WEBVTT 00:00:00.499 --> 00:00:02.760 The following content is provided under a Creative 00:00:02.760 --> 00:00:04.280 Commons license. 00:00:04.280 --> 00:00:06.620 Your support will help MIT OpenCourseWare 00:00:06.620 --> 00:00:10.980 continue to offer high quality educational resources for free. 00:00:10.980 --> 00:00:13.600 To make a donation or view additional materials 00:00:13.600 --> 00:00:17.496 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.496 --> 00:00:18.121 at ocw.mit.edu. 00:00:23.580 --> 00:00:24.920 PROFESSOR: OK let's get started. 00:00:27.790 --> 00:00:31.480 I once taught with a professor who 00:00:31.480 --> 00:00:36.100 was lamenting the fact that as the term progresses attendance 00:00:36.100 --> 00:00:37.830 in lecture tends to drop off. 00:00:37.830 --> 00:00:40.640 And gets pretty dramatic by the end of the term 00:00:40.640 --> 00:00:43.760 when you're lecturing, and nobody's there. 00:00:43.760 --> 00:00:45.980 And I asked him what he did about it. 00:00:45.980 --> 00:00:48.180 And he thought about it and he said, 00:00:48.180 --> 00:00:51.030 there's only two things that can get students to come 00:00:51.030 --> 00:00:55.840 to lecture, candy and sex. 00:00:55.840 --> 00:01:01.510 Now we've already tried candy, so today we're 00:01:01.510 --> 00:01:04.410 going to talk about sex. 00:01:04.410 --> 00:01:07.600 In fact we're going to use graph theory 00:01:07.600 --> 00:01:10.630 to address a decades old debate concerning 00:01:10.630 --> 00:01:15.240 the relative promiscuity of men versus women. 00:01:15.240 --> 00:01:19.410 Now graphs are incredibly useful structures in computer science, 00:01:19.410 --> 00:01:22.270 and we're going to be studying them for the next five or six 00:01:22.270 --> 00:01:23.520 lectures. 00:01:23.520 --> 00:01:27.800 They come up in all sorts of applications, scheduling, 00:01:27.800 --> 00:01:31.070 optimization, communications, the design 00:01:31.070 --> 00:01:32.710 and analysis of algorithms. 00:01:32.710 --> 00:01:35.310 In fact next week, you're going to see how to Stanford 00:01:35.310 --> 00:01:38.990 graduate students became gazillionaires 00:01:38.990 --> 00:01:43.200 because they use graph theoretic in a clever way. 00:01:43.200 --> 00:01:46.470 But let's talk about sex. 00:01:46.470 --> 00:01:48.640 The issue that we're going to address today 00:01:48.640 --> 00:01:52.660 is one of the most talked about, and most well studied, 00:01:52.660 --> 00:01:55.900 questions in all of human sociology. 00:01:55.900 --> 00:02:00.400 On average, who has more opposite gender partners, 00:02:00.400 --> 00:02:02.715 men or women? 00:02:02.715 --> 00:02:05.310 Now opposite gender is going to be important. 00:02:05.310 --> 00:02:09.035 And by this I mean, one boy, and one girl. 00:02:09.035 --> 00:02:11.035 All right, I'm not making a political statement. 00:02:11.035 --> 00:02:16.770 It's just that the math is a lot easier that way, as you'll see. 00:02:16.770 --> 00:02:18.860 Now I'd like to start by taking a pole here 00:02:18.860 --> 00:02:21.270 to see what you think about that. 00:02:21.270 --> 00:02:24.830 So raise your and if you think men, on average, 00:02:24.830 --> 00:02:27.910 have more opposite gender partners than women do. 00:02:30.420 --> 00:02:31.095 Only a few. 00:02:31.095 --> 00:02:33.401 AUDIENCE: In life or [INAUDIBLE] 00:02:33.401 --> 00:02:34.400 PROFESSOR: Um, you can-- 00:02:34.400 --> 00:02:39.040 [LAUGHTER] 00:02:39.040 --> 00:02:40.640 PROFESSOR: One on one. 00:02:40.640 --> 00:02:43.970 OK, so let's say over the course of their lives, let's say, 00:02:43.970 --> 00:02:48.200 or over the course of 2010, that men in America 00:02:48.200 --> 00:02:51.460 have more opposite-gender partners than women in America, 00:02:51.460 --> 00:02:52.830 say in 2010. 00:02:52.830 --> 00:02:55.030 Raise your hand if you think men have more going on. 00:02:55.030 --> 00:02:57.040 All right a bunch of you. 00:02:57.040 --> 00:02:58.730 Raise your hand if you think women have 00:02:58.730 --> 00:03:00.890 more opposite-gender partners? 00:03:00.890 --> 00:03:02.490 This is unusual. 00:03:02.490 --> 00:03:04.870 Maybe even more voted for women, but it's close. 00:03:04.870 --> 00:03:07.744 Raise your hand if you think it's equal. 00:03:07.744 --> 00:03:09.680 All right, about the same. 00:03:09.680 --> 00:03:12.550 Raise your hand if you think there's no way to know, 00:03:12.550 --> 00:03:15.210 that it's hopeless to really figure it out. 00:03:15.210 --> 00:03:17.210 All right, nobody goes for that. 00:03:17.210 --> 00:03:18.565 All right, good. 00:03:18.565 --> 00:03:20.460 All right well now in the popular literature, 00:03:20.460 --> 00:03:23.620 I think the feelings are different than expressed here. 00:03:23.620 --> 00:03:25.450 Pretty much universally, in the literature, 00:03:25.450 --> 00:03:28.000 it's believed that men have more opposite-gender partners 00:03:28.000 --> 00:03:29.216 than women. 00:03:29.216 --> 00:03:30.590 And in fact, you could even think 00:03:30.590 --> 00:03:33.120 about that, if you think about literature, 00:03:33.120 --> 00:03:35.780 the leader of the harem is always a man. 00:03:35.780 --> 00:03:37.680 And he's got lots of women. 00:03:37.680 --> 00:03:40.540 In polygamist cultures, it's always 00:03:40.540 --> 00:03:44.890 the man that has multiple wives, not the reverse. 00:03:44.890 --> 00:03:47.410 Now not surprisingly, this issue has 00:03:47.410 --> 00:03:49.620 been studied "scientifically," I'll 00:03:49.620 --> 00:03:54.520 put in quotes, extensively, in one of the largest studies ever 00:03:54.520 --> 00:03:56.220 done. 00:03:56.220 --> 00:03:58.210 Researchers from University of Chicago 00:03:58.210 --> 00:04:03.200 interviewed 2,500 people, at random, over several years. 00:04:03.200 --> 00:04:05.800 They brought them in, on many occasions, 00:04:05.800 --> 00:04:08.920 to try to get the answer for the question once and for all. 00:04:08.920 --> 00:04:11.560 And they wrote this 700 page book, 00:04:11.560 --> 00:04:14.680 called The soul of Social Organization of Sexuality: 00:04:14.680 --> 00:04:18.050 Sexual Practices in the US. 00:04:18.050 --> 00:04:19.670 Actually walking around with this book 00:04:19.670 --> 00:04:22.720 has proved to be a little embarrassing. 00:04:22.720 --> 00:04:25.890 Last week my 11-year-old daughter saw it, 00:04:25.890 --> 00:04:29.410 and she goes dad, why do you have this sex book. 00:04:29.410 --> 00:04:32.700 And I grabbed it back and said, well that's for the course. 00:04:32.700 --> 00:04:34.910 I'm teaching. 00:04:34.910 --> 00:04:37.040 And I thought I'd gotten away with it, 00:04:37.040 --> 00:04:38.930 and everything was fine. 00:04:38.930 --> 00:04:42.150 And then later that day she texted all of our friends 00:04:42.150 --> 00:04:44.330 about the new news that what do you know, 00:04:44.330 --> 00:04:48.960 her dad teaches sex ed at MIT. 00:04:48.960 --> 00:04:53.950 Anyway this study concludes that on average men 00:04:53.950 --> 00:05:00.430 have 74% more opposite-gender partners than women. 00:05:00.430 --> 00:05:01.925 There's one other central claims. 00:05:10.389 --> 00:05:11.305 And this is in the US. 00:05:36.420 --> 00:05:43.350 OK now, when you think about it that sounds maybe reasonable, 00:05:43.350 --> 00:05:45.390 might be OK. 00:05:45.390 --> 00:05:49.750 But not according to ABC News. 00:05:49.750 --> 00:05:54.315 They did a poll of 1,500 people in the country, in 2004, 00:05:54.315 --> 00:05:59.090 and concluded that the average disparity is much greater. 00:05:59.090 --> 00:06:01.950 In particular, in this study, they 00:06:01.950 --> 00:06:05.210 said that the average man has 20 partners-- 00:06:05.210 --> 00:06:07.100 I'm assuming over their lifetime-- 00:06:07.100 --> 00:06:10.050 and the average woman has six. 00:06:10.050 --> 00:06:12.636 And this gives a disparity 233%. 00:06:21.220 --> 00:06:30.630 So ABC News did a smaller survey says that it's 233% here, 00:06:30.630 --> 00:06:33.540 much more than 74%. 00:06:33.540 --> 00:06:36.800 Now ABC News claimed this is one of the most 00:06:36.800 --> 00:06:39.060 scientific studies ever done. 00:06:39.060 --> 00:06:42.430 And there was a 2.5% margin of error. 00:06:42.430 --> 00:06:45.540 Now we'll actually talk about what that means mathematically 00:06:45.540 --> 00:06:47.570 later in the term when we do probability, and do 00:06:47.570 --> 00:06:49.512 study polling. 00:06:49.512 --> 00:06:50.970 Now of course I should also mention 00:06:50.970 --> 00:06:52.770 that ABC News is the one that said 00:06:52.770 --> 00:06:57.720 Al Gore won the presidential election in 2000. 00:06:57.720 --> 00:07:01.930 Now the study is called American Sex Survey, 00:07:01.930 --> 00:07:04.740 a Peak Between the Sheets. 00:07:04.740 --> 00:07:06.530 That doesn't sound so scientific. 00:07:06.530 --> 00:07:09.990 And it was on TV, on Primetime Live in 2004. 00:07:09.990 --> 00:07:12.300 The promo for this is really good. 00:07:12.300 --> 00:07:16.740 It says, a groundbreaking ABC News Primetime Live survey 00:07:16.740 --> 00:07:20.030 finds a range of eye popping sexual activities, fantasies, 00:07:20.030 --> 00:07:22.460 and attitudes in this country, confirming 00:07:22.460 --> 00:07:25.700 some conventional wisdom, exploding some myths, 00:07:25.700 --> 00:07:28.560 and venturing where few scientific surveys have 00:07:28.560 --> 00:07:29.972 gone before. 00:07:29.972 --> 00:07:31.680 By the end of today, we're going to agree 00:07:31.680 --> 00:07:34.390 with that last statement. 00:07:34.390 --> 00:07:39.250 OK now who do you think's right? 00:07:39.250 --> 00:07:41.980 University of Chicago. 00:07:41.980 --> 00:07:45.010 Who votes for 74% as being pretty close? 00:07:45.010 --> 00:07:47.190 A few of you. 00:07:47.190 --> 00:07:48.890 I've already slammed these guys. 00:07:48.890 --> 00:07:51.830 Who votes for ABC News as being more accurate? 00:07:51.830 --> 00:07:54.980 Yeah, nobody. 00:07:54.980 --> 00:07:58.210 Who votes for no way to tell? 00:07:58.210 --> 00:08:00.360 I got some votes there, all right. 00:08:00.360 --> 00:08:03.230 So how do you tackle this problem? 00:08:03.230 --> 00:08:06.414 In theory we could do our own 6.042 survey. 00:08:06.414 --> 00:08:08.080 I don't know how much we'd really learn, 00:08:08.080 --> 00:08:10.430 and for sure I'd get fired. 00:08:10.430 --> 00:08:14.030 So I don't think we're going to do that. 00:08:14.030 --> 00:08:16.310 But fortunately, this is the kind of question 00:08:16.310 --> 00:08:17.940 that could be handled, and actually 00:08:17.940 --> 00:08:21.430 answered, by graph theory, even though it might 00:08:21.430 --> 00:08:23.680 be more interesting to interview thousands of people, 00:08:23.680 --> 00:08:25.640 and find out what's going on. 00:08:25.640 --> 00:08:28.760 That's not as efficient as using graphs. 00:08:28.760 --> 00:08:33.070 So let me start by defining what a graph is. 00:08:33.070 --> 00:08:37.210 Informally graph is just a bunch of dots and lines 00:08:37.210 --> 00:08:40.374 connecting the dots, it's actually very simple. 00:08:40.374 --> 00:08:41.165 So here's to graph. 00:08:46.830 --> 00:08:49.460 These are the nodes, and they're connected 00:08:49.460 --> 00:08:51.215 with these lines, called edges. 00:08:55.480 --> 00:08:57.530 And often the nodes, and sometimes the edges, 00:08:57.530 --> 00:08:58.770 are labeled. 00:08:58.770 --> 00:09:10.323 For example, we might call this x1, x2, x3, x4, x5, x6, and x7. 00:09:10.323 --> 00:09:13.090 So that's an example of a graph. 00:09:13.090 --> 00:09:14.940 Now this being a math class, we got 00:09:14.940 --> 00:09:17.747 to give a formal definition of a graph. 00:09:17.747 --> 00:09:19.580 And we'll usually use the formal definition. 00:09:23.910 --> 00:09:41.310 A graph G is a pair of sets often called V and E. 00:09:41.310 --> 00:09:45.850 Where V is a set of elements called vertices or nodes. 00:09:45.850 --> 00:09:48.330 And it has to be non-empty here in this class. 00:10:06.140 --> 00:10:08.820 And we'll go back and forth between vertices and nodes. 00:10:08.820 --> 00:10:13.270 Even the text we use both words interchangeably. 00:10:13.270 --> 00:10:26.590 And E is a set of 2-item subsets and V, 00:10:26.590 --> 00:10:27.950 and they're called edges. 00:10:32.970 --> 00:10:37.060 So for example, over here in this picture, 00:10:37.060 --> 00:10:48.950 V is the set of nodes is x1, x2, x3, up to x7, that's the nodes. 00:10:48.950 --> 00:10:54.430 And E, the set of edges, is pairs, 00:10:54.430 --> 00:10:58.700 unordered pairs of vertices. 00:10:58.700 --> 00:11:04.615 So for example x1, x2 is an edge. 00:11:04.615 --> 00:11:07.660 And it's the same as the set x2, x1, 00:11:07.660 --> 00:11:09.210 doesn't matter the order here. 00:11:09.210 --> 00:11:11.680 Later in a week or so, we'll talk about directed graphs 00:11:11.680 --> 00:11:14.600 where the order matters. 00:11:14.600 --> 00:11:21.750 x1, x3 is also an edge here, and so one. 00:11:21.750 --> 00:11:25.231 Think we've got, let's see, 1, 2, 3, 4, 5, 6, 00:11:25.231 --> 00:11:27.570 7 edges in this graph. 00:11:27.570 --> 00:11:31.250 And the last one would be x5, x7. 00:11:35.300 --> 00:11:41.190 Edges are also sometimes written with this notation, x1 line x2, 00:11:41.190 --> 00:11:42.877 is another notation. 00:11:42.877 --> 00:11:44.960 And then later when you talk about directed edges, 00:11:44.960 --> 00:11:49.710 we'll put a little arrowhead on one end of this. 00:11:49.710 --> 00:11:52.110 Now the definition of a graph is really pretty simple. 00:11:52.110 --> 00:11:55.290 Just think of it as dots and lines, if you want. 00:11:55.290 --> 00:11:59.440 But there's often differences in how people define graphs. 00:11:59.440 --> 00:12:04.550 For example, in this class we don't allow the empty graph, 00:12:04.550 --> 00:12:06.742 i.e. the graph with no nodes. 00:12:06.742 --> 00:12:08.950 So we're going to insist that every graph has to have 00:12:08.950 --> 00:12:10.270 at least one node in it. 00:12:10.270 --> 00:12:11.770 And that's just to make the theorems 00:12:11.770 --> 00:12:13.010 we're going prove be true. 00:12:13.010 --> 00:12:14.510 Otherwise there's some theorems that 00:12:14.510 --> 00:12:16.880 are false for the special case of the empty graph. 00:12:16.880 --> 00:12:19.270 But we don't require the graph to have any edges. 00:12:19.270 --> 00:12:23.330 In fact, it's possible you have a graph with nodes, 00:12:23.330 --> 00:12:23.945 but no edges. 00:12:26.500 --> 00:12:28.560 For example, this graph. 00:12:28.560 --> 00:12:29.310 Three-node graph. 00:12:33.560 --> 00:12:42.730 So here G equals VE, V equals x1, x2, x3. 00:12:42.730 --> 00:12:44.210 And E is just the empty set. 00:12:48.110 --> 00:12:53.060 Now for a general graph, when you do have edges, 00:12:53.060 --> 00:13:04.005 we say that two nodes, call them xi and xj, 00:13:04.005 --> 00:13:13.120 are adjacent if they're connected by an edge, namely 00:13:13.120 --> 00:13:18.160 if xi xj is an edge. 00:13:18.160 --> 00:13:24.630 All right so for example, x5 is adjacent to x7, 00:13:24.630 --> 00:13:29.040 but it's not adjacent to x4, there's no edge there. 00:13:33.160 --> 00:13:37.810 Closely related is the definition of the incidence. 00:13:37.810 --> 00:13:48.410 An edge E, which is xixj, is said 00:13:48.410 --> 00:13:55.560 to be incident to its end points, xi and xj. 00:13:59.120 --> 00:14:06.400 OK so, for example, if I labeled that edge as E, E 00:14:06.400 --> 00:14:11.435 is the edge x1, x2, and this incident to x1, and incident 00:14:11.435 --> 00:14:11.935 x2. 00:14:18.320 --> 00:14:20.600 Then we can talk about the degree of a node. 00:14:23.410 --> 00:14:38.180 The number of edges incident to a node 00:14:38.180 --> 00:14:40.030 is called the degree of the node. 00:14:51.510 --> 00:14:57.520 So for example, what's the degree of x5 over here? 00:14:57.520 --> 00:15:05.010 3, so in this case, the degree of x5 equals 3. 00:15:05.010 --> 00:15:10.920 The degree of x7 is 1. 00:15:10.920 --> 00:15:14.190 These guys all have degree 0, there's 00:15:14.190 --> 00:15:17.980 no edges incident to them. 00:15:17.980 --> 00:15:21.790 Now in this class, we're going to look at only simple graphs, 00:15:21.790 --> 00:15:23.000 at lease for a while. 00:15:26.570 --> 00:15:40.950 A graph is simple if it has no loops, or multiple edges. 00:15:48.200 --> 00:15:53.750 Now a loop is an edge that only connects up one node, that's 00:15:53.750 --> 00:15:56.250 a loop and we don't allow it. 00:15:56.250 --> 00:16:00.070 A multiple edge is we've got two edges that are really the same, 00:16:00.070 --> 00:16:02.980 they connect the same endpoints. 00:16:02.980 --> 00:16:04.040 Also called a multi-edge. 00:16:08.290 --> 00:16:10.620 And those we're not going to have in simple graphs. 00:16:10.620 --> 00:16:11.975 We don't allow this. 00:16:11.975 --> 00:16:15.200 We don't allow that. 00:16:15.200 --> 00:16:17.920 Any questions so far about what a graph is? 00:16:22.490 --> 00:16:26.390 So how are we going to use a graph 00:16:26.390 --> 00:16:32.470 to model the problem of opposite-gender partners? 00:16:32.470 --> 00:16:34.890 That's the question we're after. 00:16:34.890 --> 00:16:38.052 So any thoughts about what the nodes of the graph 00:16:38.052 --> 00:16:39.010 are going to represent? 00:16:41.840 --> 00:16:42.440 What is it? 00:16:42.440 --> 00:16:43.930 AUDIENCE: Males and females? 00:16:43.930 --> 00:16:45.370 PROFESSOR: People. 00:16:45.370 --> 00:16:47.670 Yeah, so we're going to have people. 00:16:47.670 --> 00:16:50.290 In fact, there's two kinds of people here. 00:16:50.290 --> 00:16:52.630 There's men, and women. 00:16:55.370 --> 00:17:01.520 All right we got nodes here for the men. 00:17:01.520 --> 00:17:05.400 And in fact in America, there's a lot of nodes here. 00:17:05.400 --> 00:17:07.750 All right, and so this might be oh I don't know, 00:17:07.750 --> 00:17:14.310 say that's Tom Cruise and Nicole Kidman. 00:17:14.310 --> 00:17:16.944 Now what's the edge going to represent? 00:17:16.944 --> 00:17:18.420 AUDIENCE: Partners. 00:17:18.420 --> 00:17:20.020 PROFESSOR: Partners. 00:17:20.020 --> 00:17:23.109 They were opposite-gender partners. 00:17:23.109 --> 00:17:26.470 And there's actually more edges probably here. 00:17:26.470 --> 00:17:34.200 We could have Penelope here, and Katie here. 00:17:34.200 --> 00:17:37.400 And well probably lots more, I probably don't know them all. 00:17:37.400 --> 00:17:41.250 And Ben's over here with Nicole. 00:17:41.250 --> 00:17:48.054 And Nicole got Jude and Keith. 00:17:48.054 --> 00:17:49.470 There's actually a website you can 00:17:49.470 --> 00:17:53.500 go to get a lot of these things here. 00:17:53.500 --> 00:17:56.830 And Katie went with Josh. 00:17:56.830 --> 00:17:59.779 It's called whosedatedwho.com, and you get big graph, 00:17:59.779 --> 00:18:01.320 you could start filling in the edges. 00:18:01.320 --> 00:18:03.690 I don't know how reliable it is. 00:18:03.690 --> 00:18:06.340 Now it's really critical that we're only looking edges 00:18:06.340 --> 00:18:08.270 from here to here. 00:18:08.270 --> 00:18:10.650 All right, so if there's an edge between Tom and Ben, 00:18:10.650 --> 00:18:13.740 I don't want to know about it. 00:18:13.740 --> 00:18:17.540 Just opposite-gender partners. 00:18:17.540 --> 00:18:22.090 OK now in the USA, the number of nodes 00:18:22.090 --> 00:18:25.590 here is about 300 million. 00:18:28.640 --> 00:18:30.220 About three million people. 00:18:30.220 --> 00:18:35.080 And the number of men nodes, male nodes, call these VM, 00:18:35.080 --> 00:18:39.950 and this is VW, by the way, I'm using cardinality notation. 00:18:39.950 --> 00:18:42.750 When I put bars around a set, that 00:18:42.750 --> 00:18:46.050 is the denoting how many are in the set. 00:18:46.050 --> 00:18:53.360 In the US there's about 147.6 men out of the 300. 00:18:53.360 --> 00:18:58.820 And the number of women-- oh we got a w here-- 00:18:58.820 --> 00:19:03.240 is about 152.4 million. 00:19:03.240 --> 00:19:05.740 So there's a little bit more nodes 00:19:05.740 --> 00:19:08.420 on this side of the graph, than that side in the US. 00:19:10.960 --> 00:19:12.130 What about the edges? 00:19:12.130 --> 00:19:14.115 Any idea of how many edges there are here? 00:19:17.097 --> 00:19:17.680 We don't know. 00:19:17.680 --> 00:19:21.070 I sure as heck don't know how many edges there are. 00:19:21.070 --> 00:19:22.410 So that we don't know. 00:19:22.410 --> 00:19:26.160 The cardinality of the edge set we don't know, 00:19:26.160 --> 00:19:27.940 and we're not likely to figure out. 00:19:27.940 --> 00:19:30.420 I don't even think these surveys, really, 00:19:30.420 --> 00:19:32.900 can estimate that. 00:19:32.900 --> 00:19:35.430 But what we're trying to figure out 00:19:35.430 --> 00:19:40.570 is the ratio of the average degree of the men, 00:19:40.570 --> 00:19:44.310 to the average degree of the women. 00:19:44.310 --> 00:19:46.280 Because the number of opposite-gender partners 00:19:46.280 --> 00:19:48.640 you have is your degree here, and you're 00:19:48.640 --> 00:19:50.770 looking for the average guy degree, 00:19:50.770 --> 00:19:54.370 compared to the average female degree here. 00:19:54.370 --> 00:19:55.390 That's what we're after. 00:19:55.390 --> 00:19:57.835 All right so let's find that quantity. 00:20:00.740 --> 00:20:10.020 Let's let A sub m equal the average number 00:20:10.020 --> 00:20:21.240 of opposite-gender partners for men. 00:20:24.830 --> 00:20:27.970 And we can let A W be the same thing for women. 00:20:37.310 --> 00:20:38.090 All right. 00:20:38.090 --> 00:20:42.440 Now we're trying to figure out the answer to this question. 00:20:42.440 --> 00:20:48.360 What is A m, the average guy degree, 00:20:48.360 --> 00:20:52.550 over the average woman degree. 00:20:52.550 --> 00:20:54.490 And in particular, the University of Chicago 00:20:54.490 --> 00:20:58.480 says, they say it's 1.74. 00:20:58.480 --> 00:21:02.770 That the average guy as 74% more opposite-gender partners 00:21:02.770 --> 00:21:04.550 than the average woman. 00:21:04.550 --> 00:21:10.830 ABC News says it's 3.33, that is 233% more for the men, 00:21:10.830 --> 00:21:12.460 than the women. 00:21:12.460 --> 00:21:16.860 Now we're going to figure this out what this ratio is. 00:21:16.860 --> 00:21:18.540 Just use a little bit of math here, 00:21:18.540 --> 00:21:21.260 and a little bit of graph theory. 00:21:21.260 --> 00:21:23.650 So let's write a formula for A m. 00:21:27.760 --> 00:21:34.230 Well we're trying to figure out the average degree over here. 00:21:34.230 --> 00:21:35.660 Well, that's pretty simple. 00:21:35.660 --> 00:21:39.370 We just add up all the degrees, and divide 00:21:39.370 --> 00:21:40.735 by the number of nodes. 00:21:40.735 --> 00:21:42.360 And that'll give us the average degree. 00:21:44.910 --> 00:21:48.390 So the average degree is the sum of the degrees, 00:21:48.390 --> 00:21:51.670 over all men, x in the set of men, 00:21:51.670 --> 00:21:56.490 of the degree of x, divided by the number of men. 00:21:59.830 --> 00:22:04.620 Can somebody give me a simpler expression for this? 00:22:04.620 --> 00:22:07.454 It doesn't have that nasty sum in it? 00:22:07.454 --> 00:22:09.210 AUDIENCE: E. 00:22:09.210 --> 00:22:12.440 PROFESSOR: E. The cardinality of E. 00:22:12.440 --> 00:22:16.960 I'm adding all the degrees here. 00:22:16.960 --> 00:22:20.940 Well that's just another way of counting all the edges, 00:22:20.940 --> 00:22:24.430 because every edge shows up once, and only once, 00:22:24.430 --> 00:22:26.360 in a degree count here. 00:22:26.360 --> 00:22:28.990 And this is where, we use the fact we 00:22:28.990 --> 00:22:30.370 have opposite-gender partners. 00:22:33.020 --> 00:22:36.130 Because if I had some edges over here 00:22:36.130 --> 00:22:40.240 they wouldn't get counted in sum of the degrees here. 00:22:40.240 --> 00:22:45.255 All right so this is just the cardinality 00:22:45.255 --> 00:22:48.970 of the number of edges, divided by the number of men. 00:22:48.970 --> 00:22:50.380 Any questions about that? 00:22:50.380 --> 00:22:53.020 Because this is an important statement about 00:22:53.020 --> 00:22:54.930 graphs in general. 00:22:54.930 --> 00:22:57.384 When I have a graph like this-- which 00:22:57.384 --> 00:22:58.800 is called a bipartite graph, we'll 00:22:58.800 --> 00:23:00.249 talk about more in a little bit. 00:23:00.249 --> 00:23:02.290 But where the edges go from the left to the right 00:23:02.290 --> 00:23:03.987 if I sum the degrees on the left, 00:23:03.987 --> 00:23:05.570 I'm just counting the number of edges. 00:23:09.060 --> 00:23:13.840 All right, let's figure out a formula for the average number 00:23:13.840 --> 00:23:16.340 of partners for the women. 00:23:16.340 --> 00:23:23.240 That simple that's just sum x over the women. 00:23:23.240 --> 00:23:27.950 The degree of x, divided by the number of women. 00:23:30.900 --> 00:23:32.530 Let me rewrite that so it's clearer. 00:23:36.150 --> 00:23:38.508 What's a simpler expression for this? 00:23:38.508 --> 00:23:39.383 AUDIENCE: [INAUDIBLE] 00:23:43.080 --> 00:23:46.310 PROFESSOR: Yeah, this sum, adding 00:23:46.310 --> 00:23:52.410 the degrees of the women, is just the number of edges, 00:23:52.410 --> 00:23:54.690 right. 00:23:54.690 --> 00:24:00.280 So that is cardinality of edges, divided by the number of women. 00:24:00.280 --> 00:24:06.700 All right, well now we can write, solve for our formula, 00:24:06.700 --> 00:24:12.320 average over men over average of the women. 00:24:12.320 --> 00:24:22.126 That's E over VM, divided by E over VW. 00:24:22.126 --> 00:24:24.900 Wow, this is nice. 00:24:24.900 --> 00:24:30.290 I don't know the number of edges is, but it just canceled out. 00:24:30.290 --> 00:24:33.300 And this is just the number of women, 00:24:33.300 --> 00:24:34.890 divided by the number of men. 00:24:37.430 --> 00:24:40.740 And in fact we know that. 00:24:40.740 --> 00:24:42.660 That's this number, divided by that number, 00:24:42.660 --> 00:24:52.090 which is about 1.0325. 00:24:52.090 --> 00:24:57.550 So we just proved, that on average, a man 00:24:57.550 --> 00:25:01.950 has 3%, or 3 and 1/4% more opposite-gender partners 00:25:01.950 --> 00:25:02.500 than women. 00:25:05.010 --> 00:25:08.800 No need to do the interviews, or spend years doing. 00:25:08.800 --> 00:25:10.390 That is the answer. 00:25:10.390 --> 00:25:14.120 And it has nothing to do with the promiscuity of men, 00:25:14.120 --> 00:25:16.345 or women, nothing at all. 00:25:19.080 --> 00:25:23.300 So the Chicago study is way off, and the ABC New study 00:25:23.300 --> 00:25:25.400 is completely nuts. 00:25:25.400 --> 00:25:30.050 It just can't be right, this is a proof. 00:25:30.050 --> 00:25:33.740 Now what happened here? 00:25:33.740 --> 00:25:37.910 Well what's going on, what's the reason for why this is true? 00:25:37.910 --> 00:25:38.410 Yeah? 00:25:38.410 --> 00:25:41.679 AUDIENCE: A male has a female partner then 00:25:41.679 --> 00:25:44.004 the female has a male partner. 00:25:44.004 --> 00:25:44.670 PROFESSOR: Yeah. 00:25:44.670 --> 00:25:47.456 AUDIENCE: You're not looking at like how many males 00:25:47.456 --> 00:25:49.786 are going to one female. 00:25:49.786 --> 00:25:53.110 The promiscuity isn't even a part of the question. 00:25:53.110 --> 00:25:54.110 PROFESSOR: That's right. 00:25:54.110 --> 00:25:56.610 It takes two to tango. 00:25:56.610 --> 00:25:58.880 Every time you got a guy, you got a women. 00:25:58.880 --> 00:26:01.920 And you have the number of relationships going. 00:26:01.920 --> 00:26:04.530 The average for the men is that number, divided by the men. 00:26:04.530 --> 00:26:06.810 Average for the women is that same number, 00:26:06.810 --> 00:26:07.699 divided by the women. 00:26:07.699 --> 00:26:09.240 And so if there's more women, they're 00:26:09.240 --> 00:26:11.080 going to have less partners on average. 00:26:11.080 --> 00:26:12.670 Has to be. 00:26:12.670 --> 00:26:14.420 So it really was a stupid question. 00:26:14.420 --> 00:26:17.970 It's very, very simple to answer. 00:26:17.970 --> 00:26:21.340 Now as it turns out there are endless studies like this, 00:26:21.340 --> 00:26:23.090 in the literature. 00:26:23.090 --> 00:26:25.940 In fact, a few years ago the Boston Globe 00:26:25.940 --> 00:26:31.140 ran an explosive story about the study habits of students 00:26:31.140 --> 00:26:33.530 on Boston-area campuses. 00:26:33.530 --> 00:26:38.320 And their surveys show that, on average, minority students 00:26:38.320 --> 00:26:41.380 tended to study with non-minority students 00:26:41.380 --> 00:26:42.685 more than the other way around. 00:26:46.350 --> 00:26:49.250 And they want on great length consulting the experts 00:26:49.250 --> 00:26:50.980 as to why this might be true. 00:26:50.980 --> 00:26:53.230 Why is it the minority students study 00:26:53.230 --> 00:26:56.870 with non-minority students more than the other way around. 00:26:56.870 --> 00:26:58.940 Now can anyone tell me why it is certainly 00:26:58.940 --> 00:27:02.205 true, and not surprising, why that's the case? 00:27:02.205 --> 00:27:04.120 AUDIENCE: Because they're the minority. 00:27:04.120 --> 00:27:05.703 PROFESSOR: Because they're a minority. 00:27:05.703 --> 00:27:07.745 There's fewer minorities than non-minorities. 00:27:07.745 --> 00:27:12.030 End of story, we don't need this sociology PhD 00:27:12.030 --> 00:27:13.780 from down the street to explain it to us. 00:27:16.780 --> 00:27:19.060 We're going to see a lot of other bogus studies later. 00:27:19.060 --> 00:27:22.270 This is not unusual, especially when we get the probability. 00:27:22.270 --> 00:27:26.410 Just every day there's a new one in probability. 00:27:26.410 --> 00:27:29.310 Any questions about this before we leave? 00:27:29.310 --> 00:27:32.160 Unfortunately that's most all we'll say about sex today. 00:27:35.590 --> 00:27:36.800 OK. 00:27:36.800 --> 00:27:42.110 But now, in this example, we used an edge in the graph 00:27:42.110 --> 00:27:46.530 to denote some kind of affinity between two nodes. 00:27:46.530 --> 00:27:48.594 The two nodes liked each other in some sense 00:27:48.594 --> 00:27:50.010 if they were connected by an edge, 00:27:50.010 --> 00:27:52.520 or they had a relationship of some kind. 00:27:52.520 --> 00:27:54.490 There's lots of examples in computer science 00:27:54.490 --> 00:27:57.453 where you use an edge to denote just the opposite. 00:27:57.453 --> 00:28:00.200 That the two nodes can't be near each other, 00:28:00.200 --> 00:28:02.750 or don't like each other. 00:28:02.750 --> 00:28:04.420 For example, consider the problem 00:28:04.420 --> 00:28:08.080 of scheduling final exams at MIT. 00:28:08.080 --> 00:28:11.310 And they do this after they find out all of your schedules, 00:28:11.310 --> 00:28:14.354 and they try to schedule the exams so that you don't 00:28:14.354 --> 00:28:16.270 have to take two at once, or there's as little 00:28:16.270 --> 00:28:19.270 of that as possible. 00:28:19.270 --> 00:28:22.730 For example, let's do an example here. 00:28:33.460 --> 00:28:35.760 Say we look at these five classes. 00:28:44.590 --> 00:28:47.220 Take 6041. 00:28:47.220 --> 00:28:49.920 And this may not be totally accurate, but roughly. 00:28:57.250 --> 00:28:59.950 So I've got five MIT classes, and I'm 00:28:59.950 --> 00:29:03.920 going to put an edge between pairs of classes that have 00:29:03.920 --> 00:29:05.400 overlapping student enrollment. 00:29:08.630 --> 00:29:11.490 So in this case, for example, we've 00:29:11.490 --> 00:29:13.160 assumed in the drawing of his graph, 00:29:13.160 --> 00:29:17.280 that you can't have our exam the same time is 6002, 00:29:17.280 --> 00:29:20.640 on the assumption there's students in both classes. 00:29:20.640 --> 00:29:24.560 But you could have our exam the same time as 6034. 00:29:24.560 --> 00:29:27.730 Because there's not an overlapping student 00:29:27.730 --> 00:29:29.350 in both classes, so the exams could 00:29:29.350 --> 00:29:31.510 be scheduled at the same time. 00:29:31.510 --> 00:29:33.720 So we've used a graph to represent 00:29:33.720 --> 00:29:38.740 which courses can't have their exam at the same time. 00:29:38.740 --> 00:29:42.100 Now let's also suppose we have a set of slots for the exam. 00:29:45.360 --> 00:29:47.670 And say they're all on a Wednesday. 00:29:47.670 --> 00:29:51.480 And the first slot is Wednesday from 5:00 to 7:00. 00:29:51.480 --> 00:29:55.430 And the next one is 7:00 to 9:00. 00:29:55.430 --> 00:30:00.070 And then, the next one is 9:00 to 11:00. 00:30:00.070 --> 00:30:05.290 And then 11:00 to 1:00 in the morning, and then 1:00 to 3:00, 00:30:05.290 --> 00:30:07.930 getting pretty late. 00:30:07.930 --> 00:30:13.230 And your job is to figure out how not to have 00:30:13.230 --> 00:30:15.290 to use these later exam slots. 00:30:15.290 --> 00:30:17.140 You'd like to use as few as possible 00:30:17.140 --> 00:30:20.180 so you're not going too late night, 00:30:20.180 --> 00:30:22.930 or come before the holidays, so you're not 00:30:22.930 --> 00:30:25.380 having exams on Christmas and New Year's, for example. 00:30:28.450 --> 00:30:33.840 So the goal is to assign slots to the nodes. 00:30:33.840 --> 00:30:37.090 Put every node in a slot so you don't 00:30:37.090 --> 00:30:42.110 have nodes hooked by an edge getting the same slot. 00:30:42.110 --> 00:30:44.400 Now this is an example of what's called 00:30:44.400 --> 00:30:46.496 a graph coloring problem. 00:30:46.496 --> 00:30:47.370 So let's define that. 00:31:17.680 --> 00:31:44.020 Given a graph G, and K colors, assign a color to each node, 00:31:44.020 --> 00:31:55.970 so that adjacent nodes get different colors. 00:32:03.154 --> 00:32:08.120 All right, and then the minimum number of colors you need 00:32:08.120 --> 00:32:11.750 is called the chromatic number of the graph. 00:32:11.750 --> 00:32:36.460 So the minimum value of K, for which such a coloring exist, 00:32:36.460 --> 00:32:48.840 is the chromatic number OF the graph. 00:32:48.840 --> 00:32:59.956 And it's denoted by this symbol chi of G. 00:32:59.956 --> 00:33:02.330 Because usually you want to use a small number of colors. 00:33:02.330 --> 00:33:07.270 Now what does a color represent when we're 00:33:07.270 --> 00:33:09.220 dealing with this problem? 00:33:09.220 --> 00:33:10.760 What's the meaning of a color? 00:33:10.760 --> 00:33:11.900 AUDIENCE: Time slot. 00:33:11.900 --> 00:33:14.080 PROFESSOR: A time slot, OK. 00:33:14.080 --> 00:33:22.690 So let's call this time slot C1, C2, C3, C4, C5, so there's 00:33:22.690 --> 00:33:23.850 five possible colors. 00:33:23.850 --> 00:33:26.800 Now of course, we could color this graph with five colors, 00:33:26.800 --> 00:33:29.150 every node could just get its own color. 00:33:29.150 --> 00:33:33.060 But then somebody's taking their exam from 1:00 to 3:00 AM, 00:33:33.060 --> 00:33:36.880 and that's a bit of a pain. 00:33:36.880 --> 00:33:41.360 Let's see if we can do less than five. 00:33:41.360 --> 00:33:46.010 Let's say I give this color one, let's give this one color 00:33:46.010 --> 00:33:50.630 one, that's OK, because they're not connected. 00:33:50.630 --> 00:33:55.880 I can't give this one color one, so I give it color two, say. 00:33:55.880 --> 00:33:58.570 Now this one I can't give color one, because this guy got it, 00:33:58.570 --> 00:34:00.570 he can't get color two, because that guy got it. 00:34:00.570 --> 00:34:03.020 So it give it color three. 00:34:03.020 --> 00:34:08.159 And well, I can't do one, two, or three here, 00:34:08.159 --> 00:34:11.020 so I gotta go to color four. 00:34:11.020 --> 00:34:17.800 All right so 6042 will get the 11:00 PM to 1:00 AM slot, 00:34:17.800 --> 00:34:19.443 not so good. 00:34:19.443 --> 00:34:21.916 Can we do any better? 00:34:21.916 --> 00:34:26.139 Can we get away with three colors. 00:34:26.139 --> 00:34:29.870 Some say yes, some say no. 00:34:29.870 --> 00:34:33.690 How many people think you can do three colors on this graph? 00:34:33.690 --> 00:34:34.830 A bunch. 00:34:34.830 --> 00:34:37.402 How many think you can't do any better? 00:34:37.402 --> 00:34:40.270 All right, the vote is mostly for three. 00:34:40.270 --> 00:34:42.060 Let's see. 00:34:42.060 --> 00:34:42.639 Any ideas? 00:34:42.639 --> 00:34:43.980 Anybody see how to do three? 00:34:46.775 --> 00:34:47.275 Yeah? 00:34:47.275 --> 00:34:50.199 AUDIENCE: Assign C4 to 6034 . 00:34:50.199 --> 00:34:52.550 PROFESSOR: Assign C4 to 6043. 00:34:52.550 --> 00:34:55.889 AUDIENCE: Or C1 to 6042. 00:34:55.889 --> 00:34:59.580 PROFESSOR: C-- I can't do see C1 to 6042. 00:34:59.580 --> 00:35:01.561 It crashes, but can I do-- yeah? 00:35:01.561 --> 00:35:02.060 Put 00:35:02.060 --> 00:35:04.434 AUDIENCE: C1 in 6003. 00:35:04.434 --> 00:35:05.350 PROFESSOR: C1 in 6003. 00:35:05.350 --> 00:35:10.222 AUDIENCE: And get rid of C1 in 6034. 00:35:10.222 --> 00:35:11.180 PROFESSOR: Get rid of-- 00:35:11.180 --> 00:35:12.580 AUDIENCE: Make it C2. 00:35:12.580 --> 00:35:15.070 PROFESSOR: Make this a C2. 00:35:15.070 --> 00:35:17.362 Oh, yeah. 00:35:17.362 --> 00:35:21.520 All right, these got C1, they're not adjacent. 00:35:21.520 --> 00:35:23.740 These got C2, they're not adjacent. 00:35:23.740 --> 00:35:26.380 This can now get C3. 00:35:26.380 --> 00:35:29.990 So we can have our exam from 9:00 to 11:00, which is better. 00:35:29.990 --> 00:35:32.375 All right, can anybody do it in two colors? 00:35:40.780 --> 00:35:44.935 Can anybody offer a reason why two colors may not be possible? 00:35:47.536 --> 00:35:48.035 Yeah? 00:35:48.035 --> 00:35:51.575 AUDIENCE: Because let's say you could do it with two colors. 00:35:51.575 --> 00:35:52.200 PROFESSOR: Yep. 00:35:52.200 --> 00:35:58.170 AUDIENCE: 6041 and 6002 have to be different colors. 00:35:58.170 --> 00:35:59.830 PROFESSOR: Yes. 00:35:59.830 --> 00:36:03.500 AUDIENCE: 6042 can't be C1, and it can't be C2. 00:36:03.500 --> 00:36:05.300 PROFESSOR: Yeah, good. 00:36:05.300 --> 00:36:09.130 So you can't in two colors, because these three guys 00:36:09.130 --> 00:36:10.670 would violate that. 00:36:10.670 --> 00:36:11.990 You've got a triangle here. 00:36:11.990 --> 00:36:15.550 Each one of these guys has to be different than the other two. 00:36:15.550 --> 00:36:16.920 So two colors can't work. 00:36:16.920 --> 00:36:19.740 You've got to have at least three in this case. 00:36:19.740 --> 00:36:20.760 So three is optimal. 00:36:20.760 --> 00:36:24.000 We have just shown for this graph, 00:36:24.000 --> 00:36:26.836 the chromatic number is three. 00:36:26.836 --> 00:36:33.300 All right, now in general doing what we just did is very hard. 00:36:33.300 --> 00:36:35.680 No one knows a fast algorithm for determining 00:36:35.680 --> 00:36:38.560 the chromatic number. 00:36:38.560 --> 00:36:40.370 In fact, it's a weird kind of problem, 00:36:40.370 --> 00:36:44.239 because it's easy enough to check that a coloring is OK. 00:36:44.239 --> 00:36:46.530 If somebody put a coloring on the board, you can check, 00:36:46.530 --> 00:36:48.420 oh that works really simply. 00:36:48.420 --> 00:36:52.070 Just check every edge, and make sure the colors are different. 00:36:52.070 --> 00:36:54.590 But figuring it out, as best we know, 00:36:54.590 --> 00:36:57.840 you've got to try an exponential number of possibilities. 00:36:57.840 --> 00:37:00.962 So if I had 100 nodes here, my running time 00:37:00.962 --> 00:37:02.920 of the algorithm to check all the possibilities 00:37:02.920 --> 00:37:04.800 would be exponential and a hundred. 00:37:04.800 --> 00:37:05.300 Yeah? 00:37:05.300 --> 00:37:08.772 AUDIENCE: Can that number just like the highest 00:37:08.772 --> 00:37:11.750 degree of each node, or nodes. 00:37:11.750 --> 00:37:13.015 PROFESSOR: Uh no. 00:37:13.015 --> 00:37:15.770 But it's no worse than something like that, 00:37:15.770 --> 00:37:17.130 as we'll see a few minutes. 00:37:17.130 --> 00:37:18.750 That's a great observation. 00:37:18.750 --> 00:37:21.210 And we're going to come back to that in a few minutes. 00:37:21.210 --> 00:37:24.180 But it's not just that. 00:37:24.180 --> 00:37:30.140 OK now in fact even figuring out for an arbitrary graph 00:37:30.140 --> 00:37:33.320 if three colors can be done, called 00:37:33.320 --> 00:37:36.290 the three-coloring problem, that's really hard. 00:37:36.290 --> 00:37:39.570 No one knows how to solve that in less than exponential time. 00:37:39.570 --> 00:37:43.000 In fact, one of these NP-complete problems 00:37:43.000 --> 00:37:44.340 is what it's called. 00:37:44.340 --> 00:37:47.870 How many people here don't know about NP-completeness? 00:37:47.870 --> 00:37:49.822 Is everybody-- all right so all of you 00:37:49.822 --> 00:37:51.030 haven't seen NP-completeness. 00:37:51.030 --> 00:37:56.120 OK so there is a class of thousands of problems-- 00:37:56.120 --> 00:38:00.230 in fact there's books list these 1,000 problems-- that are all 00:38:00.230 --> 00:38:03.000 NP-complete, somebody's proved they belong in the class. 00:38:03.000 --> 00:38:06.440 And what that means is that if somebody gave you 00:38:06.440 --> 00:38:09.300 a solution, like a coloring here, 00:38:09.300 --> 00:38:13.850 it's easy to check really quickly if it's valid. 00:38:13.850 --> 00:38:16.040 But figuring it out is really hard. 00:38:16.040 --> 00:38:17.790 And if you figured out how to solve 00:38:17.790 --> 00:38:19.761 one of those thousands of problems, 00:38:19.761 --> 00:38:22.260 like suddenly you figured out how to tell if any graph could 00:38:22.260 --> 00:38:25.610 work with three colors, you would solve automatically 00:38:25.610 --> 00:38:28.429 all other thousands in the book. 00:38:28.429 --> 00:38:30.470 So it's this book of problems you will constantly 00:38:30.470 --> 00:38:33.952 run into in your career in computer science. 00:38:33.952 --> 00:38:35.410 And it's bad when you run into one, 00:38:35.410 --> 00:38:38.200 because there's no good algorithm to solve it known. 00:38:38.200 --> 00:38:40.065 But if you just solved one of them, 00:38:40.065 --> 00:38:43.620 the other thousands would suddenly be solvable quickly. 00:38:43.620 --> 00:38:46.540 Even better, you win a million dollar prize. 00:38:46.540 --> 00:38:48.000 One of these Millennium Prizes we 00:38:48.000 --> 00:38:50.860 talked about the first lecture. 00:38:50.860 --> 00:38:53.880 Even if you show you can't find a fast algorithm for one 00:38:53.880 --> 00:38:57.000 of them, that means that known of them have fast algorithms, 00:38:57.000 --> 00:38:59.397 and you also get a million dollars. 00:38:59.397 --> 00:39:01.980 So this is the central problem in computer science, and theory 00:39:01.980 --> 00:39:04.410 computing, is whether or not you could solve 00:39:04.410 --> 00:39:07.060 these NP-complete problems. 00:39:07.060 --> 00:39:09.770 Now actually lots of people have claim to do it. 00:39:09.770 --> 00:39:12.300 And in fact, there was a lot of buzz in the community 00:39:12.300 --> 00:39:15.110 about a month ago when actually a reputable researcher 00:39:15.110 --> 00:39:17.370 at HP Labs said he'd done it. 00:39:17.370 --> 00:39:21.630 He proved that you can't solve NP-complete problems. 00:39:21.630 --> 00:39:24.100 And he got people going for probably at least a week, 00:39:24.100 --> 00:39:26.420 until they discovered a fatal flaw. 00:39:26.420 --> 00:39:29.550 And the proof was actually bogus. 00:39:29.550 --> 00:39:31.770 So no one still knows if you can solve 00:39:31.770 --> 00:39:34.770 these NP-complete problems quickly. 00:39:34.770 --> 00:39:38.620 Now the problem is, in practice, you run into these things 00:39:38.620 --> 00:39:40.430 all the time, like MIT really does 00:39:40.430 --> 00:39:42.430 have to schedule the exams. 00:39:42.430 --> 00:39:43.680 So you've got to do something. 00:39:43.680 --> 00:39:46.690 You can't just go say, hey it's NP-complete, so no exams 00:39:46.690 --> 00:39:48.870 this year, or whatever. 00:39:48.870 --> 00:39:52.970 That's not going to fly, so you got to do something. 00:39:52.970 --> 00:39:55.320 So now this is a problem-- many of you 00:39:55.320 --> 00:39:58.510 when you go into careers, you're going to be faced with this. 00:39:58.510 --> 00:40:00.250 You got to do something. 00:40:00.250 --> 00:40:06.010 Any thoughts about an algorithm for coloring graphs that might 00:40:06.010 --> 00:40:07.880 use a small number of colors? 00:40:07.880 --> 00:40:10.335 It doesn't have to always work, or you're 00:40:10.335 --> 00:40:11.960 going to win a lot of money if it does. 00:40:11.960 --> 00:40:14.300 But a simple algorithm, you can't 00:40:14.300 --> 00:40:16.590 take either the 100 steps. 00:40:16.590 --> 00:40:19.690 You got to be linear, probably, or quadratic time. 00:40:19.690 --> 00:40:22.800 That could get you a small number of colors. 00:40:22.800 --> 00:40:25.860 Any thoughts about what you'd do? 00:40:25.860 --> 00:40:26.360 Yeah? 00:40:26.360 --> 00:40:28.770 AUDIENCE: The number of degrees and nodes? 00:40:28.770 --> 00:40:30.590 PROFESSOR: The number-- what about it? 00:40:30.590 --> 00:40:33.220 AUDIENCE: The highest degree and that node, 00:40:33.220 --> 00:40:34.799 the 6042 is [INAUDIBLE]. 00:40:34.799 --> 00:40:35.465 PROFESSOR: Yeah. 00:40:35.465 --> 00:40:38.700 AUDIENCE: So you could use that. 00:40:38.700 --> 00:40:40.210 PROFESSOR: Good, all right. 00:40:40.210 --> 00:40:42.200 So what do I do with that-- so I found 00:40:42.200 --> 00:40:45.130 a node with a high degree, there's 00:40:45.130 --> 00:40:47.030 three of them have degree three here. 00:40:47.030 --> 00:40:48.806 What do I do with them? 00:40:48.806 --> 00:40:52.324 AUDIENCE: Pick a different color to. 00:40:52.324 --> 00:40:53.740 PROFESSOR: Pick a different color, 00:40:53.740 --> 00:40:55.890 that means I've colored some of the others. 00:40:55.890 --> 00:40:59.040 If I pick a different color, do I start with them, 00:40:59.040 --> 00:41:03.030 or do I finish with a high degree nodes? 00:41:03.030 --> 00:41:05.807 Because you've got to assign the colors to them. 00:41:05.807 --> 00:41:07.890 And high degree is important to be thinking about. 00:41:07.890 --> 00:41:10.431 We're going to prove a theorem in just a minute about related 00:41:10.431 --> 00:41:11.730 to degree and coloring. 00:41:11.730 --> 00:41:14.800 AUDIENCE: Start with them. 00:41:14.800 --> 00:41:17.310 PROFESSOR: Start with them, and do what with it? 00:41:17.310 --> 00:41:18.498 Color? 00:41:18.498 --> 00:41:22.890 AUDIENCE: Yeah, and then assign the ones that aren't connected 00:41:22.890 --> 00:41:26.310 [INAUDIBLE] to the same slots. 00:41:26.310 --> 00:41:29.250 PROFESSOR: OK, so I could-- here's a degree of theory 00:41:29.250 --> 00:41:32.630 now I can start with color one for that. 00:41:32.630 --> 00:41:34.190 And then what do I do next? 00:41:34.190 --> 00:41:38.372 I pick-- its neighbors have to get different colors, I guess. 00:41:38.372 --> 00:41:39.830 You'd start coloring the neighbors. 00:41:39.830 --> 00:41:42.305 AUDIENCE: My first instinct would be 00:41:42.305 --> 00:41:44.780 to color all the [INAUDIBLE]. 00:41:48.250 --> 00:41:49.530 PROFESSOR: OK. 00:41:49.530 --> 00:41:51.320 And what color would use for them? 00:41:51.320 --> 00:41:52.702 AUDIENCE: Different ones. 00:41:52.702 --> 00:41:54.660 PROFESSOR: Different ones if they're connected, 00:41:54.660 --> 00:41:57.790 or if they're not connected you'd still use different ones? 00:41:57.790 --> 00:42:00.670 AUDIENCE: Only if they're connected. 00:42:00.670 --> 00:42:02.940 PROFESSOR: Only they're connected use different ones. 00:42:02.940 --> 00:42:06.901 And so if they're not connected, you'd use the same colors? 00:42:06.901 --> 00:42:07.401 Yeah? 00:42:10.190 --> 00:42:14.620 You're going close, and it actually works pretty well. 00:42:14.620 --> 00:42:17.910 The underlying principle you're sort of thinking about here 00:42:17.910 --> 00:42:21.160 is you've got some notion of the order in which you're 00:42:21.160 --> 00:42:22.985 going to process your graph. 00:42:22.985 --> 00:42:25.110 And you're going to start with a high degree nodes, 00:42:25.110 --> 00:42:26.530 in your case. 00:42:26.530 --> 00:42:28.200 And as you go along, you're going 00:42:28.200 --> 00:42:30.950 to start coloring the nodes. 00:42:30.950 --> 00:42:33.990 And you're going to make sure you color them legally. 00:42:33.990 --> 00:42:37.920 And it sounds like you're going to color them with a low color 00:42:37.920 --> 00:42:39.270 as you go along. 00:42:39.270 --> 00:42:44.860 And that is probably the most basic graph coloring approach. 00:42:44.860 --> 00:42:48.560 And almost you could almost say is a generic approach. 00:42:48.560 --> 00:42:52.091 So let's define that, and then see prove some facts about it. 00:42:56.710 --> 00:42:58.860 Most of the graph coloring algorithms in practice 00:42:58.860 --> 00:43:01.070 are based on this approach. 00:43:01.070 --> 00:43:05.320 And we're going to call it the basic graph coloring algorithm. 00:43:14.240 --> 00:43:20.840 And for our graph G, with vertices V, and edges E. 00:43:20.840 --> 00:43:31.080 So the first step is going to be to order the nodes from 1 to n. 00:43:34.110 --> 00:43:35.850 Now in your case, you were suggesting 00:43:35.850 --> 00:43:39.700 an ordering where I have the high degree nodes first. 00:43:39.700 --> 00:43:41.217 All right. 00:43:41.217 --> 00:43:43.050 But for now we're not going to specify that. 00:43:43.050 --> 00:43:47.290 We're going to make it any ordering you want. 00:43:47.290 --> 00:43:50.080 And then we're going to have a notion of an order 00:43:50.080 --> 00:43:51.490 on the colors, as well. 00:43:57.815 --> 00:43:59.190 And I don't know how many colors, 00:43:59.190 --> 00:44:02.480 but they're going to be numbered 1, 2, and so forth. 00:44:02.480 --> 00:44:08.773 And then we're going to process the nodes one at a time, 00:44:08.773 --> 00:44:13.860 to N. We color the nodes, what is step I, 00:44:13.860 --> 00:44:18.430 we color the Ith node V sub i with the lowest legal color. 00:44:25.240 --> 00:44:28.490 And by the legal I mean you don't color at the same node 00:44:28.490 --> 00:44:31.640 as another node that's already been colored the same that it's 00:44:31.640 --> 00:44:32.602 adjacent to. 00:44:40.790 --> 00:44:43.290 All right so let's try this. 00:44:43.290 --> 00:44:47.480 In fact, this is sort of the algorithm I used initially 00:44:47.480 --> 00:44:50.700 to color exam graph over there. 00:44:54.508 --> 00:44:58.230 All right, so let's look at that. 00:44:58.230 --> 00:45:04.310 So let's say we-- let me erase the colors here, and put 00:45:04.310 --> 00:45:05.465 an ordering on the nodes. 00:45:08.100 --> 00:45:12.230 So let's say I ordered them with 6034 first, 00:45:12.230 --> 00:45:15.060 so this would be V1. 00:45:15.060 --> 00:45:18.390 Then 6041 is V2. 00:45:18.390 --> 00:45:23.380 Then V3, V4, V5. 00:45:26.290 --> 00:45:30.552 If that's my ordering, what color would I assign to 6034? 00:45:30.552 --> 00:45:31.730 AUDIENCE: One. 00:45:31.730 --> 00:45:35.140 PROFESSOR: One, C1, I'd color it first to get C1. 00:45:35.140 --> 00:45:38.860 What color does 6041 get? 00:45:38.860 --> 00:45:41.420 C1, as well, it's the lowest possible color 00:45:41.420 --> 00:45:47.110 that's legal, and is not hooked to this guy, so C1 is legal. 00:45:47.110 --> 00:45:49.810 What color do I give here? 00:45:49.810 --> 00:45:52.480 C2. 00:45:52.480 --> 00:45:59.170 Then I color this one next C-- can't do C2, can't do C1, 00:45:59.170 --> 00:46:01.460 so I pick C3. 00:46:01.460 --> 00:46:03.500 And then I get to 6042 last, and I 00:46:03.500 --> 00:46:07.220 can't do one, two, or three, so I do four. 00:46:07.220 --> 00:46:09.150 All right so algorithm, with that ordering, 00:46:09.150 --> 00:46:10.860 gave four colors. 00:46:10.860 --> 00:46:14.970 However we know there's a way to do a different ordering that 00:46:14.970 --> 00:46:19.220 gives us three colors. 00:46:19.220 --> 00:46:20.840 In particular, let's see if we do 00:46:20.840 --> 00:46:23.150 this what happens if we use this other ordering. 00:46:23.150 --> 00:46:26.050 Let me erase these. 00:46:26.050 --> 00:46:33.960 Say that's V1, V2, V3, V4, V5. 00:46:33.960 --> 00:46:40.140 Now I get C1, this will be C2, C1. 00:46:43.920 --> 00:46:45.670 What's this one get? 00:46:45.670 --> 00:46:46.320 C2. 00:46:46.320 --> 00:46:49.350 Ah, much better. 00:46:49.350 --> 00:46:50.800 C3. 00:46:50.800 --> 00:46:54.880 So different orderings result in different numbers of colors 00:46:54.880 --> 00:46:56.380 here. 00:46:56.380 --> 00:47:00.850 So the whole art now becomes finding a clever ordering. 00:47:00.850 --> 00:47:03.210 And so many people have already had good ideas, 00:47:03.210 --> 00:47:05.850 pick the largest degree nodes first. 00:47:05.850 --> 00:47:11.740 And in fact, if you simulate the algorithm on lots of graphs, 00:47:11.740 --> 00:47:16.140 you do better on average when you color the larger degree 00:47:16.140 --> 00:47:17.820 nodes first. 00:47:17.820 --> 00:47:20.850 And then if you start to use more exotic orderings, 00:47:20.850 --> 00:47:22.762 you can do even better. 00:47:22.762 --> 00:47:24.720 If you take a lot of graphs that are out there, 00:47:24.720 --> 00:47:26.719 and run your algorithm, and see how well you do, 00:47:26.719 --> 00:47:29.540 you do better with more sophisticated orderings. 00:47:29.540 --> 00:47:34.550 In fact, this was my senior thesis back 00:47:34.550 --> 00:47:36.290 when I was undergraduate student. 00:47:36.290 --> 00:47:39.650 I was trying to figure out better and better orderings 00:47:39.650 --> 00:47:41.690 that worked for graphs. 00:47:41.690 --> 00:47:44.820 And at the time it caused a bit of a problem. 00:47:44.820 --> 00:47:47.470 I was a undergraduate at Princeton. 00:47:47.470 --> 00:47:49.120 And Princeton, to this day I think, 00:47:49.120 --> 00:47:52.090 still has exams after the holidays, the Christmas 00:47:52.090 --> 00:47:54.010 holidays, New Year's holidays. 00:47:54.010 --> 00:47:57.810 And the students wanted to have the exams before Christmas, 00:47:57.810 --> 00:47:59.770 because they hated going home for the holiday, 00:47:59.770 --> 00:48:01.790 and then you've got to worry about your exams 00:48:01.790 --> 00:48:03.270 when you come back. 00:48:03.270 --> 00:48:06.050 And the faculty said no, there's no way 00:48:06.050 --> 00:48:09.570 to get them all compressed into a small number of days. 00:48:09.570 --> 00:48:12.010 Now I wasn't aware of all that of the time. 00:48:12.010 --> 00:48:14.255 But my thesis was go figure out good ordering. 00:48:14.255 --> 00:48:15.880 So I tried lots of different orderings. 00:48:15.880 --> 00:48:17.850 And I tried the largest degree first, 00:48:17.850 --> 00:48:21.730 and recursive versions of that actually worked very well. 00:48:21.730 --> 00:48:25.620 And then tried it on the Princeton exam graph. 00:48:25.620 --> 00:48:28.310 And lo and behold, you could actually squish it down, 00:48:28.310 --> 00:48:30.200 so you could give all the exams, I think was, 00:48:30.200 --> 00:48:33.960 4 and 1/2 days, plenty of time to give them before Christmas. 00:48:33.960 --> 00:48:36.330 Which caused a fair of scandal at the time, 00:48:36.330 --> 00:48:38.670 because then the faculty had to come clean 00:48:38.670 --> 00:48:40.730 that they just didn't want to bother having 00:48:40.730 --> 00:48:44.180 the exams before Christmas. 00:48:44.180 --> 00:48:46.770 Now this algorithm is an example of what's 00:48:46.770 --> 00:48:48.780 known as a greedy algorithm. 00:48:48.780 --> 00:48:52.930 Now in a greedy algorithm it's always simple. 00:48:52.930 --> 00:48:55.360 You just go one step after the next, 00:48:55.360 --> 00:48:57.760 taking the best you can do at each stop. 00:48:57.760 --> 00:49:00.670 You never go back and try to make things better. 00:49:00.670 --> 00:49:03.950 You never do hill climbing, if you're familiar with that term. 00:49:03.950 --> 00:49:07.010 You just always keep it simple, one thing after the next, 00:49:07.010 --> 00:49:08.830 very fast. 00:49:08.830 --> 00:49:10.770 Sometimes it works great in practice. 00:49:10.770 --> 00:49:12.570 Sometimes it doesn't. 00:49:12.570 --> 00:49:16.860 But it's always where you start, some simple approach like this. 00:49:16.860 --> 00:49:20.770 Now this algorithm actually, even 00:49:20.770 --> 00:49:22.850 if you don't try to monkey with the ordering, 00:49:22.850 --> 00:49:25.140 even for a worst case ordering of the nodes, 00:49:25.140 --> 00:49:28.930 that actually does pretty good for a lot of graphs. 00:49:28.930 --> 00:49:30.730 And in fact, it does really well-- 00:49:30.730 --> 00:49:33.040 as somebody already asked about-- 00:49:33.040 --> 00:49:35.510 if all the nodes have low degree. 00:49:35.510 --> 00:49:37.060 So let's state that as a theorem. 00:49:37.060 --> 00:49:38.518 And then we're going to prove that. 00:49:44.329 --> 00:49:54.330 So if every node in a graph G has degree, at most, 00:49:54.330 --> 00:49:58.550 d-- so that's the biggest degree in the graph, D-- then 00:49:58.550 --> 00:50:16.480 this basic algorithm uses, at most, d plus 1 colors for G. 00:50:16.480 --> 00:50:19.920 No matter what the ordering is, you'll 00:50:19.920 --> 00:50:23.490 never do worse than d plus 1 colors. 00:50:23.490 --> 00:50:28.840 So what's the value of d for our exam graph over here? 00:50:28.840 --> 00:50:30.180 d is 3. 00:50:30.180 --> 00:50:33.240 Every node has degree, at most, three. 00:50:33.240 --> 00:50:37.120 And so it says, that no matter what ordering you picked here, 00:50:37.120 --> 00:50:39.350 you'd get at most four colors. 00:50:39.350 --> 00:50:40.350 Now you might do better. 00:50:40.350 --> 00:50:42.620 In fact, we found an ordering that got three. 00:50:42.620 --> 00:50:45.940 So it's possible to do better. 00:50:45.940 --> 00:50:49.040 So let's prove this fact because this makes a difference. 00:50:49.040 --> 00:50:53.260 Say you have a graph with hundreds of nodes. 00:50:53.260 --> 00:50:55.716 But every node has degree, at most, three. 00:50:55.716 --> 00:50:58.560 Well that says you only need four colors even, 00:50:58.560 --> 00:51:01.840 if the graph has 1,000 nodes, and that's very useful. 00:51:01.840 --> 00:51:04.580 So in that kind of situation it does very well. 00:51:04.580 --> 00:51:05.525 So let's prove that. 00:51:09.570 --> 00:51:12.733 Any ideas as to what proof technique we're going to use? 00:51:12.733 --> 00:51:14.705 AUDIENCE: Invariant. 00:51:14.705 --> 00:51:17.120 PROFESSOR: Invariant, close. 00:51:17.120 --> 00:51:19.560 Not quite an invariant, but close. 00:51:19.560 --> 00:51:22.450 AUDIENCE: [INAUDIBLE] 00:51:22.450 --> 00:51:23.404 PROFESSOR: What? 00:51:23.404 --> 00:51:24.820 AUDIENCE: Well ordering principle. 00:51:24.820 --> 00:51:26.420 PROFESSOR: You know well ordering principle, yeah, 00:51:26.420 --> 00:51:28.503 we're going to use the equivalent version of that. 00:51:28.503 --> 00:51:31.024 We're going to use induction. 00:51:31.024 --> 00:51:33.190 If you like well-- it's equivalent to well ordering. 00:51:33.190 --> 00:51:35.314 If you like well ordering you could do it that way. 00:51:35.314 --> 00:51:38.400 I think it's easier using induction here. 00:51:38.400 --> 00:51:39.765 So the proof is by induction. 00:51:45.110 --> 00:51:46.720 All right so the first thing we need 00:51:46.720 --> 00:51:47.845 is an induction hypothesis. 00:51:50.950 --> 00:51:54.450 Any thoughts about what the induction hypothesis should be? 00:51:54.450 --> 00:51:55.202 Yeah? 00:51:55.202 --> 00:52:01.270 AUDIENCE: If you have a graph with n nodes then 00:52:01.270 --> 00:52:06.150 where the degree of any nodes is less than [INAUDIBLE] 00:52:06.150 --> 00:52:08.090 then you can do it. 00:52:08.090 --> 00:52:09.090 PROFESSOR: That's great. 00:52:09.090 --> 00:52:12.030 You're going to do really well on the midterm, 00:52:12.030 --> 00:52:14.450 because you put an n into this thing, 00:52:14.450 --> 00:52:16.570 but there's not an n here to start. 00:52:16.570 --> 00:52:18.320 What are most people going to do-- 00:52:18.320 --> 00:52:20.210 we used to ask this actually. 00:52:20.210 --> 00:52:22.830 We asked this once on a test many years ago, 00:52:22.830 --> 00:52:25.820 and it was an utter disaster, because did everybody do? 00:52:25.820 --> 00:52:28.832 May be one student, or two, put an n into there. 00:52:28.832 --> 00:52:31.290 But what's the naturally thing to do to induct on here when 00:52:31.290 --> 00:52:34.560 you look at this statement? 00:52:34.560 --> 00:52:37.760 You're going to induct on d, because the first thing you do 00:52:37.760 --> 00:52:39.680 is you make this be your induction hypothesis. 00:52:39.680 --> 00:52:41.950 There's only one thing to use, so you're 00:52:41.950 --> 00:52:44.910 going to have your predicate be p of d, 00:52:44.910 --> 00:52:47.014 and it's going to be that. 00:52:47.014 --> 00:52:49.680 Now It didn't occur to us that's what everybody was going to do, 00:52:49.680 --> 00:52:50.560 but it should have. 00:52:50.560 --> 00:52:53.220 They all did that and it was a disaster. 00:52:53.220 --> 00:52:56.010 Because if you do this, well you've 00:52:56.010 --> 00:53:00.280 got to take a graph with maximum degree d, or d plus 1 00:53:00.280 --> 00:53:03.200 in the inductive step, pull out all the nodes 00:53:03.200 --> 00:53:07.021 with degree d plus 1 to get a graph with now degree d. 00:53:07.021 --> 00:53:07.770 And that's a mess. 00:53:07.770 --> 00:53:10.410 You just pulled out a lot of nodes, potentially. 00:53:10.410 --> 00:53:14.650 Color that in d plus 1 colors, now put all that junk back in. 00:53:14.650 --> 00:53:17.180 And say only used one more color. 00:53:17.180 --> 00:53:18.520 Nightmare. 00:53:18.520 --> 00:53:21.300 And these were MIT students under pressure. 00:53:21.300 --> 00:53:23.330 It was a nightmare. 00:53:23.330 --> 00:53:24.410 So that does not work. 00:53:24.410 --> 00:53:27.410 And in fact, we will ask an induction question 00:53:27.410 --> 00:53:29.960 on graphs on every test you take in this course. 00:53:29.960 --> 00:53:31.190 It will happen. 00:53:31.190 --> 00:53:34.850 And so usually, with induction, you 00:53:34.850 --> 00:53:36.580 take this as your induction hypothesis. 00:53:36.580 --> 00:53:38.650 With graphs, you have to be careful. 00:53:38.650 --> 00:53:41.830 And worst part about this is we tell people 00:53:41.830 --> 00:53:45.950 when this doesn't work, use a stronger induction hypothesis. 00:53:45.950 --> 00:53:48.280 So students tried to make a stronger, 00:53:48.280 --> 00:53:51.900 but they're still stuck on d, and it was still a disaster. 00:53:51.900 --> 00:53:53.930 With graphs, you do something different. 00:53:53.930 --> 00:53:56.240 And the first thing you do with a graph, usually, 00:53:56.240 --> 00:53:58.385 is put n in here. 00:53:58.385 --> 00:54:00.510 And if it doesn't work with n, the number of nodes, 00:54:00.510 --> 00:54:03.410 you put in e the number of edges. 00:54:03.410 --> 00:54:05.120 And induct on that. 00:54:05.120 --> 00:54:08.460 And so what you said is exactly the right thing to do. 00:54:08.460 --> 00:54:11.270 Don't do this, or least don't spend too much time on it. 00:54:11.270 --> 00:54:13.360 Pretty quickly try this. 00:54:13.360 --> 00:54:17.970 If every end node graph-- if every node in an n 00:54:17.970 --> 00:54:23.840 node graph G has degree at most degree, 00:54:23.840 --> 00:54:27.070 then the basic algorithm uses at most d, plus one colors. 00:54:27.070 --> 00:54:30.350 And now you induct on n. 00:54:30.350 --> 00:54:33.100 And almost always on graphs, that's the first thing to try. 00:54:33.100 --> 00:54:37.270 Even if it's not in your theorem statement. 00:54:37.270 --> 00:54:39.900 Any questions about that? 00:54:46.940 --> 00:54:49.570 Well let's start with this, and see 00:54:49.570 --> 00:54:51.256 if we can make this one work. 00:55:02.230 --> 00:55:04.140 So what's the next step in our proof? 00:55:04.140 --> 00:55:06.560 What do we got to do? 00:55:06.560 --> 00:55:07.380 Base case. 00:55:11.890 --> 00:55:14.750 And the base case will be, not n equals 0, 00:55:14.750 --> 00:55:19.300 because we can't have a zero node graph, but n equals 1. 00:55:19.300 --> 00:55:22.940 And how many edges do we have? 00:55:22.940 --> 00:55:23.440 Zero. 00:55:23.440 --> 00:55:25.190 If there's one node, we don't allow loops, 00:55:25.190 --> 00:55:33.140 so it's zero edges, which means that the degree of our graph 00:55:33.140 --> 00:55:36.380 has to be zero. 00:55:36.380 --> 00:55:37.780 There's no edges. 00:55:37.780 --> 00:55:39.900 And of course there's only one node, 00:55:39.900 --> 00:55:44.140 so one color is going to work, and that 00:55:44.140 --> 00:55:46.340 happens to equal d plus 1. 00:55:49.460 --> 00:55:52.110 All right, so the base case is true. 00:55:52.110 --> 00:55:55.110 For one node graphs, you can always 00:55:55.110 --> 00:55:59.010 use d plus 1 colors, where d is the max degree. 00:55:59.010 --> 00:56:01.210 All right, next we have the inductive step. 00:56:05.500 --> 00:56:14.425 So here we assume P n is true for the induction. 00:56:20.330 --> 00:56:23.130 And now we look at an n plus 1 node graph 00:56:23.130 --> 00:56:26.620 to show P n plus 1 is true. 00:56:26.620 --> 00:56:35.240 So we let G be any N plus 1 node graph. 00:56:35.240 --> 00:56:39.152 We got to show you can color it in d plus 1 colors. 00:56:41.810 --> 00:56:48.790 And let's let d be the max degree, the largest 00:56:48.790 --> 00:56:56.020 degree in G. 00:56:56.020 --> 00:57:01.180 We've got to show we can color it in d plus 1 colors. 00:57:01.180 --> 00:57:03.120 Well the basic algorithm, let's say. 00:57:03.120 --> 00:57:05.680 First thing we do is we order the nodes 00:57:05.680 --> 00:57:06.954 in an arbitrary order. 00:57:10.363 --> 00:57:13.700 And we're going to show whatever order you pick is OK. 00:57:22.920 --> 00:57:28.185 All right so what are the nodes? 00:57:38.740 --> 00:57:41.200 Anyway at all. 00:57:41.200 --> 00:57:45.690 Now how am I going to use the induction hypothesis? 00:57:45.690 --> 00:57:49.900 I know, I can assume, the for any N node graph 00:57:49.900 --> 00:57:54.550 I can color it in the max degree plus 1 colors. 00:57:54.550 --> 00:57:56.940 How am I going to use that to help me color G here, 00:57:56.940 --> 00:57:58.260 the n plus 1 node graph? 00:57:58.260 --> 00:58:00.690 Any thoughts? 00:58:00.690 --> 00:58:01.190 Yeah? 00:58:01.190 --> 00:58:03.860 AUDIENCE: [INAUDIBLE] 00:58:03.860 --> 00:58:06.920 PROFESSOR: Yeah, let's create an n node graph 00:58:06.920 --> 00:58:09.410 by looking at these nodes, and taking 00:58:09.410 --> 00:58:12.310 this one out of the time being. 00:58:12.310 --> 00:58:15.910 Remove the last V n plus 1 node in the order. 00:58:15.910 --> 00:58:19.400 That leaves an n node graph. 00:58:19.400 --> 00:58:22.310 So let's write that down. 00:58:22.310 --> 00:58:33.120 We remove the n plus 1 from G. And that creates a new graph, 00:58:33.120 --> 00:58:38.530 call it G prime with vertices, V prime and edges, E prime. 00:58:38.530 --> 00:58:40.925 So we create a new graph by removing that node. 00:58:43.640 --> 00:58:46.240 And we remove all the edges tied to that node. 00:58:46.240 --> 00:58:50.360 So for example over here, the last node 00:58:50.360 --> 00:58:56.150 was 6042, so we take out 6042, and all these edges. 00:58:56.150 --> 00:58:59.116 And this is a graph that we're left with. 00:59:02.370 --> 00:59:06.190 That graph has n nodes. 00:59:06.190 --> 00:59:08.637 What's the maximum degree in G prime? 00:59:11.910 --> 00:59:18.054 When I pull out a node, can the degree of any node go up? 00:59:18.054 --> 00:59:21.320 No, I'm just taking stuff out. 00:59:21.320 --> 00:59:27.930 So I know that G prime has maximum degree, at most, d. 00:59:27.930 --> 00:59:30.370 The degree didn't go up of any node. 00:59:35.080 --> 00:59:38.430 Might have gone down, but it didn't go up. 00:59:38.430 --> 00:59:44.045 So G prime has max degree, at most, d, and it has n nodes. 00:59:47.440 --> 00:59:50.440 So we can use the induction hypothesis P n. 00:59:53.360 --> 01:00:01.320 It says that the basic algorithm uses d plus 1, 01:00:01.320 --> 01:00:09.758 at most, d plus 1 colors for nodes V1 to V n. 01:00:13.104 --> 01:00:14.260 Any questions about that? 01:00:17.140 --> 01:00:22.140 So if this were the n plus first node, last node in the ordering 01:00:22.140 --> 01:00:24.090 take it out. 01:00:24.090 --> 01:00:26.490 The basic algorithm now, take the same order here, 01:00:26.490 --> 01:00:33.850 V1, V2, V3, V4, basic, we'll color that in d plus 1 colors. 01:00:33.850 --> 01:00:36.950 And all I have left is to give this guy color, 01:00:36.950 --> 01:00:41.360 and I'll have color G. Question? 01:00:41.360 --> 01:00:43.640 No. 01:00:43.640 --> 01:00:45.600 All right. 01:00:45.600 --> 01:00:48.820 So by induction I've colored these guys, V1 to V2, 01:00:48.820 --> 01:00:51.760 and d plus 1 colors, all that I have left to do 01:00:51.760 --> 01:00:56.000 is color V n plus 1. 01:00:56.000 --> 01:00:59.585 And hopefully we're not going to use color d plus 2, 01:00:59.585 --> 01:01:02.450 because then we sort of-- it wouldn't work. 01:01:02.450 --> 01:01:04.953 We got to use one of the first d plus 1. 01:01:12.330 --> 01:01:14.370 All right, so let's look at V n plus 1. 01:01:17.110 --> 01:01:25.780 And let's call its neighbors in G, U1, U2, Ud. 01:01:25.780 --> 01:01:31.630 It has, at most d neighbors, because every node in G has, 01:01:31.630 --> 01:01:34.320 at most, degree d. 01:01:34.320 --> 01:01:35.980 A neighbor's a node you're adjacent to. 01:01:38.840 --> 01:01:45.950 All right so, V n plus 1 has at most d neighbors, is adjacent 01:01:45.950 --> 01:01:48.061 to, at most, d other nodes. 01:01:51.220 --> 01:01:54.470 Now what does that mean about the color I 01:01:54.470 --> 01:01:56.990 can use on V n plus 1? 01:01:59.540 --> 01:02:04.605 What do I know about what color I can use for that? 01:02:04.605 --> 01:02:05.105 Yeah? 01:02:05.105 --> 01:02:10.070 AUDIENCE: It can't be any of the colors of U1, U2, and so on. 01:02:10.070 --> 01:02:12.850 PROFESSOR: It can't be any one of these colors that 01:02:12.850 --> 01:02:14.860 were assigned here. 01:02:14.860 --> 01:02:16.110 That's true. 01:02:16.110 --> 01:02:19.440 So how many colors got ruled out? 01:02:19.440 --> 01:02:23.300 At most d, and how many am I working with? 01:02:23.300 --> 01:02:25.560 d Plus 1. 01:02:25.560 --> 01:02:29.580 So I got one left that I can use safely. 01:02:29.580 --> 01:02:30.920 OK. 01:02:30.920 --> 01:02:35.660 So this means there exists at least one color 01:02:35.660 --> 01:02:37.890 in my set of d plus 1 colors. 01:02:42.560 --> 01:02:47.575 It's not used by any neighbor. 01:02:53.660 --> 01:02:56.390 And we're going to give V n plus 1 that color. 01:03:07.150 --> 01:03:07.650 All right. 01:03:07.650 --> 01:03:12.350 So now I've colored every node in G, the n plus 1 node graph, 01:03:12.350 --> 01:03:15.315 safely using a total of d plus 1 colors. 01:03:18.692 --> 01:03:25.050 So that means the basic algorithm uses, 01:03:25.050 --> 01:03:33.340 at most, d plus 1 colors, on G. That means P n plus 1 01:03:33.340 --> 01:03:37.815 is true-- whoops-- and the induction is complete. 01:03:43.050 --> 01:03:44.853 Any questions? 01:03:44.853 --> 01:03:45.353 Yeah. 01:03:45.353 --> 01:03:49.281 AUDIENCE: Could you also start from the other way, 01:03:49.281 --> 01:03:55.664 and start 1, go to 2 nodes, 3 nodes at each step keeping 01:03:55.664 --> 01:03:59.101 all nodes at all other nodes. 01:03:59.101 --> 01:04:02.080 [INAUDIBLE] 01:04:02.080 --> 01:04:05.308 PROFESSOR: What do you mean by keeping all nodes connected? 01:04:05.308 --> 01:04:16.264 AUDIENCE: [INAUDIBLE] each node has an edge connecting 01:04:16.264 --> 01:04:17.804 to each other one. 01:04:17.804 --> 01:04:19.720 PROFESSOR: OK so, then I get a specific graph. 01:04:19.720 --> 01:04:24.240 I start with this, I add a node and make it adjacent. 01:04:24.240 --> 01:04:25.680 I add a node and make it adjacent. 01:04:25.680 --> 01:04:26.554 AUDIENCE: [INAUDIBLE] 01:04:32.410 --> 01:04:33.904 PROFESSOR: Yeah. 01:04:33.904 --> 01:04:36.620 So you've constructed a particular graph. 01:04:36.620 --> 01:04:40.260 This is actually called, for the n nodes, it's called Kn, 01:04:40.260 --> 01:04:45.010 is the n node complete graph, also called a clique, 01:04:45.010 --> 01:04:49.100 like a clique of friends, where everybody likes everybody, 01:04:49.100 --> 01:04:52.350 in a clique. 01:04:52.350 --> 01:04:55.660 And in fact for n here, for those n nodes, 01:04:55.660 --> 01:04:58.940 what's the max degree? 01:04:58.940 --> 01:05:01.440 Max degree is n minus 1. 01:05:01.440 --> 01:05:03.420 What's the chromatic number of this graph? 01:05:05.986 --> 01:05:07.485 What's the minimum number of colors? 01:05:07.485 --> 01:05:09.156 [INTERPOSING VOICES] 01:05:09.156 --> 01:05:11.030 PROFESSOR: And they all have to be different, 01:05:11.030 --> 01:05:12.890 which is d plus 1. 01:05:12.890 --> 01:05:16.420 So you have built a special graph 01:05:16.420 --> 01:05:19.560 for which the optimum of number colors is d plus 1. 01:05:19.560 --> 01:05:23.630 But that is not a proof that this is true for all graphs. 01:05:23.630 --> 01:05:26.384 Because you've looked at a particular graph here. 01:05:26.384 --> 01:05:27.259 AUDIENCE: [INAUDIBLE] 01:05:30.222 --> 01:05:31.180 PROFESSOR: What's that? 01:05:31.180 --> 01:05:34.760 AUDIENCE: [INAUDIBLE] It means that you can still use your 01:05:34.760 --> 01:05:36.138 less than or equal to sign. 01:05:36.138 --> 01:05:38.490 PROFESSOR: I see, so you'd add a node, 01:05:38.490 --> 01:05:41.182 and it's only connected to a few of them. 01:05:41.182 --> 01:05:43.015 AUDIENCE: No, it's connected to all of them, 01:05:43.015 --> 01:05:45.930 but it still implies that you need less than 01:05:45.930 --> 01:05:47.050 or equal to the colors. 01:05:47.050 --> 01:05:49.083 It turns out it happens to be equal to. 01:05:49.083 --> 01:05:50.832 PROFESSOR: Yes, in this case that's right. 01:05:50.832 --> 01:05:52.510 So you've made an argument for this case 01:05:52.510 --> 01:05:54.060 where it actually is equal, but that 01:05:54.060 --> 01:05:55.200 only worked for this graph. 01:05:55.200 --> 01:05:59.310 AUDIENCE: [INAUDIBLE] worse case. 01:05:59.310 --> 01:06:02.666 PROFESSOR: It is the worst case, so it meets the bound. 01:06:02.666 --> 01:06:04.290 It shows you cannot improve this bound. 01:06:04.290 --> 01:06:06.052 Yeah, is there a question up there? 01:06:06.052 --> 01:06:09.290 AUDIENCE: All I was going to say is that you've 01:06:09.290 --> 01:06:10.620 proved it's the worst case. 01:06:10.620 --> 01:06:12.369 PROFESSOR: Right, so what you've done here 01:06:12.369 --> 01:06:15.587 is you've shown that I could not make that theorem any stronger. 01:06:15.587 --> 01:06:18.531 I could not replace it with d here. 01:06:18.531 --> 01:06:19.030 All right. 01:06:19.030 --> 01:06:20.571 Because you've given an example where 01:06:20.571 --> 01:06:23.440 I can't get d colors, where the maximum degree is d. 01:06:23.440 --> 01:06:25.880 But that doesn't-- To get a proof for a theorem, 01:06:25.880 --> 01:06:27.110 I got to go through all this. 01:06:27.110 --> 01:06:30.480 That wouldn't give me a proof of the theorem. 01:06:30.480 --> 01:06:32.130 They're not equivalent. 01:06:32.130 --> 01:06:35.660 One's an upper bound, one's an existence of a lower bound. 01:06:35.660 --> 01:06:40.940 This shows that for any graph, you need at most d plus 1. 01:06:40.940 --> 01:06:43.990 So any graph, at most. 01:06:43.990 --> 01:06:50.430 That shows there is a graph that you need at least. 01:06:50.430 --> 01:06:52.640 And they are not equivalent. 01:06:52.640 --> 01:06:53.230 All right. 01:06:53.230 --> 01:06:56.610 One is for all, and upper bound. 01:06:56.610 --> 01:07:00.830 The other is there exists a lower bound. 01:07:00.830 --> 01:07:05.590 So different in two ways that are important. 01:07:05.590 --> 01:07:08.980 This kind of proof is very typical for what you'll 01:07:08.980 --> 01:07:11.390 see with induction in graphs. 01:07:11.390 --> 01:07:13.340 And you'll get a lot of practice with it. 01:07:13.340 --> 01:07:17.380 Are there any other questions on this proof? 01:07:20.200 --> 01:07:21.870 OK. 01:07:21.870 --> 01:07:24.800 All right, see we've seen now, by that example, 01:07:24.800 --> 01:07:27.330 we can't improve the theorem. 01:07:27.330 --> 01:07:30.200 In some cases, though, the theorem 01:07:30.200 --> 01:07:32.020 is way off, for some graphs. 01:07:32.020 --> 01:07:34.530 Can anybody think of a graph where 01:07:34.530 --> 01:07:37.850 the bound we get from the theorem, of d plus 1 colors, 01:07:37.850 --> 01:07:40.680 is way off from the actual chromatic number you need, 01:07:40.680 --> 01:07:41.962 the number of colors you need? 01:07:41.962 --> 01:07:42.462 Yeah? 01:07:42.462 --> 01:07:44.000 AUDIENCE: [INAUDIBLE] 01:07:44.000 --> 01:07:44.916 PROFESSOR: What is it? 01:07:44.916 --> 01:07:51.282 AUDIENCE: A graph [INAUDIBLE] two sets of [INAUDIBLE] 01:07:55.130 --> 01:07:56.470 PROFESSOR: Good, OK. 01:07:56.470 --> 01:07:58.220 Yes, so what if we did this graph. 01:07:58.220 --> 01:08:00.042 Let me draw it out. 01:08:03.310 --> 01:08:09.600 So you've got a bunch of nodes here, bunch of nodes here. 01:08:09.600 --> 01:08:12.300 And every node here is connected to every node 01:08:12.300 --> 01:08:15.600 over the other side. 01:08:15.600 --> 01:08:18.050 And if this is an n no graph, and I've got n over 2 01:08:18.050 --> 01:08:21.100 on each side, what's my degree here? 01:08:25.922 --> 01:08:27.380 What's my max degree of this graph? 01:08:27.380 --> 01:08:28.560 AUDIENCE: N over 2. 01:08:28.560 --> 01:08:30.319 PROFESSOR: N over 2. 01:08:30.319 --> 01:08:32.680 So d is n over 2. 01:08:32.680 --> 01:08:34.060 What's the chromatic number? 01:08:34.060 --> 01:08:36.520 How many colors do I need for this? 01:08:36.520 --> 01:08:38.250 Two. 01:08:38.250 --> 01:08:41.380 All right, so d plus 1 is way off of two. 01:08:41.380 --> 01:08:43.466 There is a even worse example. 01:08:43.466 --> 01:08:43.966 Yeah? 01:08:43.966 --> 01:08:48.130 AUDIENCE: That graph where you have one node center that's 01:08:48.130 --> 01:08:51.126 connected to a bunch of nodes regularly distributed about. 01:08:51.126 --> 01:08:52.564 PROFESSOR: Yeah, the star graph. 01:08:52.564 --> 01:08:56.300 All right, so I got one of the center, I got n minus 1 01:08:56.300 --> 01:08:58.330 outside. 01:08:58.330 --> 01:09:03.050 So here the maximum degree is n minus 1, 01:09:03.050 --> 01:09:05.460 just like a complete graph. 01:09:05.460 --> 01:09:08.649 But how many colors do I need? 01:09:08.649 --> 01:09:10.069 Two. 01:09:10.069 --> 01:09:11.500 So it's even worse here. 01:09:14.319 --> 01:09:18.920 All right now what about the basic algorithm? 01:09:18.920 --> 01:09:21.599 How well does the basic algorithm do on this graph? 01:09:24.649 --> 01:09:27.790 Or to the vertices some way? 01:09:27.790 --> 01:09:29.630 Color on one [INAUDIBLE] lowest color. 01:09:29.630 --> 01:09:31.620 How many colors is it going to use? 01:09:31.620 --> 01:09:33.840 AUDIENCE: Two. 01:09:33.840 --> 01:09:35.240 PROFESSOR: Two. 01:09:35.240 --> 01:09:38.109 It doesn't matter the vertices. 01:09:38.109 --> 01:09:46.619 V1, V2, V3, V4, because I'll color this one 1. 01:09:46.619 --> 01:09:49.668 What am I going to call that one? 01:09:49.668 --> 01:09:51.950 1. 01:09:51.950 --> 01:09:54.770 Then I get to the center, what am I going to color it? 01:09:54.770 --> 01:09:56.360 2. 01:09:56.360 --> 01:10:00.130 And now all the arms, what do they get colored? 01:10:00.130 --> 01:10:01.240 They all get 1. 01:10:01.240 --> 01:10:04.275 Whatever order you pick, you get two colors. 01:10:07.270 --> 01:10:08.892 All right so now there's a difference 01:10:08.892 --> 01:10:11.350 between the theorem just gives you an upper bound, it says, 01:10:11.350 --> 01:10:13.150 at most, d plus 1 colors. 01:10:13.150 --> 01:10:15.700 But in fact the algorithm can do a lot better 01:10:15.700 --> 01:10:19.050 than that, as on this example. 01:10:19.050 --> 01:10:22.420 So the algorithm might be a lot better. 01:10:22.420 --> 01:10:24.770 Everybody see that what we're doing here? 01:10:24.770 --> 01:10:26.520 How the algorithm is better than the bound 01:10:26.520 --> 01:10:28.644 we proved by the theorem, even though the bound was 01:10:28.644 --> 01:10:31.180 pretty good for some graphs. 01:10:31.180 --> 01:10:33.620 Now it turns out-- I mean we're not 01:10:33.620 --> 01:10:36.110 going to win a million dollars for this algorithm. 01:10:36.110 --> 01:10:39.880 And in fact, this algorithm is sometimes very bad. 01:10:39.880 --> 01:10:44.370 And a really bad example it's very close to this. 01:10:44.370 --> 01:10:48.470 In fact actually this one, let's look at how well does basic 01:10:48.470 --> 01:10:51.310 do one this one here. 01:10:54.240 --> 01:10:56.160 Make some ordering. 01:10:56.160 --> 01:10:59.950 V1, V2, V3. 01:11:02.715 --> 01:11:05.520 What's the basic algorithm going to do on this complete-- 01:11:05.520 --> 01:11:08.780 it's called a complete bipartite graph, is what's this called. 01:11:08.780 --> 01:11:10.300 I'll define bipartite in a minute-- 01:11:10.300 --> 01:11:12.001 but what's the basic algorithm do here? 01:11:15.690 --> 01:11:20.860 Any idea-- does it take n over 2 colors, or does it take 2? 01:11:20.860 --> 01:11:23.170 Any ideas? 01:11:23.170 --> 01:11:24.370 2. 01:11:24.370 --> 01:11:29.650 So take a vertex, and the first one, say V1s here, get C1. 01:11:29.650 --> 01:11:32.790 As long as I keep picking vertices over on this side, 01:11:32.790 --> 01:11:33.790 they're going to get C1. 01:11:36.430 --> 01:11:39.560 As soon as I get to a vertex over here, 01:11:39.560 --> 01:11:41.363 what color does it have to get? 01:11:41.363 --> 01:11:42.130 AUDIENCE: C2. 01:11:42.130 --> 01:11:43.620 PROFESSOR: C2 because it's touching 01:11:43.620 --> 01:11:46.296 the very first one we had here. 01:11:46.296 --> 01:11:47.670 So when I get vertices over here, 01:11:47.670 --> 01:11:48.795 they're all going to be C2. 01:11:48.795 --> 01:11:52.270 When I go back over here, they're going to be back to C1. 01:11:52.270 --> 01:11:55.910 So actually basic does good here too, gives you two colors. 01:11:55.910 --> 01:11:56.410 Yeah? 01:11:56.410 --> 01:11:57.408 AUDIENCE: [INAUDIBLE] 01:12:06.870 --> 01:12:08.620 PROFESSOR: Ah, those two aren't connected. 01:12:08.620 --> 01:12:11.480 But this case, if I've got a vertex over here 01:12:11.480 --> 01:12:14.540 it is, by definition, connected to the vertex over here. 01:12:14.540 --> 01:12:17.820 Because every possible edge is here. 01:12:17.820 --> 01:12:18.870 But that's a great idea. 01:12:18.870 --> 01:12:22.447 What if they weren't all connected, 01:12:22.447 --> 01:12:23.655 that's actually a great idea. 01:12:25.590 --> 01:12:29.270 In fact, the nasty example for the basic algorithm 01:12:29.270 --> 01:12:30.270 is very much like that. 01:12:30.270 --> 01:12:31.446 Let's draw it. 01:12:35.748 --> 01:12:38.200 Because so far, the basic algorithm 01:12:38.200 --> 01:12:42.080 is pretty much done perfectly on all the graphs 01:12:42.080 --> 01:12:48.870 we looked at even when the theorem wasn't tight. 01:12:48.870 --> 01:12:50.445 So here is a nasty graph. 01:12:56.140 --> 01:12:58.930 And it is very close to the graph we just look like, 01:12:58.930 --> 01:13:00.910 where all the edges are there. 01:13:00.910 --> 01:13:03.080 In this case, all the edges are there, 01:13:03.080 --> 01:13:07.040 except for the one straight across. 01:13:07.040 --> 01:13:10.980 So if this is-- the edge denotes likes, 01:13:10.980 --> 01:13:14.410 this is a world where you like everybody but your spouse. 01:13:16.950 --> 01:13:19.390 All right, so you have an edge to every one, 01:13:19.390 --> 01:13:21.240 except the one directly across from you. 01:13:25.190 --> 01:13:29.370 No edge there, and so forth. 01:13:29.370 --> 01:13:32.460 So it has almost every edge, but it's missing these edges. 01:13:37.620 --> 01:13:42.440 Now the basic algorithm might do well here. 01:13:42.440 --> 01:13:46.120 What would be a good ordering for this graph 01:13:46.120 --> 01:13:48.010 to label these V1 through Vn? 01:13:48.010 --> 01:13:48.510 Yeah? 01:13:48.510 --> 01:13:51.630 AUDIENCE: Go through everything on the left side, 01:13:51.630 --> 01:13:54.818 and then the right side. 01:13:54.818 --> 01:13:56.312 PROFESSOR: Yeah, that's right. 01:14:03.800 --> 01:14:12.480 Because then color 1, color 1, color 1, all the way down. 01:14:12.480 --> 01:14:17.460 One color for the left, what does this one get? 01:14:17.460 --> 01:14:19.540 Color 2, because it's hooked up against. 01:14:19.540 --> 01:14:25.670 And these all get color 2, so I've used two colors. 01:14:25.670 --> 01:14:26.260 Really good. 01:14:26.260 --> 01:14:28.630 Basic algorithm's looking great. 01:14:28.630 --> 01:14:31.470 Now here's a harder question. 01:14:31.470 --> 01:14:34.800 Can you figure out a bad ordering 01:14:34.800 --> 01:14:37.995 for this graph, where I use a lot more than two colors. 01:14:37.995 --> 01:14:38.870 AUDIENCE: [INAUDIBLE] 01:14:42.124 --> 01:14:43.040 PROFESSOR: What is it? 01:14:43.040 --> 01:14:47.286 AUDIENCE: It starts at the top of the cross, and then 01:14:47.286 --> 01:14:48.685 the next level then across. 01:14:48.685 --> 01:14:49.560 PROFESSOR: Very good. 01:14:49.560 --> 01:14:50.740 V1, V2. 01:14:50.740 --> 01:14:53.210 Just as natural, really, if think about it, 01:14:53.210 --> 01:14:54.490 to order it this way. 01:15:02.670 --> 01:15:03.860 All right. 01:15:03.860 --> 01:15:06.650 What color does V1 get? 01:15:06.650 --> 01:15:07.410 C1. 01:15:07.410 --> 01:15:08.890 What color does V2 get? 01:15:08.890 --> 01:15:09.910 AUDIENCE: C1. 01:15:09.910 --> 01:15:12.250 PROFESSOR: C1 because it's not hooked up here. 01:15:12.250 --> 01:15:13.735 What color does V3 get? 01:15:13.735 --> 01:15:14.387 AUDIENCE: C2. 01:15:14.387 --> 01:15:14.970 PROFESSOR: C2. 01:15:14.970 --> 01:15:16.480 What about V4? 01:15:16.480 --> 01:15:17.887 AUDIENCE: C2. 01:15:17.887 --> 01:15:18.470 PROFESSOR: C2. 01:15:18.470 --> 01:15:19.670 It's not hooked up. 01:15:19.670 --> 01:15:21.560 It can't get one, because that's up here. 01:15:21.560 --> 01:15:25.310 And it's not the two, so it gets two What color does V5 get? 01:15:25.310 --> 01:15:26.600 AUDIENCE: C3. 01:15:26.600 --> 01:15:27.490 PROFESSOR: C3. 01:15:27.490 --> 01:15:29.450 Because it's hooked up to one to two. 01:15:29.450 --> 01:15:30.806 V6 ? 01:15:30.806 --> 01:15:31.610 AUDIENCE: C3. 01:15:31.610 --> 01:15:34.900 PROFESSOR: C3, it's hooked up to one and two, but not three. 01:15:34.900 --> 01:15:38.560 And you can see what's happening here. 01:15:38.560 --> 01:15:40.230 All the way down here he's hooked up 01:15:40.230 --> 01:15:43.165 to all the n over 2 minus 1 colors. 01:15:43.165 --> 01:15:47.110 So he also takes C n over 2. 01:15:47.110 --> 01:15:50.120 So if you pick that ordering, not so good. 01:15:50.120 --> 01:15:52.600 You use n over two colors. 01:15:52.600 --> 01:15:55.510 So it really matters the ordering. 01:15:55.510 --> 01:15:58.030 Now I should say graphs like-- actually any questions 01:15:58.030 --> 01:15:59.700 about what we did here? 01:15:59.700 --> 01:16:02.430 About this? 01:16:02.430 --> 01:16:04.560 All right, now I should say that graphs like this 01:16:04.560 --> 01:16:09.470 have a special name, they're called bipartite graphs. 01:16:09.470 --> 01:16:11.249 And that's important to remember. 01:16:24.520 --> 01:16:43.780 All right, so a graph G is said to be bipartite 01:16:43.780 --> 01:16:53.316 if the vertices can be split into two sets, or partitioned, 01:16:53.316 --> 01:17:01.360 and we'll call them a left set, and a right set, 01:17:01.360 --> 01:17:17.330 so that all the edges connect a node in the left set, 01:17:17.330 --> 01:17:21.190 to a node in the right set. 01:17:23.759 --> 01:17:25.300 So in fact, a lot of today we've been 01:17:25.300 --> 01:17:28.500 looking at bipartite graphs, because the nodes are here. 01:17:28.500 --> 01:17:32.660 Like the men, and the women, and the edges only go from the left 01:17:32.660 --> 01:17:33.900 to the right. 01:17:33.900 --> 01:17:35.335 And that is called bipartite. 01:17:35.335 --> 01:17:37.290 And it's called bipartite because you 01:17:37.290 --> 01:17:41.820 can do it with two colors, or in two pieces. 01:17:41.820 --> 01:17:43.860 So you don't win a million dollars 01:17:43.860 --> 01:17:45.630 for deciding whether or not a graph can 01:17:45.630 --> 01:17:47.600 be colored in two colors. 01:17:47.600 --> 01:17:48.950 That's easy. 01:17:48.950 --> 01:17:51.730 You'll even do it for homework one of these times. 01:17:51.730 --> 01:17:54.330 You do win the million dollars for deciding if a graph can 01:17:54.330 --> 01:17:56.700 be colored in three colors. 01:17:56.700 --> 01:17:58.450 That's really hard to do. 01:18:01.090 --> 01:18:04.800 Now coloring problems come up in all sorts of applications. 01:18:04.800 --> 01:18:08.950 You know with this company, Akamai, that came out of MIT, 01:18:08.950 --> 01:18:10.070 we've talked about. 01:18:10.070 --> 01:18:14.490 We run a network of 75,000 servers. 01:18:14.490 --> 01:18:16.860 And they're used to distribute content on the internet, 01:18:16.860 --> 01:18:18.280 and so forth. 01:18:18.280 --> 01:18:20.210 And we have to deploy a new version 01:18:20.210 --> 01:18:21.800 of our software on those servers, 01:18:21.800 --> 01:18:23.420 pretty much every week. 01:18:23.420 --> 01:18:25.560 We're pushing new software out. 01:18:25.560 --> 01:18:31.070 And you can't deploy on every server at the same time, 01:18:31.070 --> 01:18:32.950 because you've got to take down a server 01:18:32.950 --> 01:18:34.450 to deploy new software on it. 01:18:34.450 --> 01:18:36.050 Got to take it out of commission. 01:18:36.050 --> 01:18:38.650 And so we can't just take down all 75,000 servers, 01:18:38.650 --> 01:18:40.960 because then all the Facebook, and Netflix, 01:18:40.960 --> 01:18:42.410 and all those sites would stop. 01:18:42.410 --> 01:18:44.840 That would be bad. 01:18:44.840 --> 01:18:48.060 And we can't do them one at a time, because there's 75,000. 01:18:48.060 --> 01:18:50.780 And it takes a few hours for each one 01:18:50.780 --> 01:18:54.070 to get the traffic off, stop it, load new software, 01:18:54.070 --> 01:18:54.980 and turn it back on. 01:18:54.980 --> 01:18:58.240 And it would take us years to do one software install, 01:18:58.240 --> 01:19:00.610 which we got to do every week. 01:19:00.610 --> 01:19:03.280 So we've got to figure out a schedule for how many servers 01:19:03.280 --> 01:19:05.530 you take down at a given time, and which ones. 01:19:05.530 --> 01:19:07.590 And it turns out pairs of servers 01:19:07.590 --> 01:19:09.639 have certain critical functions. 01:19:09.639 --> 01:19:11.930 So there's certain pairs of servers you can't take down 01:19:11.930 --> 01:19:14.750 at the same time. 01:19:14.750 --> 01:19:18.536 So we have a gigantic 75,000 node coloring problem, 01:19:18.536 --> 01:19:20.035 where there's edges between servers. 01:19:20.035 --> 01:19:21.618 Nodes are servers, and there's an edge 01:19:21.618 --> 01:19:25.634 between if you can't install new software at the same time. 01:19:25.634 --> 01:19:27.050 And so when it turns out, when you 01:19:27.050 --> 01:19:30.410 run one of these graph coloring algorithms on it, 01:19:30.410 --> 01:19:32.120 you could do it with eight colors. 01:19:32.120 --> 01:19:34.160 It just turns out that way. 01:19:34.160 --> 01:19:37.070 So that means there's eight waves of install 01:19:37.070 --> 01:19:38.920 that go on to the network. 01:19:38.920 --> 01:19:41.450 And now eight times a few hours each 01:19:41.450 --> 01:19:44.672 means that we can do it in a day, and you can manage it. 01:19:44.672 --> 01:19:47.780 You know on a much smaller scale, 01:19:47.780 --> 01:19:50.310 the same problem exists for register allocation, 01:19:50.310 --> 01:19:52.120 for variables. 01:19:52.120 --> 01:19:55.486 Here you've got to assign every variable to register. 01:19:55.486 --> 01:19:56.860 But you can't have variables that 01:19:56.860 --> 01:19:59.360 are active at the same time associated 01:19:59.360 --> 01:20:00.900 with the same register. 01:20:00.900 --> 01:20:03.850 And you want to minimize the number of registers you need. 01:20:03.850 --> 01:20:07.110 So again, you have the graph coloring problem. 01:20:07.110 --> 01:20:10.220 The number of colors is the number of registers you need. 01:20:10.220 --> 01:20:12.920 And two variables can't get the same color if their active 01:20:12.920 --> 01:20:15.810 at the same time, so you put an edge between them. 01:20:15.810 --> 01:20:17.720 The most famous example of graph coloring 01:20:17.720 --> 01:20:21.670 is the map coloring problem, with the four coloring theorem. 01:20:21.670 --> 01:20:25.800 And so here, every country is a node. 01:20:25.800 --> 01:20:27.727 Adjacent countries have an edge between them, 01:20:27.727 --> 01:20:29.810 because you don't want to color adjacent countries 01:20:29.810 --> 01:20:31.480 the same color, or you can't tell 01:20:31.480 --> 01:20:34.730 they're different countries. 01:20:34.730 --> 01:20:36.990 Now the last example we can talk about 01:20:36.990 --> 01:20:40.000 is an important problem in communication theory, 01:20:40.000 --> 01:20:43.630 communication networks, where again coloring comes up. 01:20:43.630 --> 01:20:48.100 Now here you need to assign frequencies to radio stations, 01:20:48.100 --> 01:20:49.035 or the cell towers. 01:20:49.035 --> 01:20:54.510 It comes up in mobile networks, or just in with radio stations. 01:20:54.510 --> 01:20:59.370 And if two towers have an overlapping area, 01:20:59.370 --> 01:21:01.290 they can't be given the same frequency, 01:21:01.290 --> 01:21:05.540 so you get collisions between the towers. 01:21:05.540 --> 01:21:07.110 And frequencies are very expensive. 01:21:07.110 --> 01:21:08.901 Companies pay the government a lot of money 01:21:08.901 --> 01:21:11.020 to get certain spectrum. 01:21:11.020 --> 01:21:13.350 So suppose you had this problem. 01:21:13.350 --> 01:21:17.205 Here's tower A, this is A's range, where it reaches. 01:21:17.205 --> 01:21:20.750 Here's tower B, so it overlaps some 01:21:20.750 --> 01:21:36.660 with A. Here's tower C. Here's tower E. And here's tower D. 01:21:36.660 --> 01:21:38.280 All right now the question would be, 01:21:38.280 --> 01:21:40.869 how many radio frequencies do you need? 01:21:40.869 --> 01:21:42.910 What's the minimum number of frequencies you need 01:21:42.910 --> 01:21:45.916 to enable all the towers here? 01:21:45.916 --> 01:21:47.480 We could make that be a graph. 01:21:47.480 --> 01:21:49.760 There's a node for each tower. 01:21:49.760 --> 01:21:54.400 And an edge between towers, if they overlap. 01:21:54.400 --> 01:21:58.650 C doesn't overlap with B, E does. 01:21:58.650 --> 01:22:00.970 E overlaps here. 01:22:00.970 --> 01:22:05.060 And then D overlaps here. 01:22:05.060 --> 01:22:09.197 So how many frequencies do you need for this graph? 01:22:09.197 --> 01:22:10.360 AUDIENCE: Four. 01:22:10.360 --> 01:22:14.580 PROFESSOR: Four would work, three is better. 01:22:14.580 --> 01:22:16.512 Can you do two? 01:22:19.900 --> 01:22:22.380 No you can't do two, because you got here. 01:22:22.380 --> 01:22:23.820 But you could do three. 01:22:23.820 --> 01:22:29.830 You could do one, two, three, two, one. 01:22:29.830 --> 01:22:30.841 This problem comes up-- 01:22:30.841 --> 01:22:31.715 AUDIENCE: [INAUDIBLE] 01:22:31.715 --> 01:22:33.010 PROFESSOR: Did I screw up? 01:22:33.010 --> 01:22:34.660 Ooh, no I can't do that. 01:22:34.660 --> 01:22:36.630 One, two, yeah much better. 01:22:36.630 --> 01:22:39.300 All right, this problem comes up all over the place. 01:22:39.300 --> 01:22:41.640 I'm certain you'll see it sometime in your career, 01:22:41.640 --> 01:22:44.015 you'll have some problem, or you're scheduling something, 01:22:44.015 --> 01:22:45.960 and it's really a graph problem in disguise. 01:22:45.960 --> 01:22:48.250 OK that's it for today.