.010 --> 00:19:31.670 That's right, 3 implies 2, exactly right. 00:19:31.670 --> 00:19:32.450 3 implies 2. 00:19:35.990 --> 00:19:41.110 So we need to do 3 implies 2. 00:19:41.110 --> 00:19:45.760 Now of course, the theorem says that, 1 implies 2, 00:19:45.760 --> 00:19:49.950 2 implies 3, 3 implies 1, et cetera. 00:19:49.950 --> 00:19:52.350 There's a lot of implications here. 00:19:52.350 --> 00:19:59.710 What we are going to do is we're going to show that 1 implies 2, 00:19:59.710 --> 00:20:03.590 2 implies 3, and 3 implies 1. 00:20:03.590 --> 00:20:06.746 And that pretty much takes care of everything. 00:20:06.746 --> 00:20:08.120 And it turns out the reason we do 00:20:08.120 --> 00:20:13.400 this is simply because it makes for the simplest proofs. 00:20:13.400 --> 00:20:18.040 1 implies 2 and 2 implies 3 are one- liners, and that 3 implies 00:20:18.040 --> 00:20:20.410 1 is a little more interesting and involved, 00:20:20.410 --> 00:20:24.060 but it's a little bit easier than directly 00:20:24.060 --> 00:20:26.532 doing 3 implies 2. 00:20:26.532 --> 00:20:28.240 So that's the way we're going to do this. 00:20:28.240 --> 00:20:31.290 You could certainly play around and do other things. 00:20:31.290 --> 00:20:32.690 So any questions so far? 00:20:35.750 --> 00:20:38.040 OK, so let's go ahead and do this. 00:20:56.780 --> 00:20:59.260 All right, we should be able to knock these two, 00:20:59.260 --> 00:21:03.930 1 implies 2 and 2 implies 3, in just about a minute each. 00:21:06.940 --> 00:21:11.370 So I want to show 1 implies 2, and essentially 00:21:11.370 --> 00:21:16.510 what I want to say here is, if I've 00:21:16.510 --> 00:21:21.150 saturated a particular cuts capacity then 00:21:21.150 --> 00:21:22.460 I have a max flow. 00:21:22.460 --> 00:21:26.350 And really, I mean this comes from the definitions, 00:21:26.350 --> 00:21:31.860 since I'm going to have f less than or equal to the c S, T, 00:21:31.860 --> 00:21:36.300 and this is simply because of edge capacity constraints. 00:21:36.300 --> 00:21:54.510 This is just edge capacities for any cut S T, the assumption 00:21:54.510 --> 00:22:07.187 that f equals c of S T implies f is a maximum flow because f 00:22:07.187 --> 00:22:07.895 can be increased. 00:22:14.460 --> 00:22:16.700 And that's basically it. 00:22:16.700 --> 00:22:19.770 Right, so this is pretty easy. 00:22:19.770 --> 00:22:23.950 Next one is easy as well. 00:22:23.950 --> 00:22:26.660 2 implies 3. 00:22:26.660 --> 00:22:33.080 If the rare and augmenting path-- 00:22:33.080 --> 00:22:37.650 so I'm going to do this by contradiction-- 00:22:37.650 --> 00:22:47.890 if the rare and augmenting path, the flow value 00:22:47.890 --> 00:22:48.860 can be increased. 00:22:48.860 --> 00:22:50.790 Because remember the augmenting path 00:22:50.790 --> 00:22:55.070 corresponds to a path with strictly positive 00:22:55.070 --> 00:22:57.560 residual capacities. 00:22:57.560 --> 00:22:59.560 The only reason there's an edge in there 00:22:59.560 --> 00:23:04.560 is because you have a greater than zero residual capacity. 00:23:04.560 --> 00:23:08.260 So that means that the flow value 00:23:08.260 --> 00:23:12.580 could be increased by some small amount corresponding 00:23:12.580 --> 00:23:17.650 to C f u v, and each of these quantities that 00:23:17.650 --> 00:23:21.130 are in here-- I'm sorry, corresponding to min C f u 00:23:21.130 --> 00:23:24.510 v, that's the residual capacity. 00:23:24.510 --> 00:23:27.910 But you know that each of these capacities, C F u v, 00:23:27.910 --> 00:23:30.140 are strictly greater than zero. 00:23:30.140 --> 00:23:32.860 So the min clearly is strictly greater than zero. 00:23:32.860 --> 00:23:36.310 It might be some tiny quantity but it's greater than zero. 00:23:36.310 --> 00:23:38.540 Which means that the flow value could 00:23:38.540 --> 00:23:46.440 be increased contradicting the maximality of f. 00:23:53.510 --> 00:23:55.880 So a little proof by contradiction 00:23:55.880 --> 00:23:58.520 gives you 2 implies 3. 00:23:58.520 --> 00:24:00.560 So, so far, so good. 00:24:00.560 --> 00:24:03.350 This is primarily based on the definitions 00:24:03.350 --> 00:24:06.819 of augmenting paths, residual capacities, et cetera. 00:24:06.819 --> 00:24:07.610 Now the fun starts. 00:24:10.310 --> 00:24:12.850 Now we want to show really, I mean 00:24:12.850 --> 00:24:16.950 this is truly what the max flow min cut is because I really 00:24:16.950 --> 00:24:18.420 didn't have cuts up there. 00:24:18.420 --> 00:24:20.450 So what's min-cut about that, right? 00:24:24.540 --> 00:24:28.600 So I had a question at the end of Tuesday's lecture 00:24:28.600 --> 00:24:33.510 on how come the algorithm didn't use cuts. 00:24:33.510 --> 00:24:37.740 And, well, the algorithm doesn't use cuts, 00:24:37.740 --> 00:24:41.770 but showing that the algorithm converges to the max flow 00:24:41.770 --> 00:24:42.850 uses the notion of cuts. 00:24:45.990 --> 00:24:50.220 So what we want to do is do 3 implies 1, 00:24:50.220 --> 00:24:53.060 and that's going to take care of what we want. 00:24:53.060 --> 00:24:59.330 And we're going to say, well, suppose 00:24:59.330 --> 00:25:04.345 f admits no augmenting paths. 00:25:08.450 --> 00:25:10.220 So what does that mean? 00:25:10.220 --> 00:25:15.140 Well, that means that you can't reach t from s. 00:25:15.140 --> 00:25:18.720 If you reach t from s, there's an augmenting path, 00:25:18.720 --> 00:25:24.100 so you can't make that move or make that path. 00:25:24.100 --> 00:25:27.730 So let's go ahead and, sort of, look at this a little more 00:25:27.730 --> 00:25:32.020 carefully and figure out what's causing 00:25:32.020 --> 00:25:36.150 this lack of connectivity. 00:25:36.150 --> 00:25:38.610 There's a lack of connectivity between s and t. 00:25:38.610 --> 00:25:40.780 You can't actually get there. 00:25:40.780 --> 00:25:43.570 So there's like a chasm, there's a gap. 00:25:43.570 --> 00:25:45.320 And so what's causing that? 00:25:45.320 --> 00:25:50.680 And what we're going to do is figure 00:25:50.680 --> 00:25:55.230 out what the cut of the partition 00:25:55.230 --> 00:26:00.740 is that could be the reason why you 00:26:00.740 --> 00:26:03.900 have this lack of connectivity. 00:26:03.900 --> 00:26:07.010 So we're going to define S which is exactly 00:26:07.010 --> 00:26:13.610 going to be our S from our favorite s t cut definition, 00:26:13.610 --> 00:26:16.110 but bear with me for just a second. 00:26:16.110 --> 00:26:19.590 Because I will tell you what S is. 00:26:19.590 --> 00:26:35.020 There exists a path in G f from s to u. 00:26:35.020 --> 00:26:37.120 So what I'm doing is I'm collecting 00:26:37.120 --> 00:26:46.080 the reachable vertices from S. And I know that I'm not 00:26:46.080 --> 00:26:50.940 going to have t, small t, in this set 00:26:50.940 --> 00:26:55.400 because f admits no augmenting paths so I can't possibly 00:26:55.400 --> 00:26:59.440 reach t from s. 00:26:59.440 --> 00:27:02.510 But I can collect all of the vertices 00:27:02.510 --> 00:27:08.530 that that can be reached from s in this given G of f 00:27:08.530 --> 00:27:14.050 and put them all into S. 00:27:14.050 --> 00:27:17.590 And then my T definition is simply 00:27:17.590 --> 00:27:23.780 V minus S. And the key observation which I already 00:27:23.780 --> 00:27:26.890 have made but important to emphasize 00:27:26.890 --> 00:27:36.680 is that small t belongs to T, and obviously s belongs to S, 00:27:36.680 --> 00:27:43.030 therefore S T is a cut. 00:27:43.030 --> 00:27:47.230 It satisfies the definition of a cut. 00:27:47.230 --> 00:27:49.350 So we've set ourselves up. 00:27:49.350 --> 00:27:55.430 And then we have a key observation 00:27:55.430 --> 00:27:59.399 to make that I'm going to try and extract out or you. 00:27:59.399 --> 00:28:00.315 Let me move over here. 00:28:17.370 --> 00:28:28.160 All right, so what I want to do is I want to draw this out, 00:28:28.160 --> 00:28:31.620 and I want to look at this as this 00:28:31.620 --> 00:28:36.700 is S, that is T. I know that S is over here, 00:28:36.700 --> 00:28:39.660 are s is over here, t is over here. 00:28:43.660 --> 00:28:51.520 I'm going to pick some vertex v that's over here, 00:28:51.520 --> 00:28:56.170 some vertex u that's over here. 00:28:56.170 --> 00:29:00.850 I know that there's a path, and I'm 00:29:00.850 --> 00:29:03.480 writing this is a dotted path because this 00:29:03.480 --> 00:29:06.470 is a path in G of f. 00:29:06.470 --> 00:29:10.340 We're going to actually move between a G of f and G 00:29:10.340 --> 00:29:11.670 in this proof. 00:29:11.670 --> 00:29:14.260 So that's the only hard part of this proof. 00:29:14.260 --> 00:29:16.780 The movement between g and G of f. 00:29:16.780 --> 00:29:20.420 The edges are different between g and G of f. 00:29:20.420 --> 00:29:21.699 Got to keep that in your head. 00:29:21.699 --> 00:29:23.740 Because we're going to make arguments about that. 00:29:23.740 --> 00:29:27.730 I know that all of the vertices that are on the left hand side 00:29:27.730 --> 00:29:31.930 are reachable from s, so I just picked an arbitrary vertex u, 00:29:31.930 --> 00:29:33.390 and I know that there's a path in G 00:29:33.390 --> 00:29:38.360 of f because that's the way I defined it from s to u. 00:29:38.360 --> 00:29:46.485 I also know that there is no path in G of f from u 00:29:46.485 --> 00:29:55.610 to v because if there was a path v would have been on this side, 00:29:55.610 --> 00:29:57.860 correct? 00:29:57.860 --> 00:30:01.590 So there's clearly no path in g of f from u 00:30:01.590 --> 00:30:04.170 to v. That's why v is over here. 00:30:04.170 --> 00:30:11.510 But let's say that I'm picking something 00:30:11.510 --> 00:30:20.000 where there's an edge in G, the original flow network from u 00:30:20.000 --> 00:30:24.280 to v. So the edge that I've drawn here, the dark edge, 00:30:24.280 --> 00:30:33.062 is an edge between u and v in G. And that edge 00:30:33.062 --> 00:30:34.560 had a certain capacity. 00:30:34.560 --> 00:30:36.230 It existed. 00:30:36.230 --> 00:30:37.880 The reason it existed is because it had 00:30:37.880 --> 00:30:40.670 a nonzero capacity, correct? 00:30:40.670 --> 00:30:43.340 Because we originally defined our flow network 00:30:43.340 --> 00:30:46.120 to be something where-- if you only put edges 00:30:46.120 --> 00:30:48.400 in there, in the original flow network 00:30:48.400 --> 00:30:51.380 if the capacities were greater than zero. 00:30:51.380 --> 00:30:56.577 So that edge in there, in G had some capacity c u 00:30:56.577 --> 00:30:57.990 v which was greater than zero. 00:31:00.720 --> 00:31:09.460 But this edge does not exist in G of f, correct? 00:31:09.460 --> 00:31:11.347 So what can I say? 00:31:11.347 --> 00:31:11.847 Go ahead. 00:31:11.847 --> 00:31:15.130 AUDIENCE: [INAUDIBLE] 00:31:17.480 --> 00:31:18.630 PROFESSOR: Exactly. 00:31:18.630 --> 00:31:26.090 In the original graph, we have saturated this edge-- that 00:31:26.090 --> 00:31:28.950 was my best throw of the term. 00:31:28.950 --> 00:31:31.030 I like that. 00:31:31.030 --> 00:31:36.090 And you guys didn't notice, but maybe it'll be on video. 00:31:36.090 --> 00:31:38.430 Actually, it won't be on video unfortunately. 00:31:38.430 --> 00:31:44.580 But so I got this edge here in the original graph 00:31:44.580 --> 00:31:47.710 and I didn't get this dotted edge here 00:31:47.710 --> 00:31:51.540 because the residual capacity was zero. 00:31:51.540 --> 00:31:53.260 The original capacity was nonzero. 00:31:53.260 --> 00:31:54.910 The residual capacity is zero. 00:31:54.910 --> 00:32:08.600 The only reason that happens is because C f u v is zero since 00:32:08.600 --> 00:32:19.900 if C f u v greater than zero then v would belong to S, 00:32:19.900 --> 00:32:23.590 not v belonging to T as assumed. 00:32:26.720 --> 00:32:30.430 So that's essentially what we've determined here. 00:32:30.430 --> 00:32:44.100 So this means that f of u v equals c of u v 00:32:44.100 --> 00:32:58.720 simply because c f u v equals c u v minus f u v which is 0. 00:32:58.720 --> 00:33:03.820 All right, so that's the statement I can make. 00:33:03.820 --> 00:33:09.620 Now, I did not have any constraint on u 00:33:09.620 --> 00:33:15.090 and v. I just said u belongs to s and v belongs to t. 00:33:15.090 --> 00:33:17.290 And for each of those things, I know 00:33:17.290 --> 00:33:23.050 that I can't have edges in the residual network between u 00:33:23.050 --> 00:33:26.506 and v which means that any edge that exists-- maybe there's 00:33:26.506 --> 00:33:28.380 no edge in the original network, but if there 00:33:28.380 --> 00:33:31.920 was an edge in that original network it's saturated 00:33:31.920 --> 00:33:33.730 according to this argument. 00:33:33.730 --> 00:33:42.640 So every edge from S to T in the original network is saturated, 00:33:42.640 --> 00:33:45.800 and that essentially means that I've gotten 00:33:45.800 --> 00:33:47.902 to the capacity of my cut. 00:33:47.902 --> 00:33:49.360 That's essentially what that means. 00:33:49.360 --> 00:33:53.230 If I've saturated every edge, I've reached that capacity. 00:33:53.230 --> 00:33:56.300 OK, so that's it. 00:33:56.300 --> 00:33:59.480 All you do once you recognize that you had an arbitrary 00:33:59.480 --> 00:34:09.260 choice of u and v, you just say summing over all u belonging 00:34:09.260 --> 00:34:31.280 to s and v belong to t yields f of S T equals c of S T. 00:34:31.280 --> 00:34:34.110 All right, I'm on a roll not just with Frisbees. 00:34:34.110 --> 00:34:38.050 I finished the proof with a finger to spare. 00:34:38.050 --> 00:34:41.679 So f of S T equals c of S T. All right, 00:34:41.679 --> 00:34:44.070 so that's exactly what we want. 00:34:44.070 --> 00:34:49.810 We are saying f of S T is obviously a cardinality of f 00:34:49.810 --> 00:34:52.080 so I've shown this thing over here. 00:34:54.639 --> 00:34:58.660 So that's why the Ford-Fulkerson algorithm works. 00:34:58.660 --> 00:35:00.830 It's because of this analysis that 00:35:00.830 --> 00:35:03.210 the Ford-Fulkerson algorithm works. 00:35:03.210 --> 00:35:05.490 So are we done? 00:35:05.490 --> 00:35:13.020 What are we missing in algorithm design, our algorithm analysis? 00:35:13.020 --> 00:35:13.770 Not you, yet. 00:35:13.770 --> 00:35:14.872 AUDIENCE: [INAUDIBLE] 00:35:14.872 --> 00:35:15.580 PROFESSOR: Sorry? 00:35:15.580 --> 00:35:16.413 AUDIENCE: A runtime? 00:35:16.413 --> 00:35:19.400 PROFESSOR: Runtime, good runtime. 00:35:19.400 --> 00:35:22.450 I know you have too many Frisbees. 00:35:22.450 --> 00:35:23.810 You probably do, too. 00:35:23.810 --> 00:35:28.690 But so we haven't done-- we've done correctness, 00:35:28.690 --> 00:35:30.080 we've done conversions. 00:35:30.080 --> 00:35:33.690 We haven't done a complexity analysis. 00:35:33.690 --> 00:35:37.730 And so there's some bad news here. 00:35:37.730 --> 00:35:51.070 And erasing a pivotal moment in 6 over 6. 00:36:27.920 --> 00:36:32.370 Why is this a pivotal moment in 6 over 6? 00:36:32.370 --> 00:36:34.680 It's on the Frisbee, exactly. 00:36:34.680 --> 00:36:37.970 This picture-- maybe not exactly that picture. 00:36:37.970 --> 00:36:40.275 It's not quite that picture but it's the same graph. 00:36:42.980 --> 00:36:45.750 It's a pivotal moment because this picture was famous 00:36:45.750 --> 00:36:47.500 because it was in a textbook but now 00:36:47.500 --> 00:36:52.206 it's iconic because it's on a Frisbee. 00:36:52.206 --> 00:36:56.230 We immortalized this picture. 00:36:56.230 --> 00:37:01.920 So this is an example of really a failure 00:37:01.920 --> 00:37:04.780 of the Ford-Fulkerson algorithm. 00:37:04.780 --> 00:37:06.750 So it's kind of a bad example on this Frisbee 00:37:06.750 --> 00:37:10.100 but it's going to lead us to better algorithms. 00:37:10.100 --> 00:37:15.060 So what I have here is a weird flow network that 00:37:15.060 --> 00:37:17.040 has these strange capacities. 00:37:17.040 --> 00:37:19.390 I mean it's a fairly straightforward flow network. 00:37:19.390 --> 00:37:22.975 But at the as of the moment it obviously satisfies 00:37:22.975 --> 00:37:25.830 a flow conservation because all of the flows 00:37:25.830 --> 00:37:28.780 are 0 and therefore edge capacities. 00:37:28.780 --> 00:37:34.860 But we're going to now take a pathological execution 00:37:34.860 --> 00:37:37.060 of the Ford-Fulkerson algorithm. 00:37:37.060 --> 00:37:40.390 Which, by the way, did not specify 00:37:40.390 --> 00:37:44.760 how you are going to select the augmenting path. 00:37:44.760 --> 00:37:47.470 It just said, find an augmenting path. 00:37:47.470 --> 00:37:51.690 And in passing, I said, overuse depth-first search, 00:37:51.690 --> 00:37:54.230 we'll use breadth-first search, and kind of hand wave my way 00:37:54.230 --> 00:37:55.280 through that. 00:37:55.280 --> 00:37:58.200 Now it turns out you can't hand wave when you 00:37:58.200 --> 00:37:59.860 want to run algorithms, right. 00:37:59.860 --> 00:38:02.680 You eventually have to code them up, you got to pick something. 00:38:02.680 --> 00:38:03.770 So people did. 00:38:03.770 --> 00:38:07.980 People picked different ways of selecting 00:38:07.980 --> 00:38:11.840 augmenting paths in the Ford-Fulkerson framework. 00:38:11.840 --> 00:38:15.790 And they discovered that some of them would work beautifully 00:38:15.790 --> 00:38:18.860 and some of them would fail miserably 00:38:18.860 --> 00:38:21.110 on networks that were as small as this, 00:38:21.110 --> 00:38:23.250 and they'd just take forever. 00:38:23.250 --> 00:38:26.680 And this is before complexity analysis, kind of maybe 00:38:26.680 --> 00:38:30.940 before asymptotic complexity really came into vogue. 00:38:30.940 --> 00:38:33.640 So they just had empirical analysis 00:38:33.640 --> 00:38:36.580 to tell them that some techniques for discovering 00:38:36.580 --> 00:38:40.150 augmenting paths work better than other techniques 00:38:40.150 --> 00:38:44.220 on an empirical basis. 00:38:44.220 --> 00:38:50.980 So what could go wrong with Ford-Fulkerson on this example? 00:38:50.980 --> 00:38:54.360 What selection strategy corresponding 00:38:54.360 --> 00:39:00.220 to the augmenting path could cause Ford-Fulkerson to have 00:39:00.220 --> 00:39:02.890 a huge number of iterations? 00:39:02.890 --> 00:39:03.392 Go ahead. 00:39:03.392 --> 00:39:05.475 AUDIENCE: So each time you pick an augmenting path 00:39:05.475 --> 00:39:07.300 you pick one that goes to the center. 00:39:07.300 --> 00:39:11.990 PROFESSOR: Exactly, and so let's walk through a couple here. 00:39:11.990 --> 00:39:15.604 So the first one, let's just say this was a and b, 00:39:15.604 --> 00:39:16.270 to make it easy. 00:39:16.270 --> 00:39:19.830 So what is the first path that we picked? 00:39:19.830 --> 00:39:20.500 What is it--? 00:39:20.500 --> 00:39:21.380 Go for it. 00:39:21.380 --> 00:39:22.658 You're not off the hook yet. 00:39:22.658 --> 00:39:24.450 AUDIENCE: You pick s to a to b to t. 00:39:24.450 --> 00:39:27.230 PROFESSOR: Yeah, and then when you pick s to a to b to t, 00:39:27.230 --> 00:39:29.670 I'm not going to draw G of f for this, 00:39:29.670 --> 00:39:36.680 it's pretty clear that G of f is going to have-- I'm sorry, 00:39:36.680 --> 00:39:40.530 you end up picking s to a to b to t corresponding 00:39:40.530 --> 00:39:42.680 to the G of f that you've created. 00:39:42.680 --> 00:39:47.860 Right, so the G of f might have something like s 00:39:47.860 --> 00:39:50.600 to a-- there's only going to be an edge this way 00:39:50.600 --> 00:39:52.450 that's going to have 10 raised to 9. 00:39:52.450 --> 00:39:55.740 And then this way it's going to have 1, and that's going to b, 00:39:55.740 --> 00:40:04.330 and over here to t it's going to have 10 raised to 9. 00:40:04.330 --> 00:40:09.170 It's also going to have 10 raised to 9 this way, 00:40:09.170 --> 00:40:12.700 and it's going to have 10 raised to 9 this way. 00:40:12.700 --> 00:40:14.530 And the reason the gentleman was laughing 00:40:14.530 --> 00:40:19.730 is because we ended up picking this path. 00:40:19.730 --> 00:40:24.010 S in G of f-- this is G of f just to be clear-- you pick 00:40:24.010 --> 00:40:27.930 s a b t. 00:40:27.930 --> 00:40:33.980 And so what you end up doing is you end up making this 1, 00:40:33.980 --> 00:40:41.720 making this 1, and coming over here making this 1. 00:40:41.720 --> 00:40:42.220 Wow! 00:40:42.220 --> 00:40:43.720 OK, you did increase the flow. 00:40:43.720 --> 00:40:46.590 Very good, made progress. 00:40:46.590 --> 00:40:48.400 Now what happens? 00:40:48.400 --> 00:40:54.330 When you do this, certainly you'll get a different G of f. 00:40:54.330 --> 00:40:56.732 And the G of f is going to turn into-- well, 00:40:56.732 --> 00:40:58.440 this gets a little more interesting here. 00:40:58.440 --> 00:41:01.050 This is 10 raised to 9 minus 1. 00:41:01.050 --> 00:41:04.440 This is 10 raised to 9-- no, that doesn't change. 00:41:04.440 --> 00:41:06.890 This is still 10 raised to 9. 00:41:06.890 --> 00:41:09.720 This is 10 raised to 9 minus 1. 00:41:09.720 --> 00:41:11.860 And that this edge here also changes 00:41:11.860 --> 00:41:16.580 because basically what ends up happening 00:41:16.580 --> 00:41:22.580 is you end up going this way with 1. 00:41:22.580 --> 00:41:24.430 Did I get that right? 00:41:24.430 --> 00:41:25.850 Yeah? 00:41:25.850 --> 00:41:26.800 OK, good. 00:41:26.800 --> 00:41:30.990 So now what's the bad path? 00:41:30.990 --> 00:41:35.260 s b a t. 00:41:35.260 --> 00:41:41.790 So s b a t would now say, oh, what I'm going to do is, 00:41:41.790 --> 00:41:44.300 I'm going to go ahead and make this 1. 00:41:44.300 --> 00:41:45.883 What do I do with that? 00:41:45.883 --> 00:41:46.830 AUDIENCE: Make it 0. 00:41:46.830 --> 00:41:49.650 PROFESSOR: Make it 0, exactly. 00:41:49.650 --> 00:41:54.440 Make it 0, and then this 1 becomes a 1. 00:41:54.440 --> 00:41:56.380 Shall we keep going? 00:41:56.380 --> 00:41:58.260 No. 00:41:58.260 --> 00:42:03.190 So 2 billion iterations for a graph with four vertices. 00:42:03.190 --> 00:42:06.060 Now that's performance for you. 00:42:06.060 --> 00:42:12.100 So this particular, potentially a depth-first Ford-Fulkerson 00:42:12.100 --> 00:42:14.830 on this example, depending on the ordering of vertices 00:42:14.830 --> 00:42:19.127 it's certainly quite possible that you 00:42:19.127 --> 00:42:24.640 would get to this particular order in the depth-first search 00:42:24.640 --> 00:42:26.760 on G of f. 00:42:26.760 --> 00:42:31.190 The computers is dumb, it does what you tell it to do, 00:42:31.190 --> 00:42:32.750 it's doing depth-first search. 00:42:32.750 --> 00:42:35.510 It's producing these paths, and you end up 00:42:35.510 --> 00:42:39.850 with 2 billion iterations for this. 00:42:39.850 --> 00:42:44.840 And how many iterations do you really 00:42:44.840 --> 00:42:48.480 need if you did it right? 00:42:48.480 --> 00:42:50.560 Two. 00:42:50.560 --> 00:42:52.860 So that's a billion factor slowdown. 00:42:52.860 --> 00:42:54.340 So this is a pathological example, 00:42:54.340 --> 00:42:56.520 a simple pathological example, to just show you 00:42:56.520 --> 00:42:57.920 what the problem is. 00:42:57.920 --> 00:43:01.320 But you can imagine that if you use depth-first search 00:43:01.320 --> 00:43:04.817 you might be a factor of five slower on average 00:43:04.817 --> 00:43:06.400 than if you use some of the technique. 00:43:06.400 --> 00:43:08.850 And a factor of 5 is nothing to be scoffed at, 00:43:08.850 --> 00:43:10.990 especially if you're running for minutes. 00:43:10.990 --> 00:43:15.270 And, you know, back in the day computers were horribly slow. 00:43:15.270 --> 00:43:19.780 So how is this problem fixed? 00:43:19.780 --> 00:43:23.310 Well, any number of ways. 00:43:23.310 --> 00:43:28.160 But the first real way that took into account 00:43:28.160 --> 00:43:32.650 asymptotitc complexity, did analysis, and did all of that 00:43:32.650 --> 00:43:34.785 was due to Edmonds and Karp, which 00:43:34.785 --> 00:43:36.650 is a few years after Ford-Fulkerson. 00:43:36.650 --> 00:43:39.080 In fact, several years after Ford-Fulkerson. 00:43:39.080 --> 00:43:45.380 And their contribution was not as much a new algorithm, 00:43:45.380 --> 00:43:47.980 though it is called Edmonds-Karp algorithm. 00:43:47.980 --> 00:43:51.484 It's really a small change to Ford-Fulkerson. 00:43:51.484 --> 00:43:53.650 You might want to think about it as the Edmonds-Karp 00:43:53.650 --> 00:43:56.550 analysis, which was kind of important, 00:43:56.550 --> 00:43:58.550 and we're not going to do that here. 00:43:58.550 --> 00:44:01.490 But you'll see it in section on Friday. 00:44:01.490 --> 00:44:03.670 But the Edmonds-Karp algorithm slash 00:44:03.670 --> 00:44:21.150 analysis is that, if you use a breadth-first augmenting path, 00:44:21.150 --> 00:44:26.220 it's obviously just as easy to discover complexity-wise 00:44:26.220 --> 00:44:28.190 as the depth-first path. 00:44:28.190 --> 00:44:32.020 Then you could get a polynomial bound that did not 00:44:32.020 --> 00:44:36.170 depend on the capacities for the complexity 00:44:36.170 --> 00:44:40.690 and therefore iteration count of Ford-Fulkerson. 00:44:40.690 --> 00:44:43.480 So that was their contribution. 00:44:43.480 --> 00:44:54.590 And so the breadth-first path is the shortest path in G f from s 00:44:54.590 --> 00:45:07.550 to t if you count where each edge has weight 1. 00:45:07.550 --> 00:45:09.840 So in that case you would get your two iterations. 00:45:09.840 --> 00:45:12.579 You'd get the shortest path of weight 2. 00:45:12.579 --> 00:45:14.370 Whenever I say weight, I mean just counting 00:45:14.370 --> 00:45:16.210 the number of edges. 00:45:16.210 --> 00:45:18.510 And these algorithms they actually 00:45:18.510 --> 00:45:22.000 recognized that breadth-first implementations 00:45:22.000 --> 00:45:26.160 of Ford-Fulkerson were the ones that ran faster 00:45:26.160 --> 00:45:28.352 than all the other ones. 00:45:28.352 --> 00:45:30.310 And then they said, wait, what's going on here? 00:45:30.310 --> 00:45:31.590 This looks pretty good. 00:45:31.590 --> 00:45:35.620 And so they went off and did a fairly sophisticated analysis 00:45:35.620 --> 00:45:42.700 that showed that you had, and as I 00:45:42.700 --> 00:45:47.810 said you'll see that in section, that you only 00:45:47.810 --> 00:45:53.230 need order VE augmentations in the worst case. 00:45:56.015 --> 00:45:57.890 So this is what we know and love with respect 00:45:57.890 --> 00:46:00.450 to algorithmic complexity. 00:46:00.450 --> 00:46:03.160 Order VE augmentations, in the worst case, 00:46:03.160 --> 00:46:06.310 provided you used breadth-first augmentation. 00:46:06.310 --> 00:46:09.700 And so what is the overall complexity of the algorithm, 00:46:09.700 --> 00:46:11.720 let's call it the Edmonds-Karp algorithm, 00:46:11.720 --> 00:46:23.850 with this breadth-first search it would be-- order? 00:46:23.850 --> 00:46:28.250 VE square, you could think about it as, 00:46:28.250 --> 00:46:30.980 let's assume that e is greater than V, 00:46:30.980 --> 00:46:34.760 and so we just say order V squared. 00:46:37.520 --> 00:46:39.520 So that was the Edmonds and Karp's contribution. 00:46:42.070 --> 00:46:45.660 This mean that an amazing amount of work over the past 30, 40 00:46:45.660 --> 00:46:48.960 years on improving that bound. 00:46:48.960 --> 00:46:52.470 [? Denic ?] independently arrived 00:46:52.470 --> 00:46:58.700 at a similar analysis in bound, along with Emonds and Karp. 00:46:58.700 --> 00:47:03.160 They were the first to give polynomial bounds on max flow. 00:47:03.160 --> 00:47:12.650 And there was an algorithm that came out of MIT partly, 00:47:12.650 --> 00:47:19.220 or at least MIT alums, where you improve that. 00:47:19.220 --> 00:47:22.236 And so until about 2011-- and there's 00:47:22.236 --> 00:47:23.860 a lot of different algorithms depending 00:47:23.860 --> 00:47:27.290 on whether you're taking into account the value 00:47:27.290 --> 00:47:31.420 of the capacity, and you want to think about this is as some max 00:47:31.420 --> 00:47:34.520 value, and if you want to include that in the complexity 00:47:34.520 --> 00:47:36.720 then things change a bit. 00:47:36.720 --> 00:47:37.790 If you can limit that. 00:47:37.790 --> 00:47:41.010 But assuming that those capacities are essentially 00:47:41.010 --> 00:47:45.500 unbounded, then the fastest algorithm in 2011 00:47:45.500 --> 00:47:50.285 was King, Raul, and Tarjan. 00:47:53.070 --> 00:48:05.364 And that runs in order VE log of E divided by V log V V. 00:48:05.364 --> 00:48:07.280 Obviously, we need something for the log here. 00:48:10.500 --> 00:48:12.340 So that's kind of, pretty good here, right, 00:48:12.340 --> 00:48:14.030 because this is pretty small. 00:48:14.030 --> 00:48:17.880 If you really think about it, if E's large for dense graphs 00:48:17.880 --> 00:48:19.960 this is going to be a small constant. 00:48:19.960 --> 00:48:21.040 So that's pretty good. 00:48:21.040 --> 00:48:23.620 So that was the fastest until 2011. 00:48:23.620 --> 00:48:27.320 Very recently, Orlin, who's at MIT, 00:48:27.320 --> 00:48:30.010 came up with an order v e algorithm 00:48:30.010 --> 00:48:36.200 and there are many variants here that use fast matrix multiply. 00:48:36.200 --> 00:48:40.220 And then just this past September, 00:48:40.220 --> 00:48:42.350 Aleksander Madry joined our department, 00:48:42.350 --> 00:48:50.680 and he has a variant algorithm that is better in some sense, 00:48:50.680 --> 00:48:56.254 not in the pure sense of V and E but there's 00:48:56.254 --> 00:48:57.670 a couple of constraints associated 00:48:57.670 --> 00:49:01.410 with it that are very mild. 00:49:01.410 --> 00:49:04.520 But it's in that sense better than order VE 00:49:04.520 --> 00:49:06.110 and I won't get to that. 00:49:06.110 --> 00:49:10.590 In the context of 6 over 6 we are obviously 00:49:10.590 --> 00:49:13.930 going to stick with Edmonds-Karp and Ford-Fulkerson 00:49:13.930 --> 00:49:16.040 in terms of what you're responsible for, 00:49:16.040 --> 00:49:20.364 but this as been a rich area of research for decades. 00:49:20.364 --> 00:49:21.030 All right, yeah. 00:49:21.030 --> 00:49:21.530 [INAUDIBLE] 00:49:26.354 --> 00:49:32.810 Oh, simply because the breadth-first augmentation 00:49:32.810 --> 00:49:33.670 takes order e. 00:49:33.670 --> 00:49:36.470 I'm just saying it takes order E time. 00:49:36.470 --> 00:49:40.730 So this was the order VE augmentations, in the worst 00:49:40.730 --> 00:49:45.090 case, each augmentation takes order E time, 00:49:45.090 --> 00:49:47.490 because breadth-first search takes order E time assuming 00:49:47.490 --> 00:49:51.290 that E is greater than V and I just added the E square factor. 00:49:51.290 --> 00:49:52.040 Thanks for asking. 00:49:52.040 --> 00:49:55.130 I'm sure a lot of other people had the same question. 00:49:55.130 --> 00:49:58.510 Yes, I just sort of skipped over that a little too quickly. 00:49:58.510 --> 00:50:01.990 That's why you get the order VE square. 00:50:01.990 --> 00:50:04.500 And so that's a significant reduction 00:50:04.500 --> 00:50:07.610 if you assume that constant factors don't explode 00:50:07.610 --> 00:50:12.300 going from Edmonds-Karp to King, Raul, and Tarjan, 00:50:12.300 --> 00:50:16.610 and certainly Orlin, So now we got algorithms, 00:50:16.610 --> 00:50:19.990 we've proven them correct, we've done some analysis. 00:50:19.990 --> 00:50:24.610 At tomorrow, in section, you're going to see, 00:50:24.610 --> 00:50:28.390 at the analysis that gives you the order VE. 00:50:28.390 --> 00:50:32.720 Which is a half hours worth of pain, should I say. 00:50:32.720 --> 00:50:35.960 Uh, no, a half hours worth of excitement! 00:50:35.960 --> 00:50:40.410 That corresponds to the order VE, that bound that we have. 00:50:40.410 --> 00:50:42.340 All right, so it's not that difficult. It's 00:50:42.340 --> 00:50:45.600 about the level of this thing over here, 00:50:45.600 --> 00:50:47.990 and maybe a little bit more involved. 00:50:47.990 --> 00:50:50.198 So, yeah? 00:50:50.198 --> 00:50:53.460 AUDIENCE: [INAUDIBLE] 00:50:56.210 --> 00:51:00.580 PROFESSOR: Breadth-first search takes order v plus e time. 00:51:00.580 --> 00:51:04.220 On a graph, breadth-first search is order v plus e. 00:51:04.220 --> 00:51:06.430 And I'm getting a little lazy here, 00:51:06.430 --> 00:51:10.470 and I'm paying for it, by putting in the v e square. 00:51:10.470 --> 00:51:12.560 But I should really have been-- if I 00:51:12.560 --> 00:51:15.081 wanted to be a stickler, I would have put in v plus e, 00:51:15.081 --> 00:51:15.580 thank you. 00:51:18.250 --> 00:51:21.570 So where was I? 00:51:21.570 --> 00:51:22.720 So we got an algorithm now. 00:51:22.720 --> 00:51:24.800 You know, we've got a bunch of algorithms. 00:51:24.800 --> 00:51:27.220 Network flow is this amazing hammer 00:51:27.220 --> 00:51:29.500 that is being used in a wide variety of things. 00:51:29.500 --> 00:51:31.880 And it's been used for matching, bipartite matching, 00:51:31.880 --> 00:51:33.920 all sorts of things. 00:51:33.920 --> 00:51:39.600 My favorite example is something that 00:51:39.600 --> 00:51:43.290 is related to sports, I should say, 00:51:43.290 --> 00:51:45.900 which is a baseball elimination. 00:51:45.900 --> 00:51:49.450 So what is this application about? 00:51:49.450 --> 00:51:54.260 You see charts that correspond to how your team is doing. 00:51:54.260 --> 00:51:55.190 Standings, right? 00:51:55.190 --> 00:52:00.530 Go to ESPN and click on standings for baseball, 00:52:00.530 --> 00:52:02.440 for example. 00:52:02.440 --> 00:52:04.860 It doesn't-- now you see pre season standings 00:52:04.860 --> 00:52:07.690 but let's assume it's August or something-- 00:52:07.690 --> 00:52:11.660 and you click on standings. 00:52:11.660 --> 00:52:17.510 And you see a chart that corresponds to games played, 00:52:17.510 --> 00:52:21.780 games won, games lost, games remaining, 00:52:21.780 --> 00:52:23.540 and so on and so forth. 00:52:23.540 --> 00:52:37.640 And back in the day when they were just 00:52:37.640 --> 00:52:45.050 winners corresponding to each division 00:52:45.050 --> 00:52:49.270 you had to win your division to make the playoffs. 00:52:49.270 --> 00:52:51.520 So you had to be the best. 00:52:51.520 --> 00:52:54.540 Now if you ended up being tied, you 00:52:54.540 --> 00:52:58.440 ended up having to play an elimination game. 00:52:58.440 --> 00:53:02.290 But the whole purpose of this analysis 00:53:02.290 --> 00:53:07.360 is deciding whether you still have a chance 00:53:07.360 --> 00:53:09.420 to make the playoffs or not. 00:53:09.420 --> 00:53:12.950 So the goal here is you want an algorithm that 00:53:12.950 --> 00:53:15.480 is going to look at the standings 00:53:15.480 --> 00:53:19.700 and decide if your team is alive or not. 00:53:19.700 --> 00:53:22.830 Is there a chance of God is great, 00:53:22.830 --> 00:53:26.970 whatever, everything goes your way, angels in the outfield, 00:53:26.970 --> 00:53:28.100 right? 00:53:28.100 --> 00:53:30.410 Are you going to make the playoffs? 00:53:30.410 --> 00:53:33.610 And so you can be very optimistic. 00:53:33.610 --> 00:53:36.160 And you might think that this is a straightforward, 00:53:36.160 --> 00:53:38.970 I'm going to add up a couple numbers and figure this out. 00:53:38.970 --> 00:53:42.200 It turns out it's not quite that simple though sportswriters 00:53:42.200 --> 00:53:44.710 would love to think that. 00:53:44.710 --> 00:53:47.280 That it's as simple as adding up a bunch of numbers. 00:53:47.280 --> 00:53:52.840 And obviously, there's network flow lurking in here somewhere. 00:53:52.840 --> 00:53:55.250 Otherwise, I wouldn't be talking about it here. 00:53:55.250 --> 00:54:02.434 So this is not quite historical in terms of the standings. 00:54:02.434 --> 00:54:03.850 I tweaked the numbers a little bit 00:54:03.850 --> 00:54:06.340 in order to make this example a little more interesting. 00:54:06.340 --> 00:54:10.950 But roughly speaking, August 30th, 1996, this 00:54:10.950 --> 00:54:14.490 was the standings corresponding to the American League East. 00:54:14.490 --> 00:54:16.890 And now this is 1996 so the teams 00:54:16.890 --> 00:54:21.060 are a little bit different from what you have. 00:54:21.060 --> 00:54:24.500 So New York, unfortunately, was leading the division even back 00:54:24.500 --> 00:54:25.550 then. 00:54:25.550 --> 00:54:27.910 Still had Baltimore. 00:54:27.910 --> 00:54:30.980 Then there was Boston. 00:54:30.980 --> 00:54:33.440 And then was Toronto. 00:54:33.440 --> 00:54:37.007 And then this is back in 1996, and Detroit was part 00:54:37.007 --> 00:54:38.215 of the [INAUDIBLE] back then. 00:54:38.215 --> 00:54:41.180 This was before realignment. 00:54:41.180 --> 00:54:46.720 So I just called these one, two, three, four, and five. 00:54:46.720 --> 00:54:48.050 Those are the positions. 00:54:48.050 --> 00:54:51.064 All right, so Boston was in third place. 00:54:51.064 --> 00:54:52.480 And so, I'm going to say these are 00:54:52.480 --> 00:54:56.940 w i, which are the wins, so far. l i, which are the losses. 00:54:59.510 --> 00:55:04.300 And r i, which are remaining games to play. 00:55:07.710 --> 00:55:10.390 And This is usually what you see. 00:55:10.390 --> 00:55:19.010 I mean this is the simple version of the standings 00:55:19.010 --> 00:55:25.090 and I'm just going to give you most of these, I should say. 00:55:25.090 --> 00:55:28.677 And I'm not going to write down what Detroit is, simply 00:55:28.677 --> 00:55:30.260 because I'm going to vary that number. 00:55:30.260 --> 00:55:32.690 Because it gets interesting as that number varies. 00:55:32.690 --> 00:55:34.990 And each of these teams had 28 games to play. 00:55:39.380 --> 00:55:42.970 OK so that's usually what you see when you see a snapshot. 00:55:42.970 --> 00:55:54.150 OK now I'm going to also add rij, 00:55:54.150 --> 00:56:01.920 which are the games that these teams play against each other. 00:56:01.920 --> 00:56:04.260 And the reason is simple. 00:56:04.260 --> 00:56:07.350 This linear correlation between win loss column, 00:56:07.350 --> 00:56:09.520 specifically corresponding to games 00:56:09.520 --> 00:56:11.710 that Boston plays with New York. 00:56:11.710 --> 00:56:14.820 No ties in baseball, so Boston wins, 00:56:14.820 --> 00:56:17.500 New York loses and vice versa. 00:56:17.500 --> 00:56:20.490 So I want those numbers as well and these 00:56:20.490 --> 00:56:24.100 are things that you typically don't see 00:56:24.100 --> 00:56:25.620 but they're going to be important, 00:56:25.620 --> 00:56:29.540 because they do determine whether your team is alive 00:56:29.540 --> 00:56:31.620 or eliminated. 00:56:31.620 --> 00:56:40.670 So 5 The reason I have dashes here 00:56:40.670 --> 00:56:45.140 is because the team doesn't play itself in the diagonal. 00:56:49.727 --> 00:56:51.310 AUDIENCE: Is that the number of games? 00:56:51.310 --> 00:56:53.518 PROFESSOR: Yes that is absolutely the number of games 00:56:53.518 --> 00:56:56.070 that they're going to play against each other. 00:56:56.070 --> 00:57:00.460 All right so I've got small number. 00:57:00.460 --> 00:57:03.120 I mean 5 plus 7, 12. 00:57:03.120 --> 00:57:04.809 Plus 4. 00:57:04.809 --> 00:57:05.350 What is that? 00:57:05.350 --> 00:57:06.130 16. 00:57:06.130 --> 00:57:08.050 16 plus 3 is 19. 00:57:08.050 --> 00:57:10.416 There's 28 games left to play, there's 00:57:10.416 --> 00:57:12.790 obviously games that are being played against other teams 00:57:12.790 --> 00:57:14.160 outside the division. 00:57:14.160 --> 00:57:17.110 And so this is just rij, OK? 00:57:17.110 --> 00:57:20.020 So that's the story here. 00:57:20.020 --> 00:57:29.640 And I should say that when I talk about this, what 00:57:29.640 --> 00:57:30.800 are these corresponding to? 00:57:30.800 --> 00:57:34.430 Well this corresponds to NY. 00:57:34.430 --> 00:57:37.880 This column corresponds to Baltimore this corresponds 00:57:37.880 --> 00:57:42.330 to Boston, same order, this corresponds to Toronto. 00:57:42.330 --> 00:57:45.830 This column corresponds to Detroit. 00:57:45.830 --> 00:57:50.010 So otherwise it's under specified. 00:57:50.010 --> 00:57:51.880 Any questions about the table here? 00:57:59.556 --> 00:58:01.180 If you want to know, the Bo Sox did not 00:58:01.180 --> 00:58:02.796 make the playoffs that year. 00:58:07.150 --> 00:58:12.550 So all that most sportswriters do, most people do, 00:58:12.550 --> 00:58:18.685 is say Team i is eliminated. 00:58:22.220 --> 00:58:37.304 If wi plus ri is strictly less than wj for sum j. 00:58:37.304 --> 00:58:39.220 So you're just going to say, OK you know what? 00:58:39.220 --> 00:58:41.240 If I've won all my games, I'm still 00:58:41.240 --> 00:58:43.440 not going to make it to this team that's 00:58:43.440 --> 00:58:47.030 already won 75 games, right? 00:58:47.030 --> 00:58:50.360 And so clearly I'm done. 00:58:50.360 --> 00:58:51.860 That team has already won more games 00:58:51.860 --> 00:58:53.234 than I could possibly win. 00:58:53.234 --> 00:58:54.150 So that's an easy one. 00:58:56.970 --> 00:58:59.330 And so this is strictly less, because we're just 00:58:59.330 --> 00:59:03.650 talking about elimination, total elimination. 00:59:03.650 --> 00:59:08.070 You get your summer vacation, or your fall vacation. 00:59:08.070 --> 00:59:13.240 So let's just say that you had Detroit, 00:59:13.240 --> 00:59:14.980 and you're talking about w 5 years. 00:59:14.980 --> 00:59:20.380 You're talking about this number here, is 46. 00:59:20.380 --> 00:59:25.020 l5 is 88. 00:59:25.020 --> 00:59:28.180 So they're having a bad season here. 00:59:28.180 --> 00:59:34.700 And you're going to say 46 plus 28 equals 00:59:34.700 --> 00:59:38.340 74, which is less than 75. 00:59:38.340 --> 00:59:42.734 Therefore in this scenario, Detroit is eliminated. 00:59:42.734 --> 00:59:43.400 All makes sense? 00:59:46.240 --> 00:59:53.610 So this is sufficient, but I claim it's not necessary. 00:59:53.610 --> 00:59:56.280 So you might actually have false hope 00:59:56.280 --> 01:00:02.130 if you are a Mirror sportswriter, who 01:00:02.130 --> 01:00:04.160 only knows to add up these numbers 01:00:04.160 --> 01:00:06.240 and compare it with others. 01:00:06.240 --> 01:00:09.370 But if you were a 6046 student, and you 01:00:09.370 --> 01:00:12.590 went through these wonderful lectures on network flow, 01:00:12.590 --> 01:00:15.080 you know much more. 01:00:15.080 --> 01:00:21.530 And we're not quite there yet in terms of requiring 01:00:21.530 --> 01:00:23.420 the power of network flow. 01:00:23.420 --> 01:00:32.360 But I ask, what if w5 were 47? 01:00:37.080 --> 01:00:42.120 Can you make an argument which clearly doesn't work here, 01:00:42.120 --> 01:00:46.260 with this naive argument, you have 75 equals 75. 01:00:46.260 --> 01:00:47.595 So you still have hope. 01:00:47.595 --> 01:00:51.900 But can you make an argument that if w5 were 47, 01:00:51.900 --> 01:00:55.750 given the whole table-- that's the hint, the whole table, 01:00:55.750 --> 01:00:59.170 that Detroit was actually eliminated. 01:00:59.170 --> 01:01:01.630 Worth the Frisbee for sure, right? 01:01:01.630 --> 01:01:02.130 Over there. 01:01:02.130 --> 01:01:03.436 I saw you first. 01:01:03.436 --> 01:01:05.792 AUDIENCE: It looks like the only way 01:01:05.792 --> 01:01:09.400 that Detroit can get more wins than New York is 01:01:09.400 --> 01:01:12.050 if they win all their games and New York loses all their games? 01:01:12.050 --> 01:01:12.740 PROFESSOR: Yes. 01:01:12.740 --> 01:01:13.726 AUDIENCE: and if New York loses all their games, 01:01:13.726 --> 01:01:15.698 they're going to lose 7 games to Boston 01:01:15.698 --> 01:01:19.650 And Boston will have 76 wins. 01:01:19.650 --> 01:01:21.190 PROFESSOR: Exactly right. 01:01:21.190 --> 01:01:22.315 So that's a great argument. 01:01:22.315 --> 01:01:25.850 All right, could you stand up? 01:01:25.850 --> 01:01:26.350 All right. 01:01:29.910 --> 01:01:32.980 You catch well too. 01:01:32.980 --> 01:01:37.490 So that's basically the essence of the issue. 01:01:37.490 --> 01:01:40.210 There's stuff going on which corresponds 01:01:40.210 --> 01:01:45.610 to teams playing each other that can affect the outcome. 01:01:45.610 --> 01:01:48.480 OK so let me write out what the gentleman just said. 01:01:52.320 --> 01:02:05.220 Still eliminated, boohoo, because w5 plus r5 01:02:05.220 --> 01:02:06.940 equals 75 now. 01:02:06.940 --> 01:02:10.710 47 plus 28 is 75. 01:02:10.710 --> 01:02:19.420 But either New York or Baltimore-- I 01:02:19.420 --> 01:02:23.650 mean there's many ways you could show this. 01:02:23.650 --> 01:02:25.760 But we'll just take-- this is not 01:02:25.760 --> 01:02:27.390 exactly what the gentleman said. 01:02:27.390 --> 01:02:31.130 But this is what I had here, so let me just write that out. 01:02:31.130 --> 01:02:33.475 Either New York or Baltimore will win 76 games. 01:02:36.700 --> 01:02:38.740 Since they play each other 5 times. 01:02:46.780 --> 01:02:47.280 OK? 01:02:51.840 --> 01:02:58.630 So now I'm going to do some more sophisticated analysis 01:02:58.630 --> 01:03:02.430 and I got what I wanted. 01:03:02.430 --> 01:03:10.780 But let's say that w5 was 48. 01:03:10.780 --> 01:03:14.330 So this was 76 and it kind of worked out 01:03:14.330 --> 01:03:16.060 for both of the examples. 01:03:16.060 --> 01:03:22.000 The one with Boston and the 7 games, that I got as an answer. 01:03:22.000 --> 01:03:25.250 And this one that I just wrote down. 01:03:25.250 --> 01:03:33.265 But what if w5 equals 48? 01:03:37.910 --> 01:03:41.340 Is Detroit eliminated? 01:03:41.340 --> 01:03:44.797 How many people think Detroit is eliminated? 01:03:44.797 --> 01:03:45.880 All right, a bunch of you. 01:03:49.400 --> 01:03:51.800 You want to explain why? 01:03:51.800 --> 01:03:54.240 AUDIENCE: So New York, Baltimore, and Boston 01:03:54.240 --> 01:03:57.690 are guaranteed to get a total of at least 14 more wins. 01:03:57.690 --> 01:03:59.865 Because of their total number of wins 01:03:59.865 --> 01:04:02.274 that they're allowed to have without any of them getting 01:04:02.274 --> 01:04:04.202 to 77 is only going to be 13. 01:04:04.202 --> 01:04:06.612 So it will be 1 plus 5 so that's too many. 01:04:06.612 --> 01:04:08.540 But they're not actually eliminated, 01:04:08.540 --> 01:04:10.903 because this is from the Wild Card round anyway. 01:04:13.669 --> 01:04:16.840 PROFESSOR: I didn't like the last part of your answer. 01:04:16.840 --> 01:04:19.610 Did, but the other question is, did people understand 01:04:19.610 --> 01:04:22.760 the first part of his answer? 01:04:22.760 --> 01:04:23.820 That was complicated. 01:04:23.820 --> 01:04:25.780 I mean it was the right analysis. 01:04:25.780 --> 01:04:27.430 Great job. 01:04:27.430 --> 01:04:30.300 So, we want a systematic way so we 01:04:30.300 --> 01:04:33.940 can run an algorithm so we need to bother 01:04:33.940 --> 01:04:37.040 with this type of analysis. 01:04:37.040 --> 01:04:39.450 And you're exactly right. 01:04:39.450 --> 01:04:42.940 It turns out that I skipped this little detail, 01:04:42.940 --> 01:04:45.450 because it was inconvenient, which 01:04:45.450 --> 01:04:48.870 is that in '95, a wild card was introduced. 01:04:48.870 --> 01:04:51.930 So we could think about this as elimination 01:04:51.930 --> 01:04:54.550 from being the division winner. 01:04:54.550 --> 01:04:56.300 So that's an aside. 01:04:56.300 --> 01:04:59.380 Thanks for pointing out, it's important to know 01:04:59.380 --> 01:05:01.590 sports history. 01:05:01.590 --> 01:05:02.320 Really important. 01:05:02.320 --> 01:05:04.810 Almost as important as 046. 01:05:04.810 --> 01:05:11.060 But basically it gets more and more complicated 01:05:11.060 --> 01:05:16.230 as you get closer and closer to these edge cases. 01:05:16.230 --> 01:05:19.860 And so you can do an analysis, and I'll actually describe 01:05:19.860 --> 01:05:22.620 roughly what your analysis was. 01:05:22.620 --> 01:05:23.340 What's your name? 01:05:23.340 --> 01:05:24.173 AUDIENCE: Alexander. 01:05:24.173 --> 01:05:26.930 PROFESSOR: Alexander's analysis was. 01:05:26.930 --> 01:05:28.310 But I don't want to do that yet. 01:05:28.310 --> 01:05:30.030 I'll do that at the end if you have time. 01:05:30.030 --> 01:05:32.460 I want to give you a general purpose technique that 01:05:32.460 --> 01:05:36.140 obviously is going to use max flow to solve this problem. 01:05:36.140 --> 01:05:36.901 All right? 01:05:36.901 --> 01:05:38.150 So we're going to set this up. 01:05:38.150 --> 01:05:40.680 And so that's actually one of the nice things 01:05:40.680 --> 01:05:42.430 about maximum flow. 01:05:42.430 --> 01:05:43.380 How you can translate. 01:05:43.380 --> 01:05:47.260 You can sort of create these networks from tables 01:05:47.260 --> 01:05:51.320 or what have you, and add capacities to edges, 01:05:51.320 --> 01:05:54.280 find max flows, and solve these problems. 01:05:54.280 --> 01:05:57.570 So this is a good example of that. 01:05:57.570 --> 01:06:01.980 So let's go ahead and do that I am going to draw a flow 01:06:01.980 --> 01:06:06.970 network based on this table. 01:06:06.970 --> 01:06:11.310 It's going to have a source and a sync that are basically 01:06:11.310 --> 01:06:13.880 these dummy nodes that are essentially going 01:06:13.880 --> 01:06:17.012 to source about infinite flow. 01:06:17.012 --> 01:06:18.470 But the capacities of the edges are 01:06:18.470 --> 01:06:19.850 going to come from that table. 01:06:19.850 --> 01:06:23.690 And the edges themselves are straightforward in the sense 01:06:23.690 --> 01:06:28.410 that the graph is going to look the same based 01:06:28.410 --> 01:06:30.860 on the number of teams, and the particular team 01:06:30.860 --> 01:06:32.300 that you're analyzing. 01:06:32.300 --> 01:06:34.880 So this graph is trying to determine-- 01:06:34.880 --> 01:06:46.178 this is a flow network to determine if team 01:06:46.178 --> 01:06:50.455 5, Detroit, is eliminated. 01:06:53.800 --> 01:06:56.590 This is just the flow network. 01:06:56.590 --> 01:07:00.490 I'm going to have a bunch of edges. 01:07:00.490 --> 01:07:06.870 I'm going to just draw this once and we'll be moving things 01:07:06.870 --> 01:07:09.220 around on this. 01:07:09.220 --> 01:07:18.680 2 to 3, 1 to 4, 2 to 4, and lastly 3 to 4. 01:07:18.680 --> 01:07:21.230 And the reason you don't see 5 here 01:07:21.230 --> 01:07:24.915 is because this is an analysis for team 5. 01:07:33.950 --> 01:07:36.250 So the other 4 teams show up here. 01:07:36.250 --> 01:07:38.700 They have edges going to t. 01:07:38.700 --> 01:07:42.710 s is over here, and it's got a bunch of edges 01:07:42.710 --> 01:07:44.600 going to all of these nodes. 01:07:44.600 --> 01:07:46.220 What are these pairs? 01:07:46.220 --> 01:07:50.940 As you can imagine, these pairs correspond to the games 01:07:50.940 --> 01:07:55.220 that each of these teams that are inside the circle 01:07:55.220 --> 01:07:56.710 play against each other. 01:07:56.710 --> 01:07:59.870 So 3 plays 4 a certain number of times. 01:07:59.870 --> 01:08:01.930 According to that table it's 4 times. 01:08:01.930 --> 01:08:04.140 So I'm going to put a 4 in here. 01:08:04.140 --> 01:08:05.420 This is 4 as well. 01:08:05.420 --> 01:08:11.940 4, 5, 7, 2. 01:08:11.940 --> 01:08:13.250 OK? 01:08:13.250 --> 01:08:16.254 And the edges in between here are 01:08:16.254 --> 01:08:18.279 going to have capacities of infinity 01:08:18.279 --> 01:08:22.240 and how are these edges structured? 01:08:22.240 --> 01:08:25.310 So far I've just explained how the left-hand side works. 01:08:25.310 --> 01:08:27.310 These edges have a capacity of infinity. 01:08:27.310 --> 01:08:30.590 1 2 goes to 1, 1 2 goes to 2. 01:08:30.590 --> 01:08:34.149 1 3 goes to 1, 1 3 goes to 3. 01:08:34.149 --> 01:08:36.450 And that's pretty much it. 01:08:36.450 --> 01:08:39.700 So that's where these edges are. 01:08:39.700 --> 01:08:43.540 That's how these edges are introduced. 01:08:43.540 --> 01:08:50.439 And all of these edges have a capacity of infinity. 01:08:50.439 --> 01:09:00.529 2 to 4, 3 to 4, and that. 01:09:00.529 --> 01:09:03.229 So far, it's pretty straightforward. 01:09:03.229 --> 01:09:05.970 There's one last thing that we need to do, 01:09:05.970 --> 01:09:09.939 which is add capacities to these edges. 01:09:09.939 --> 01:09:11.670 This is actually crucial. 01:09:11.670 --> 01:09:14.432 It turns out that we have to add capacities such 01:09:14.432 --> 01:09:17.490 that this max flow is going to represent elimination. 01:09:17.490 --> 01:09:19.490 So these capacities have to be chosen. 01:09:19.490 --> 01:09:22.910 And in particular the equation for this 01:09:22.910 --> 01:09:28.700 is simply w5 plus r5 minus w1. 01:09:31.640 --> 01:09:35.310 In this case if it's 48, I'm asking the question for 48 01:09:35.310 --> 01:09:40.939 for w5 plus 28 minus 75. 01:09:40.939 --> 01:09:43.979 So that's 1. 01:09:43.979 --> 01:09:48.920 And then this thing here would be w5. 01:09:48.920 --> 01:09:54.090 I'll just write that out. w5 plus r5 minus w2, hopefully 01:09:54.090 --> 01:09:57.440 you can read that, and that's a 5 and then similarly this 01:09:57.440 --> 01:10:00.520 is a 7, and that's 13. 01:10:00.520 --> 01:10:03.280 And you might ask, what does this represent? 01:10:03.280 --> 01:10:04.940 Well it kind of represents what you 01:10:04.940 --> 01:10:07.510 think it represents if you just look at that equation. 01:10:07.510 --> 01:10:14.430 It says how much can I push through here 01:10:14.430 --> 01:10:18.780 before I get greater than w1. 01:10:18.780 --> 01:10:19.780 So what is the capacity? 01:10:19.780 --> 01:10:20.905 What's the difference here? 01:10:23.680 --> 01:10:30.980 if I have a 1 here, that means that I still see w5 plus r5, 01:10:30.980 --> 01:10:32.930 I'm 1 greater than w1. 01:10:32.930 --> 01:10:36.190 So in particular, if I want to write this out, 01:10:36.190 --> 01:10:47.430 the capacities are the number of games team 01:10:47.430 --> 01:11:05.730 i can win, and not have more wins than team 5. 01:11:10.970 --> 01:11:15.770 So this is a statement about team 1 01:11:15.770 --> 01:11:18.090 when I see that over there. 01:11:18.090 --> 01:11:19.960 Our team 2 for the other edge. 01:11:19.960 --> 01:11:23.320 And I'm just trying to figure out 01:11:23.320 --> 01:11:29.540 how much can I allow, with respect to this other team, 01:11:29.540 --> 01:11:32.080 to determine elimination for the particular team 01:11:32.080 --> 01:11:34.430 that I'm looking at, team 5 or not. 01:11:34.430 --> 01:11:41.740 So in particular, let's say that team i 01:11:41.740 --> 01:11:44.120 wins one particular game. 01:11:44.120 --> 01:11:46.850 So it goes up to it goes up to 76. 01:11:46.850 --> 01:11:53.850 And if it goes up to 76, it does not necessarily 01:11:53.850 --> 01:11:56.910 have more wins then team 5, because team 5 01:11:56.910 --> 01:11:59.000 can go up to 76 as well. 01:11:59.000 --> 01:12:00.940 That's pretty much it, and the same thing 01:12:00.940 --> 01:12:03.950 for team 2, team 3, and team 4. 01:12:07.410 --> 01:12:15.040 So the intuition here is the following. 01:12:15.040 --> 01:12:19.670 So what we're going to do is compute the max flow 01:12:19.670 --> 01:12:22.280 and look for a certain property on that network. 01:12:22.280 --> 01:12:25.210 And that's going to tell us if team 5 is eliminated 01:12:25.210 --> 01:12:29.180 or not for the particular choice of wi of 48, 01:12:29.180 --> 01:12:32.560 because that's how I constructed this particular example. 01:12:32.560 --> 01:12:45.530 So the intuition is, assume team 5 wins all remaining games. 01:12:45.530 --> 01:12:47.890 That makes perfect sense. 01:12:47.890 --> 01:12:49.200 This is the best case scenario. 01:12:49.200 --> 01:12:51.830 You're trying to figure out, in the best case scenario, 01:12:51.830 --> 01:12:53.340 are you eliminated or not? 01:12:53.340 --> 01:12:55.520 Is team 5 eliminated or not? 01:12:55.520 --> 01:13:00.620 And what you want to do is divvy up the remaining games. 01:13:03.440 --> 01:13:07.830 And that corresponds to sending flow through these edges. 01:13:07.830 --> 01:13:12.780 So flow through these edges corresponds to assigning wins 01:13:12.780 --> 01:13:16.440 to teams 1, 2, 3, and 4. 01:13:16.440 --> 01:13:18.190 That's the key. 01:13:18.190 --> 01:13:28.330 Divvy up the remaining games, so all teams have less than 01:13:28.330 --> 01:13:32.440 or equal to w5 plus r5 wins. 01:13:32.440 --> 01:13:35.030 So what does it mean if all teams have less than 01:13:35.030 --> 01:13:37.620 or equal to w5 plus r5 wins? 01:13:37.620 --> 01:13:39.620 If you can divvied up such that that's the case, 01:13:39.620 --> 01:13:44.080 what does that mean for team 5? 01:13:44.080 --> 01:13:45.650 Team five is still in the game. 01:13:45.650 --> 01:13:48.250 Team 5 is not eliminated. 01:13:48.250 --> 01:14:01.850 So if you can do this, then team 5 is not eliminated 01:14:01.850 --> 01:14:03.910 If you can't team 5 is eliminated. 01:14:03.910 --> 01:14:11.760 Because some team is going to get 77 wins. 01:14:11.760 --> 01:14:14.400 And so that analysis that Alexander carried out 01:14:14.400 --> 01:14:16.240 wasn't a flow analysis. 01:14:16.240 --> 01:14:18.732 But somehow, that's got to be in here. 01:14:18.732 --> 01:14:20.190 That type of analysis has got to be 01:14:20.190 --> 01:14:24.210 in here to show that some team is going to get to 77 wins. 01:14:24.210 --> 01:14:28.440 Because you know, well thanks to Alexander, 01:14:28.440 --> 01:14:31.870 that Detroit is going to be eliminated here. 01:14:31.870 --> 01:14:35.470 But we want to show that in terms of the max flow 01:14:35.470 --> 01:14:37.010 on this network. 01:14:39.730 --> 01:14:50.586 So there is an associated theorem that I'll write. 01:14:50.586 --> 01:14:52.400 We've got enough time here. 01:14:52.400 --> 01:14:55.710 We just have to find the max flow for this. 01:14:55.710 --> 01:14:58.340 And that should take just a minute. 01:14:58.340 --> 01:15:05.040 But I want to set it up so that's pretty much the end. 01:15:05.040 --> 01:15:08.400 Finding the max flow, what that means. 01:15:08.400 --> 01:15:10.490 But the theorem is, which is essentially 01:15:10.490 --> 01:15:15.210 a more precise restatement of this observation, 01:15:15.210 --> 01:15:21.170 is that team 5-- obviously you can do this for any team-- 01:15:21.170 --> 01:15:22.020 is eliminated. 01:15:24.590 --> 01:15:37.760 If and only if the max flow does not saturate 01:15:37.760 --> 01:15:42.705 all edges leaving the source. 01:15:46.394 --> 01:15:56.090 I.e. in this case, the max flow is strictly less than 26, 01:15:56.090 --> 01:15:58.470 which corresponds to the sum of those things. 01:15:58.470 --> 01:16:01.310 5 plus 7 plus, yadda yadda yadda. 01:16:01.310 --> 01:16:05.490 And so what does it mean to saturate all of the edges? 01:16:05.490 --> 01:16:07.970 It means that all the games have been played. 01:16:07.970 --> 01:16:12.330 All the relevant games here that affect team 5 have been played. 01:16:12.330 --> 01:16:14.350 So you want to play all the games, 01:16:14.350 --> 01:16:18.600 because that's the end game, if you will. 01:16:18.600 --> 01:16:21.980 But basically what that intuition here 01:16:21.980 --> 01:16:26.060 is that saturation of those edges 01:16:26.060 --> 01:16:34.005 corresponds to playing all remaining games. 01:16:39.930 --> 01:16:48.380 And the argument is not a proof if you 01:16:48.380 --> 01:17:01.690 can't play all the remaining games, 01:17:01.690 --> 01:17:15.660 without exceeding the capacity of i to t edges, 01:17:15.660 --> 01:17:16.535 team 5 is eliminated. 01:17:25.040 --> 01:17:29.400 So you have to saturate, because you have to play all the games. 01:17:29.400 --> 01:17:33.140 And when you do that saturation, obviously your flow network 01:17:33.140 --> 01:17:37.350 has to satisfy its laws corresponding to the capacity 01:17:37.350 --> 01:17:38.750 that we have in there. 01:17:38.750 --> 01:17:42.040 And it's quite possible that this 01:17:42.040 --> 01:17:46.420 is going to cause a restriction over here that corresponds 01:17:46.420 --> 01:17:49.240 to requiring that not all the games 01:17:49.240 --> 01:17:54.850 we play in order for Detroit to survive. 01:17:54.850 --> 01:17:56.660 and that's basically the game. 01:17:56.660 --> 01:17:59.880 So saturating the network means playing all the games. 01:17:59.880 --> 01:18:05.000 And if you get a max flow, to be more concrete, 01:18:05.000 --> 01:18:12.380 if you can get a max flow here that's less than 26, 01:18:12.380 --> 01:18:16.260 then you've saturated all the edges 01:18:16.260 --> 01:18:21.260 and this means that you played all the games. 01:18:21.260 --> 01:18:26.550 And you found this team that beats you. 01:18:26.550 --> 01:18:28.960 That beats team 5. 01:18:28.960 --> 01:18:33.950 So what happens over here? 01:18:33.950 --> 01:18:38.510 Let's take a look at this and all you have to do here, 01:18:38.510 --> 01:18:41.180 is to find the max flow and then we 01:18:41.180 --> 01:18:43.590 can go back to the statement of the theorem. 01:18:43.590 --> 01:18:48.680 So I want to find the min cut corresponding 01:18:48.680 --> 01:18:51.470 to this, which is going to tell me what the max flow is. 01:18:51.470 --> 01:18:54.450 I could go around for focus in on this, and start with 0 01:18:54.450 --> 01:18:56.530 and augment, and so on and so forth. 01:18:56.530 --> 01:18:59.850 But there's another way which is not necessarily algorithmic, 01:18:59.850 --> 01:19:01.330 but eyeballs. 01:19:01.330 --> 01:19:08.140 Use your eyeballs and find me the min cut here. 01:19:08.140 --> 01:19:11.430 Which has the minimum capacity. 01:19:11.430 --> 01:19:13.700 The min cut, the minimum capacity 01:19:13.700 --> 01:19:16.720 is going to tell me what my max flow is. 01:19:16.720 --> 01:19:19.880 We know that from the max-flow min-cut theorem. 01:19:19.880 --> 01:19:25.680 So I want to cut this network into capital S and capital T 01:19:25.680 --> 01:19:30.680 and find the max flow. 01:19:30.680 --> 01:19:33.210 So any ideas? 01:19:33.210 --> 01:19:35.110 It's a little hard. 01:19:35.110 --> 01:19:38.910 So I see some people waving their hands. 01:19:38.910 --> 01:19:40.670 All right, let me just do that since we're 01:19:40.670 --> 01:19:41.630 running out of time. 01:19:41.630 --> 01:19:43.270 It's kind of cool. 01:19:43.270 --> 01:19:45.150 So I'm going to use a different color. 01:19:45.150 --> 01:19:48.980 We're going to go like that, yeah. 01:19:48.980 --> 01:19:57.060 and then veer over this way, and then jump up like that, 01:19:57.060 --> 01:20:00.510 and then come over like that. 01:20:00.510 --> 01:20:09.450 And this is S. And on this side is T. 01:20:09.450 --> 01:20:14.330 So I got small s and capital S, small t and cap T. 01:20:14.330 --> 01:20:18.250 But I got this 4 in cap T. Notice that I got 3 to 4, 01:20:18.250 --> 01:20:24.500 2 to 4, 1 to 4 in cap T and I got these other ones in cap S. 01:20:24.500 --> 01:20:27.250 So all I've done here, forget baseball elimination, 01:20:27.250 --> 01:20:29.160 is find a min-cut. 01:20:29.160 --> 01:20:33.740 And if you look at what the value of this min-cut is, 01:20:33.740 --> 01:20:37.075 the value of the min-cut is simply 01:20:37.075 --> 01:20:42.330 the capacities of the edges that go from S to T. 01:20:42.330 --> 01:20:47.650 So it's simply the S to T edges and summing 01:20:47.650 --> 01:20:49.030 over the capacities. 01:20:49.030 --> 01:20:50.995 And if you take a look, obviously S to T, 01:20:50.995 --> 01:20:52.810 you've got 4 plus 4 plus 4. 01:20:52.810 --> 01:20:56.020 So you've got 4 plus 4 plus 4, corresponding 01:20:56.020 --> 01:20:59.860 to this one, this one, and that one. 01:20:59.860 --> 01:21:02.630 Do I need to add this edge in here? 01:21:02.630 --> 01:21:03.130 Over here? 01:21:07.050 --> 01:21:08.290 Where does this edge go? 01:21:08.290 --> 01:21:10.540 From this to over there? 01:21:10.540 --> 01:21:12.824 That goes from T to S. So that's good 01:21:12.824 --> 01:21:14.490 because that has a capacity of infinity. 01:21:14.490 --> 01:21:16.680 That would cause trouble. 01:21:16.680 --> 01:21:21.580 And so the other edges are I got 1, 5, and 7. 01:21:21.580 --> 01:21:25.680 So I need this one, that one, and that one. 01:21:25.680 --> 01:21:33.400 So 4 plus 4 plus 4, plus 1 plus 5 plus 7 equals 25. 01:21:33.400 --> 01:21:41.350 And so this implies elimination, because what I've done 01:21:41.350 --> 01:21:46.050 is found the max flow, which is strictly less than 26. 01:21:46.050 --> 01:21:49.280 Which means that I have not saturated all of the edges that 01:21:49.280 --> 01:21:50.540 come out of s. 01:21:50.540 --> 01:21:53.730 I have not been able to play all the games. 01:21:53.730 --> 01:21:55.860 And so if I'd played all the games, 01:21:55.860 --> 01:21:59.100 I'd be exceeding some capacity constraint. 01:21:59.100 --> 01:22:00.800 And if I push more flow in there, 01:22:00.800 --> 01:22:03.830 I'd be exceeding some capacity constraint on this side. 01:22:03.830 --> 01:22:07.880 And that would imply that Detroit 01:22:07.880 --> 01:22:11.190 is eliminated, because some team would have 77 wins. 01:22:11.190 --> 01:22:16.140 All right, so hopefully you've got the gist of it. 01:22:16.140 --> 01:22:17.510 I'll put this in the notes. 01:22:17.510 --> 01:22:19.510 Take a long, hard look at it. 01:22:19.510 --> 01:22:22.340 That particular values aren't important. 01:22:22.340 --> 01:22:25.100 The framework of translation is important. 01:22:25.100 --> 01:22:28.780 And you can certainly ask questions of your TAs 01:22:28.780 --> 01:22:29.330 on Friday. 01:22:29.330 --> 01:22:32.220 And I'll stick around here for questions.