WEBVTT 00:00:09.409 --> 00:00:11.378 PATRICK WINSTON: It's too bad, in a way, that we can't paint 00:00:11.378 --> 00:00:14.214 everything black, because this map coloring problem sure 00:00:14.214 --> 00:00:17.383 would be a lot easier. 00:00:17.383 --> 00:00:21.888 So I don't know what we're going to do about that. 00:00:21.888 --> 00:00:23.890 How long is this going to take? 00:00:23.890 --> 00:00:24.724 Here's what we're going to do. 00:00:24.724 --> 00:00:26.459 We're going to wait till either all the laptops are 00:00:26.459 --> 00:00:28.895 closed, or this program terminates, 00:00:28.895 --> 00:00:30.130 whichever comes first. 00:00:32.499 --> 00:00:34.667 So how long is this going to take? 00:00:34.667 --> 00:00:36.569 I actually don't know. 00:00:36.569 --> 00:00:38.671 I think it'll take more than the life time of the universe, 00:00:38.671 --> 00:00:39.706 but I'm not sure. 00:00:39.706 --> 00:00:42.709 So let's take a look at a slightly easier 00:00:42.709 --> 00:00:44.511 map coloring problem. 00:00:44.511 --> 00:00:47.247 We'll stop this one and change the map to something I call 00:00:47.247 --> 00:00:49.315 [? simplicia. ?] 00:00:49.315 --> 00:00:53.753 There's 26 states, one for each letter in the alphabet. 00:00:53.753 --> 00:00:55.889 And what we're going to do is we're going to do a depth 00:00:55.889 --> 00:01:01.327 first search for a suitable coloring of this map. 00:01:01.327 --> 00:01:04.330 We're going to go in order, A, B, C, D, E. And as I've 00:01:04.330 --> 00:01:08.134 suggested here, at each level we're going to rotate the 00:01:08.134 --> 00:01:11.871 color choices so we don't over use any one color. 00:01:11.871 --> 00:01:14.807 So if we launch this particular search, this depth 00:01:14.807 --> 00:01:17.777 first attempt to color this map. 00:01:17.777 --> 00:01:18.478 There it goes. 00:01:18.478 --> 00:01:21.381 And maybe we should wait until it terminates. 00:01:21.381 --> 00:01:25.184 Or maybe we should just let it run and it'll terminate, 00:01:25.184 --> 00:01:27.320 perhaps, sometime within the lecture. 00:01:27.320 --> 00:01:30.923 Or maybe we should just let it run over the weekend. 00:01:30.923 --> 00:01:33.960 Or how long do you think we would have to wait, if we want 00:01:33.960 --> 00:01:37.196 to come back and watch it terminate? 00:01:37.196 --> 00:01:39.232 At roughly 30 frames a second-- 00:01:39.232 --> 00:01:41.934 I'm calculating it all in my head, I don't want to bet my 00:01:41.934 --> 00:01:46.506 life on it-- but I think about 5,000 years. 00:01:46.506 --> 00:01:50.276 And if we want to use as many states as there are in the 00:01:50.276 --> 00:01:53.946 United States, in this demonstration, you get up to 00:01:53.946 --> 00:01:58.918 numbers like 10 to the 17th years-- 00:01:58.918 --> 00:02:01.254 17th, 18th, 16th, I'm not exactly sure. 00:02:01.254 --> 00:02:03.656 I did a rough calculation. 00:02:03.656 --> 00:02:06.092 And, of course, you could do some parallels into that, and 00:02:06.092 --> 00:02:07.994 it's not as bad as chess, where you need all the atoms 00:02:07.994 --> 00:02:09.328 in the universe and you still can't do it, 00:02:09.328 --> 00:02:10.562 and things like that. 00:02:10.562 --> 00:02:11.497 Acting as computers. 00:02:11.497 --> 00:02:14.400 But still, it's pretty horrendous. 00:02:14.400 --> 00:02:20.573 The problem is, well the problem is illustrated by this 00:02:20.573 --> 00:02:22.809 diagram I put in back of me. 00:02:22.809 --> 00:02:27.914 If you do a depth first search and you have a 00:02:27.914 --> 00:02:31.517 problem like Texas. 00:02:31.517 --> 00:02:36.189 If you label... if you assign a color to Arizona, 00:02:36.189 --> 00:02:40.893 Oklahoma, Arkansas, and Louisiana first, and then wait 00:02:40.893 --> 00:02:47.733 around to Texas last, then you get yourself into a bind by 00:02:47.733 --> 00:02:50.803 your fourth choice, that you don't discover until you're 00:02:50.803 --> 00:02:52.972 48th choice. 00:02:52.972 --> 00:02:56.042 So what happens then, is that you start developing this tree 00:02:56.042 --> 00:03:00.913 like this, you get to those states surrounding Texas. 00:03:00.913 --> 00:03:04.150 Texas is last state assigned a color and there's 00:03:04.150 --> 00:03:05.718 nothing left for it. 00:03:05.718 --> 00:03:08.521 And that problem was there on the fourth choice, and you 00:03:08.521 --> 00:03:10.923 don't discover it until your 50th choice. 00:03:10.923 --> 00:03:15.227 So you develop a horrendous, impossible, impossible search. 00:03:15.227 --> 00:03:19.265 So you simply can't do it that way. 00:03:19.265 --> 00:03:22.468 But now, not to worry. 00:03:22.468 --> 00:03:24.603 I've come equipped with the idea of constraint 00:03:24.603 --> 00:03:26.605 propagation. 00:03:26.605 --> 00:03:28.374 So we could just take a country with four 00:03:28.374 --> 00:03:30.042 states, like this. 00:03:30.042 --> 00:03:34.380 Each can be labeled red, green, blue and yellow. 00:03:34.380 --> 00:03:37.283 So just like in a case of line drawings, we'll pile up all 00:03:37.283 --> 00:03:41.487 the possible things that the value can be-- red, green, 00:03:41.487 --> 00:03:42.954 blue and yellow. 00:03:42.954 --> 00:03:48.627 Red, green, blue and yellow. 00:03:48.627 --> 00:03:51.497 And we start up constraint propagation. 00:03:51.497 --> 00:03:55.401 So we say for the upper left hand corner state, is there 00:03:55.401 --> 00:03:58.504 any reason to believe that R is impossible? 00:03:58.504 --> 00:04:00.372 Well we look at our neighbors and see what kind of 00:04:00.372 --> 00:04:02.508 constraint flows in from them. 00:04:02.508 --> 00:04:04.276 And sure, this guy could be green, and this 00:04:04.276 --> 00:04:05.511 guy could be blue. 00:04:05.511 --> 00:04:09.248 They don't have to be red, so there's nothing 00:04:09.248 --> 00:04:11.984 that rules out red. 00:04:11.984 --> 00:04:13.519 And there's nothing the rules out blue. 00:04:13.519 --> 00:04:16.454 And there's nothing that rules out yellow. 00:04:16.454 --> 00:04:18.624 And there's nothing that rules out green. 00:04:18.624 --> 00:04:22.595 So constraint propagation just sits there with its finger up 00:04:22.595 --> 00:04:25.564 its nose, doing nothing. 00:04:25.564 --> 00:04:31.804 So it doesn't look like we can use either depth first search, 00:04:31.804 --> 00:04:34.673 or constraint propagation. 00:04:34.673 --> 00:04:37.843 So we could just give up and cry. 00:04:37.843 --> 00:04:41.714 But maybe there's some other approach that will help. 00:04:41.714 --> 00:04:45.017 So let me actually work the Texas problem. 00:04:45.017 --> 00:04:50.890 So with apologies to Houston and Tyler, 00:04:50.890 --> 00:04:52.157 here's a map of Texas. 00:04:57.863 --> 00:04:58.764 There we are. 00:04:58.764 --> 00:05:01.834 And here's, roughly speaking, Arizona 00:05:01.834 --> 00:05:03.702 is over here somewhere. 00:05:03.702 --> 00:05:05.404 And we've got Oklahoma in there. 00:05:05.404 --> 00:05:09.642 And old Bill Clinton's state, Arkansas. 00:05:09.642 --> 00:05:12.611 And then Louisiana sticks out there a little bit. 00:05:12.611 --> 00:05:15.914 So there's our map of that 00:05:15.914 --> 00:05:18.384 particular part of the country. 00:05:18.384 --> 00:05:25.524 So we've got Arizona here, Oklahoma here, Arkansas here, 00:05:25.524 --> 00:05:29.528 and Louisiana here, and Texas here. 00:05:29.528 --> 00:05:34.233 And we have elected to assign colors to these states, in 00:05:34.233 --> 00:05:35.100 that order. 00:05:35.100 --> 00:05:39.271 So this is one, two, three, four. 00:05:42.174 --> 00:05:44.710 And we're going to do that. 00:05:44.710 --> 00:05:46.912 We're going to rotate our color choices, just so we 00:05:46.912 --> 00:05:50.649 don't over use any one color. 00:05:50.649 --> 00:05:53.819 But we're going to also have a look at Texas as we go around. 00:05:53.819 --> 00:05:57.122 Because Texas is a state that borders on the States they 00:05:57.122 --> 00:06:00.926 were choosing colors for. 00:06:00.926 --> 00:06:05.230 So the only possible colors that Texas could be are red, 00:06:05.230 --> 00:06:10.702 green, blue, and yellow. 00:06:10.702 --> 00:06:15.707 So as we make our choices around here, we'll say that-- 00:06:15.707 --> 00:06:18.744 we don't have to adhere to any particular style-- we can say 00:06:18.744 --> 00:06:26.018 that Arizona is going to get the colored red, R. That's 00:06:26.018 --> 00:06:29.454 going to rule out R over here for Texas, because no adjacent 00:06:29.454 --> 00:06:32.925 states can have the same color. 00:06:32.925 --> 00:06:35.194 Then we go over to Oklahoma, and we're rotating our color 00:06:35.194 --> 00:06:38.864 choices, so we'll say that can be green. 00:06:38.864 --> 00:06:41.800 And that's fine, because it's consistent with the red here, 00:06:41.800 --> 00:06:43.368 but it rules out the possibility that 00:06:43.368 --> 00:06:46.004 Texas could be green. 00:06:46.004 --> 00:06:52.377 And then we go over here to Arkansas, red, green, blue. 00:06:52.377 --> 00:06:55.647 That's fine, that's consistent with the green on Oklahoma, 00:06:55.647 --> 00:06:58.483 but if we look at its neighbors we know that Texas 00:06:58.483 --> 00:07:02.788 is forever forbidden to be blue, now. 00:07:02.788 --> 00:07:05.324 So now we go over to Louisiana, and remember, we're 00:07:05.324 --> 00:07:07.459 rotating our color choices because we don't want to 00:07:07.459 --> 00:07:09.428 overuse them. 00:07:09.428 --> 00:07:11.229 So this means that the first choice we're going to make 00:07:11.229 --> 00:07:12.864 here, for Louisiana, is yellow. 00:07:15.600 --> 00:07:18.570 And that's fine because it's consistent with Arkansas, but 00:07:18.570 --> 00:07:22.307 it's not so fine because it's now ruled out the last 00:07:22.307 --> 00:07:25.410 possibility for Texas. 00:07:25.410 --> 00:07:30.248 So even though Texas is going to be the 48th state that we 00:07:30.248 --> 00:07:33.985 color, we're going to say, at this point, there's no 00:07:33.985 --> 00:07:35.020 need in going on. 00:07:35.020 --> 00:07:36.888 We'd better back up. 00:07:36.888 --> 00:07:39.124 Because there's nothing left for Texas when we get around 00:07:39.124 --> 00:07:40.692 to coloring it. 00:07:40.692 --> 00:07:45.163 So that means that this yellow is ruled out here. 00:07:45.163 --> 00:07:47.732 This yellow reappears. 00:07:47.732 --> 00:07:51.937 We select the next color in my line for Louisiana, which 00:07:51.937 --> 00:07:53.839 happens to be red. 00:07:53.839 --> 00:07:57.509 And now that's consistent with this yellow that's 00:07:57.509 --> 00:07:59.010 still left for Texas. 00:07:59.010 --> 00:08:00.445 And it's also consistent with the blue 00:08:00.445 --> 00:08:03.482 that's up here for Arkansas. 00:08:03.482 --> 00:08:04.416 So that's cool. 00:08:04.416 --> 00:08:07.786 I wonder if maybe we could make an algorithm out of that, 00:08:07.786 --> 00:08:09.187 and solve problems like this. 00:08:09.187 --> 00:08:12.057 Do you see, sort of, the intuition of what we're doing? 00:08:12.057 --> 00:08:14.159 We're actually using the martial arts principle, again. 00:08:14.159 --> 00:08:17.696 Because the whole problem is that local constraints, 00:08:17.696 --> 00:08:19.798 undiscovered local constraints, are causing 00:08:19.798 --> 00:08:21.900 downstream problems. 00:08:21.900 --> 00:08:24.970 So we're going to use the enemy's powers against him, 00:08:24.970 --> 00:08:26.938 and we're going to look at those local constraints as we 00:08:26.938 --> 00:08:30.108 go and make sure they're not going downstream, not going to 00:08:30.108 --> 00:08:32.177 get us later on. 00:08:32.177 --> 00:08:37.249 So now I'm going to look like I'm getting a little formal, 00:08:37.249 --> 00:08:39.084 but I'm just getting a little bit more formal. 00:08:39.084 --> 00:08:44.289 Because I want to have some language that I can use to 00:08:44.289 --> 00:08:46.324 describe what's going on, so that it's clear what the 00:08:46.324 --> 00:08:48.360 choices are. 00:08:48.360 --> 00:08:50.762 So to start off with, we're going to have to have some 00:08:50.762 --> 00:08:52.497 vocabulary. 00:08:52.497 --> 00:08:55.867 So let's start up our vocabulary here. 00:08:55.867 --> 00:09:04.209 We're going to have a notion of a variable v. And that's 00:09:04.209 --> 00:09:05.443 something that can have an assignment. 00:09:24.029 --> 00:09:25.463 There's nothing complicated about that. 00:09:25.463 --> 00:09:26.731 And a value. 00:09:30.902 --> 00:09:36.241 A value x is something that can be in assignment. 00:09:39.544 --> 00:09:41.079 It's a little bit circular, but we're all in computer 00:09:41.079 --> 00:09:43.515 science so you know what I mean. 00:09:43.515 --> 00:09:47.986 So the next thing is a little slightly less obvious, and 00:09:47.986 --> 00:09:54.058 that's the notion of a domain, d. 00:09:54.058 --> 00:09:57.428 And that's going to be a bag of values. 00:10:01.366 --> 00:10:01.699 OK. 00:10:01.699 --> 00:10:03.334 So one more thing. 00:10:03.334 --> 00:10:04.602 A constraint. 00:10:10.008 --> 00:10:18.283 That's a constraint c, is a limit on-- 00:10:18.283 --> 00:10:20.885 in our examples it's mostly going to be pairs of 00:10:20.885 --> 00:10:23.121 variables, pairs of variable values. 00:10:23.121 --> 00:10:25.189 But in general, it could be variable values. 00:10:33.765 --> 00:10:37.235 So if we go back here to Texas we could say, OK, how does our 00:10:37.235 --> 00:10:40.371 vocabulary drape itself over that configuration? 00:10:40.371 --> 00:10:45.610 And the answer is, the states have the role of variables, 00:10:45.610 --> 00:10:50.882 the colors have the role of values. 00:10:50.882 --> 00:10:53.985 And the domains are the remaining color possibilities 00:10:53.985 --> 00:10:56.721 that we can still use on a particular state. 00:10:56.721 --> 00:10:58.990 And the constraint, in this case, is the simple map 00:10:58.990 --> 00:11:02.960 coloring constraint that no states that share a boundary 00:11:02.960 --> 00:11:05.296 can have the same color. 00:11:05.296 --> 00:11:09.033 So states are variables, colors are values, domains are 00:11:09.033 --> 00:11:11.936 bags of colors, and constraints-- 00:11:11.936 --> 00:11:13.504 there's only one-- 00:11:13.504 --> 00:11:16.507 adjacent states can't have the same color. 00:11:16.507 --> 00:11:19.644 So that's how it fits with this vocabulary. 00:11:19.644 --> 00:11:22.013 So now, what did I actually do here? 00:11:22.013 --> 00:11:25.149 Well what I actually did here, I'm going to now formalize a 00:11:25.149 --> 00:11:29.720 little by writing it down in pseudo code. 00:11:29.720 --> 00:11:33.324 So here we are, we're going to have a look at what we did 00:11:33.324 --> 00:11:35.793 here with our intuition, and we're going to 00:11:35.793 --> 00:11:37.895 reduce it to a procedure. 00:11:37.895 --> 00:11:39.130 And here's the procedure. 00:11:42.900 --> 00:11:45.036 Remember, we're doing depth first search on this stuff. 00:11:45.036 --> 00:11:46.838 I did a depth first search. 00:11:46.838 --> 00:11:52.410 We're going to do a depth first search, and for each 00:11:52.410 --> 00:11:55.780 depth first search assignment-- 00:12:01.385 --> 00:12:05.823 OK, so here I am, I'm labeling Arizona, and then Oklahoma, 00:12:05.823 --> 00:12:08.426 and then Arkansas, and then Louisiana. 00:12:08.426 --> 00:12:12.697 When I give each one of those a label, a color, I'm going to 00:12:12.697 --> 00:12:14.398 do this procedure. 00:12:14.398 --> 00:12:15.666 Every time I make one of those assignments. 00:12:15.666 --> 00:12:18.836 The last one that caused trouble was 00:12:18.836 --> 00:12:20.404 coloring Louisiana yellow. 00:12:20.404 --> 00:12:23.708 Each time I put one of those colors down, each time I make 00:12:23.708 --> 00:12:25.276 an assignment, I'm going to do this procedure. 00:12:25.276 --> 00:12:38.889 So for each depth first search assignment, for each variable 00:12:38.889 --> 00:12:40.290 v, considered. 00:12:45.896 --> 00:12:48.566 Now you don't know what I mean by considered. 00:12:48.566 --> 00:12:54.505 But when I put a label, when I put up a value, show something 00:12:54.505 --> 00:12:59.543 as a color for Louisiana, I thought about Texas. 00:12:59.543 --> 00:13:02.546 So I was considering the variable, Texas, when I made 00:13:02.546 --> 00:13:04.849 the assignment for Louisiana. 00:13:04.849 --> 00:13:07.084 Now I'm going to be a little bit vague about what I mean by 00:13:07.084 --> 00:13:07.918 considered, right now. 00:13:07.918 --> 00:13:09.720 Because there are lots of choices about how much stuff 00:13:09.720 --> 00:13:11.455 you actually consider. 00:13:11.455 --> 00:13:13.591 So let me just say consider, and then we'll open that up 00:13:13.591 --> 00:13:18.095 and talk about the options in a moment, so for each variable 00:13:18.095 --> 00:13:22.733 v considered for-- 00:13:22.733 --> 00:13:26.237 let's call that variable v sub i-- 00:13:26.237 --> 00:13:35.546 for each x sub i, for each value in the domain of that 00:13:35.546 --> 00:13:40.951 variable, consider each of the things that still surviving. 00:13:40.951 --> 00:13:55.699 For each of those, for each constraint c, that's between x 00:13:55.699 --> 00:14:08.579 sub i, and some x sub j, where x sub j is an element of the 00:14:08.579 --> 00:14:11.248 domain of j. 00:14:11.248 --> 00:14:14.218 Now that sounds awfully fancy, but this just says, in the 00:14:14.218 --> 00:14:18.155 case of Texas up there, whenever I consider one of the 00:14:18.155 --> 00:14:22.359 values that's still remaining as a choice for Texas, I want 00:14:22.359 --> 00:14:24.895 to consider all of the constraints between that 00:14:24.895 --> 00:14:29.733 variable and the adjacent states. 00:14:29.733 --> 00:14:34.805 And I want to be sure that anything I leave in the domain 00:14:34.805 --> 00:14:38.542 is OK for some selection in the other states, some 00:14:38.542 --> 00:14:41.145 remaining choices in the other states. 00:14:41.145 --> 00:14:43.280 So that's why we're getting pretty nested here. 00:14:43.280 --> 00:14:46.750 But we're doing depth first search. 00:14:46.750 --> 00:14:49.953 We are considering the variables in a certain 00:14:49.953 --> 00:14:54.825 collection of variables. 00:14:54.825 --> 00:14:57.327 For each one of those where considering all the values 00:14:57.327 --> 00:15:00.531 that still remain in the domains of those variables. 00:15:00.531 --> 00:15:03.133 And then for each of those values, we're checking to see 00:15:03.133 --> 00:15:06.203 if it satisfies this some constraint, satisfies the 00:15:06.203 --> 00:15:09.473 constraint that are placed upon it. 00:15:09.473 --> 00:15:14.778 So for each of these constraints if there does not 00:15:14.778 --> 00:15:23.120 exist an x sub j, such that, the constraint between x sub i 00:15:23.120 --> 00:15:33.263 and x sub j is satisfied, well if in that adjacent place 00:15:33.263 --> 00:15:36.967 there's nothing that is consistent with a value, then 00:15:36.967 --> 00:15:39.570 we've got to get rid of it. 00:15:39.570 --> 00:15:40.537 If that's true. 00:15:40.537 --> 00:15:46.009 If there does not exist some value in an adjacent variable 00:15:46.009 --> 00:15:48.579 such that that constraint is satisfied, we're hosed. 00:15:48.579 --> 00:15:50.981 We've got to get rid of that value. 00:15:50.981 --> 00:16:01.558 So we're going to remove x sub i from d sub i. 00:16:05.162 --> 00:16:05.429 OK. 00:16:05.429 --> 00:16:07.264 Now, that's fine. 00:16:07.264 --> 00:16:09.900 That's sort of what I did with Texas. 00:16:09.900 --> 00:16:13.804 As soon as I plopped down a value for Louisiana I said, 00:16:13.804 --> 00:16:15.973 well what are the possible values in Texas? 00:16:15.973 --> 00:16:17.808 Red, green, blue and yellow. 00:16:17.808 --> 00:16:19.810 Let's consider red. 00:16:19.810 --> 00:16:21.678 Let's consider the constraints between Texas and 00:16:21.678 --> 00:16:23.347 all adjacent states. 00:16:23.347 --> 00:16:24.848 One of those constraints says it can't be the 00:16:24.848 --> 00:16:26.850 same color as Arizona. 00:16:26.850 --> 00:16:28.885 The only color I've got available for Arizona, since 00:16:28.885 --> 00:16:31.088 I've already made the assignment is red. 00:16:31.088 --> 00:16:32.789 Red is not consistent with red, so I've got 00:16:32.789 --> 00:16:35.258 to get rid of it. 00:16:35.258 --> 00:16:39.262 So it looks complicated, but it's just intuition. 00:16:39.262 --> 00:16:41.965 So what do we do if we get to a situation where 00:16:41.965 --> 00:16:43.900 the domain is empty? 00:16:43.900 --> 00:16:46.169 That means whenever we get around to making assignment to 00:16:46.169 --> 00:16:48.739 it, there won't be anything left. 00:16:48.739 --> 00:17:03.520 So if that ever happens, if the domain ever becomes empty, 00:17:03.520 --> 00:17:04.954 then what do we do? 00:17:04.954 --> 00:17:06.189 We've got to back up. 00:17:13.195 --> 00:17:14.297 So the intuition is clear. 00:17:14.297 --> 00:17:15.265 This is the algorithm. 00:17:15.265 --> 00:17:18.135 The algorithm when you work through it, think about 00:17:18.135 --> 00:17:19.536 whether it makes sense and what not. 00:17:19.536 --> 00:17:20.569 How it fits with Texas. 00:17:20.569 --> 00:17:21.371 Yeah, it sure does. 00:17:21.371 --> 00:17:24.340 All we're doing is we're making these death first 00:17:24.340 --> 00:17:24.907 assignments. 00:17:24.907 --> 00:17:27.477 And in the neighborhood of those depth first assignments 00:17:27.477 --> 00:17:31.181 we're looking around to see if the values that are possible 00:17:31.181 --> 00:17:32.148 include something. 00:17:32.148 --> 00:17:35.385 And if they don't include anything, we know we made and 00:17:35.385 --> 00:17:39.589 irrevocable blunder, and we have to back up. 00:17:39.589 --> 00:17:43.226 So that's the essence of the idea. 00:17:43.226 --> 00:17:46.229 Now, how well does it work? 00:17:46.229 --> 00:17:53.770 Well a little bit depends on what we choose for considered. 00:17:53.770 --> 00:17:55.739 There are lots of choices for what we consider. 00:17:58.708 --> 00:18:02.545 So let me enumerate some of those choices and then we'll 00:18:02.545 --> 00:18:03.914 have a look and see what they do. 00:18:18.128 --> 00:18:21.898 Oh I guess one possibility is to consider nothing. 00:18:32.275 --> 00:18:33.510 Let's try it out. 00:18:39.115 --> 00:18:42.685 So our type of search is going to be no checks. 00:18:42.685 --> 00:18:44.087 What do you think is going to happen? 00:18:46.790 --> 00:18:50.093 We're not even checking the assignment. 00:18:50.093 --> 00:18:51.561 That's pretty fast. 00:18:51.561 --> 00:18:53.730 Unfortunately, since we haven't even check the most 00:18:53.730 --> 00:18:56.666 recent assignment, we get lots of places where there are 00:18:56.666 --> 00:18:58.668 states that are adjacent to each other that 00:18:58.668 --> 00:19:00.303 have the same color. 00:19:00.303 --> 00:19:01.571 That's no good. 00:19:05.141 --> 00:19:09.646 So another thing we can do is consider everything 00:19:09.646 --> 00:19:10.880 everything. 00:19:18.254 --> 00:19:20.824 That's no good, because that would say, as soon as we color 00:19:20.824 --> 00:19:25.195 our first state, we check to make sure that all 47 other 00:19:25.195 --> 00:19:27.463 states can be colored. 00:19:27.463 --> 00:19:31.267 That seems like it over doing it a little bit. 00:19:31.267 --> 00:19:34.971 But in any case, at least we want to check the assignment. 00:19:43.413 --> 00:19:47.417 So if we go back here and check the assignment, let's 00:19:47.417 --> 00:19:48.685 see what happens. 00:19:54.190 --> 00:19:57.126 Type assignments, assignments only. 00:19:57.126 --> 00:19:57.827 Boom. 00:19:57.827 --> 00:20:00.763 Aw, gees, that's where I got in trouble before. 00:20:00.763 --> 00:20:08.338 This is the thing is going to run for 17 billion years at 00:20:08.338 --> 00:20:10.506 nanosecond or something like that. 00:20:10.506 --> 00:20:12.809 It's only a billion years if you run it at nanosecond 00:20:12.809 --> 00:20:15.945 speed, so I guess maybe you could do that. 00:20:15.945 --> 00:20:17.313 Have a fast computer. 00:20:17.313 --> 00:20:19.549 But this isn't going to work because of our unfortunate 00:20:19.549 --> 00:20:25.154 choice of Texas as the last state to be considered, and 00:20:25.154 --> 00:20:27.890 the unfortunate coloring of the four surrounding states 00:20:27.890 --> 00:20:29.459 right up front. 00:20:29.459 --> 00:20:32.295 And our unfortunate decision to rotate the color so as to 00:20:32.295 --> 00:20:36.499 avoid overdoing any one color. 00:20:36.499 --> 00:20:37.400 So this doesn't work. 00:20:37.400 --> 00:20:40.937 We know we went to the trouble of working out the business 00:20:40.937 --> 00:20:46.142 with Texas by hand, using the domain reduction algorithm. 00:20:46.142 --> 00:20:47.677 Better make a note that this is the 00:20:47.677 --> 00:20:48.945 domain reduction algorithm. 00:20:58.554 --> 00:21:00.456 And what we're going to do is we're going to check the 00:21:00.456 --> 00:21:03.292 neighbors of the assignments. 00:21:03.292 --> 00:21:04.160 Just like we did here. 00:21:04.160 --> 00:21:06.529 We checked Texas each time we made one of those four 00:21:06.529 --> 00:21:10.233 choices, because it's a neighbor of all of the choices 00:21:10.233 --> 00:21:12.668 of the states that we made. 00:21:12.668 --> 00:21:14.837 So one thing to do is to check neighbors. 00:21:20.109 --> 00:21:23.713 This is one, two, three, now let's see what happens. 00:21:32.455 --> 00:21:35.024 Check neighbors only, go. 00:21:53.643 --> 00:21:54.577 Shoot, I don't know. 00:21:54.577 --> 00:21:57.747 It's OK with Texas, right? 00:21:57.747 --> 00:22:02.718 Because it didn't color the states around Texas with all 00:22:02.718 --> 00:22:04.954 of the four color choices. 00:22:04.954 --> 00:22:07.990 But it's still getting into trouble in other places. 00:22:10.660 --> 00:22:15.097 Like the states like Missouri, Kentucky, Virginia, Tennessee, 00:22:15.097 --> 00:22:17.266 states with lots of boundaries. 00:22:19.869 --> 00:22:22.271 So I don't know whether this is going to-- 00:22:22.271 --> 00:22:25.541 oh, there it finally worked. 00:22:25.541 --> 00:22:28.210 It went through a lot of effort, though. 00:22:28.210 --> 00:22:30.746 For the sake of comparison, we might make a note that it ran 00:22:30.746 --> 00:22:43.426 into 9,139 dead ends. 00:22:43.426 --> 00:22:44.894 But it did do some good. 00:22:44.894 --> 00:22:49.398 It didn't take a length of time longer than the remaining 00:22:49.398 --> 00:22:51.801 part of the universe. 00:22:51.801 --> 00:22:55.337 But if it's a good idea to check the neighbors, if we 00:22:55.337 --> 00:22:58.774 make a change to the neighbors, what might that 00:22:58.774 --> 00:23:03.145 suggests that we do in addition? 00:23:03.145 --> 00:23:04.880 Well it might suggest that if we make a change to a 00:23:04.880 --> 00:23:08.984 neighbor, that we check its neighbors, too, make sure 00:23:08.984 --> 00:23:11.253 they're all right. 00:23:11.253 --> 00:23:15.691 So another choice is to propagate. 00:23:45.654 --> 00:23:50.559 So propagate through variables with reduced domains. 00:23:50.559 --> 00:23:51.961 Let's see how that works. 00:24:03.706 --> 00:24:04.940 Wait a minute. 00:24:06.775 --> 00:24:08.410 I must have made a mistake. 00:24:08.410 --> 00:24:10.679 Let's try that again. 00:24:10.679 --> 00:24:13.115 Oh, maybe we better slow it down. 00:24:17.953 --> 00:24:20.989 All that grey stuff is showing the limit of the propagation. 00:24:27.463 --> 00:24:31.967 Man it's, let's see at four second of 40, 00:24:31.967 --> 00:24:37.706 that's about 10 seconds. 00:24:37.706 --> 00:24:39.374 Boom. 00:24:39.374 --> 00:24:40.542 Not bad. 00:24:40.542 --> 00:24:42.311 Zero dead ends. 00:24:42.311 --> 00:24:43.278 And it was a lot faster. 00:24:43.278 --> 00:24:45.047 I didn't happen to notice how many constraints were checked 00:24:45.047 --> 00:24:48.317 on that other thing, I think it was around 20,000 or so. 00:24:48.317 --> 00:24:50.919 This is a lot less. 00:24:50.919 --> 00:24:53.522 So this looks like a good idea. 00:24:53.522 --> 00:24:56.425 But why did I label it number five? 00:24:56.425 --> 00:25:00.128 Well because there's something between this and number three. 00:25:00.128 --> 00:25:14.509 So number four is, through v with d reduced to one value. 00:25:18.213 --> 00:25:20.916 So we're not going to propagate through all of the 00:25:20.916 --> 00:25:24.953 variables which have their domains shrunk a little bit. 00:25:24.953 --> 00:25:26.822 We're only going to propagate through those that have the 00:25:26.822 --> 00:25:31.693 greater shrinkage, all the way down to a single value. 00:25:31.693 --> 00:25:33.395 So let's see how that might work. 00:25:44.773 --> 00:25:48.043 Anybody want to place any bets on this one? 00:25:48.043 --> 00:25:48.477 Let's see. 00:25:48.477 --> 00:25:52.681 We checked 2,623 constraints last time. 00:25:52.681 --> 00:25:53.949 Let's see what happens this time. 00:25:56.785 --> 00:26:00.122 You can see that the extent of the gray is less, because it's 00:26:00.122 --> 00:26:01.656 not propagating so far. 00:26:10.799 --> 00:26:22.878 And as we breathily await the answer, I'd say we've found 00:26:22.878 --> 00:26:23.712 our winner. 00:26:23.712 --> 00:26:27.215 As this does a couple of dead ends, but the number of 00:26:27.215 --> 00:26:31.219 constraint checked is less than 1,000. 00:26:31.219 --> 00:26:33.555 So in general, with problems this, you have all of these 00:26:33.555 --> 00:26:35.891 choices for what you consider. 00:26:35.891 --> 00:26:37.492 You don't want to consider nothing, because then you're 00:26:37.492 --> 00:26:39.628 not honoring your constraints. 00:26:39.628 --> 00:26:41.496 You'll certainly want to consider the things you just 00:26:41.496 --> 00:26:47.869 made assignments for, because otherwise you'll construct a 00:26:47.869 --> 00:26:51.106 solution that violates a constraint. 00:26:51.106 --> 00:26:54.376 You don't want to do everything, because that's 00:26:54.376 --> 00:26:56.244 excessive work. 00:26:56.244 --> 00:27:03.018 And so checking the neighbors is a good idea, but it's 00:27:03.018 --> 00:27:05.520 always better in practice. 00:27:05.520 --> 00:27:08.356 In practice, inevitably it's the case that it's better to 00:27:08.356 --> 00:27:09.658 do some propagation through the 00:27:09.658 --> 00:27:11.192 things that you've changed. 00:27:11.192 --> 00:27:12.694 How much propagation? 00:27:12.694 --> 00:27:14.729 It doesn't seem to do much good to propagate through 00:27:14.729 --> 00:27:16.264 things are just changed. 00:27:16.264 --> 00:27:18.800 But it does seem to do some good to propagate through the 00:27:18.800 --> 00:27:19.901 things that have changed and been 00:27:19.901 --> 00:27:22.771 reduced to a single value. 00:27:22.771 --> 00:27:25.607 So as soon as you get a neighbor of some assignment 00:27:25.607 --> 00:27:28.476 you just made that has its domain reduced to a single 00:27:28.476 --> 00:27:30.912 value, then you check its neighbors, too. 00:27:30.912 --> 00:27:32.614 So you check the neighbors, of the neighbors, of the 00:27:32.614 --> 00:27:36.484 neighbors, of the neighbors, on and on and on, as long as 00:27:36.484 --> 00:27:40.155 you've found a domain being reduced. 00:27:40.155 --> 00:27:43.358 And not only being reduced, but reduced to a signal value. 00:27:43.358 --> 00:27:43.758 All right? 00:27:43.758 --> 00:27:45.393 So that's the demand reduction algorithm. 00:27:45.393 --> 00:27:49.030 And I guarantee you a problem like that. 00:27:49.030 --> 00:27:52.500 And I know you don't know how to work those problems yet, 00:27:52.500 --> 00:27:54.235 because this is a little bit abstract. 00:27:54.235 --> 00:27:58.540 And to work these problems in the exam setting you need to 00:27:58.540 --> 00:28:01.443 know a little bit about how to keep track of the variable 00:28:01.443 --> 00:28:03.878 values that remain in the domain, and 00:28:03.878 --> 00:28:04.512 that sort of thing. 00:28:04.512 --> 00:28:07.015 And you'll learn more about that in your recitations, and 00:28:07.015 --> 00:28:10.985 in this mega recitation, and in the tutorials. 00:28:10.985 --> 00:28:13.455 So we could go home except that there are few little 00:28:13.455 --> 00:28:16.057 flourishes to deal with here. 00:28:16.057 --> 00:28:25.400 And those flourishes, include some dirty, 00:28:25.400 --> 00:28:27.702 filthy little secrets. 00:28:27.702 --> 00:28:31.005 For example, I've chosen, as my classroom 00:28:31.005 --> 00:28:33.575 example, to pick on Texas. 00:28:33.575 --> 00:28:35.810 And arranged for this situation to 00:28:35.810 --> 00:28:38.313 be especially ugly. 00:28:38.313 --> 00:28:43.284 So I could arrange the states in a different way. 00:28:43.284 --> 00:28:45.620 We have highly constrained states, that have a lot of 00:28:45.620 --> 00:28:48.456 bordering states around them. 00:28:48.456 --> 00:28:54.128 And we have other states, like Maine, up there, that only 00:28:54.128 --> 00:28:56.831 borders on one other state. 00:28:56.831 --> 00:28:59.734 So I don't know. 00:28:59.734 --> 00:29:01.869 Will, what do you think? 00:29:05.173 --> 00:29:09.043 Should we arrange the states for our death first search in 00:29:09.043 --> 00:29:12.680 the order of least constrained to most constrained, or most 00:29:12.680 --> 00:29:13.948 constrained to least constrained? 00:29:17.885 --> 00:29:21.289 In other words, should we start with Missouri, or 00:29:21.289 --> 00:29:23.891 Tennessee, or Kentucky, or something like that? 00:29:23.891 --> 00:29:26.427 Or should we start with Maine? 00:29:26.427 --> 00:29:28.396 What do you think? 00:29:28.396 --> 00:29:31.332 You have a 50% chance of getting it right, just by 00:29:31.332 --> 00:29:33.835 [? looking ?] at points. 00:29:33.835 --> 00:29:36.137 WILL: Start with the most first. 00:29:36.137 --> 00:29:36.170 PROF. 00:29:36.170 --> 00:29:37.105 PATRICK WINSTON: He thinks we should start with the most 00:29:37.105 --> 00:29:38.806 constraint first. 00:29:38.806 --> 00:29:41.676 Do we have a volunteer who wants to suggest that we start 00:29:41.676 --> 00:29:44.412 with the least constraint first? 00:29:44.412 --> 00:29:47.782 That's the way I always work on stuff. 00:29:47.782 --> 00:29:51.252 I'm working on a book or something, I have 500 things 00:29:51.252 --> 00:29:54.122 to fix, I'll always choose to work on the easiest stuff 00:29:54.122 --> 00:29:58.159 first, so that I feel like I'm making the list a lot smaller. 00:29:58.159 --> 00:30:00.595 Leave the hardest things to last. 00:30:00.595 --> 00:30:04.098 But we don't have any volunteers who want to bet on 00:30:04.098 --> 00:30:07.235 that idea of least constraint first? 00:30:07.235 --> 00:30:08.903 OK. 00:30:08.903 --> 00:30:11.205 Jason wants to suggest that we should work on least 00:30:11.205 --> 00:30:12.240 constraint first. 00:30:12.240 --> 00:30:13.574 Well we have ground truth, because we 00:30:13.574 --> 00:30:14.809 can just try it out. 00:30:19.514 --> 00:30:21.215 I guess we'll stick with our shrinking to one 00:30:21.215 --> 00:30:24.152 value thing, here. 00:30:24.152 --> 00:30:29.090 But we will reorder things so that we have the least 00:30:29.090 --> 00:30:30.324 constrained first. 00:30:34.128 --> 00:30:37.098 So right away, we got a color for Maine. 00:30:40.334 --> 00:30:41.602 Maybe we ought to speed this up a little bit. 00:30:46.741 --> 00:30:48.643 Well, that's a good idea. 00:30:48.643 --> 00:30:54.215 Jason suggested this and we only 1,732 constraints and we 00:30:54.215 --> 00:30:55.216 had 59 dead ends. 00:30:55.216 --> 00:31:01.122 So let's try the other way around, and we'll go back to 00:31:01.122 --> 00:31:02.356 four frames a second. 00:31:15.069 --> 00:31:17.138 So we're working, kind of from the middle of the country out, 00:31:17.138 --> 00:31:18.506 with this one. 00:31:18.506 --> 00:31:20.041 We're going to deal with Maine, I guess, last. 00:31:26.080 --> 00:31:28.716 Which is better. 00:31:28.716 --> 00:31:31.852 Too bad, I think it looks like this is better. 00:31:31.852 --> 00:31:35.056 In fact, let's not be so aggressive with the use of 00:31:35.056 --> 00:31:36.157 constraint propagation. 00:31:36.157 --> 00:31:38.893 Let's just check the assignments only. 00:31:38.893 --> 00:31:41.228 If we go back to an arrangement where we have 00:31:41.228 --> 00:31:45.166 least constrained first, and we'll crank up the speed. 00:31:48.335 --> 00:31:50.371 Well actually, we would have to crank it up pretty big, 00:31:50.371 --> 00:31:54.909 because the states like Missouri, Tennessee, Kentucky, 00:31:54.909 --> 00:31:58.446 they're going to be like Texas. 00:31:58.446 --> 00:32:00.915 And so were kind of back to the length of the universe 00:32:00.915 --> 00:32:01.615 type problem, here. 00:32:01.615 --> 00:32:04.852 With the least constraint first, and no use of 00:32:04.852 --> 00:32:10.291 constraints, other than to check the current assignment. 00:32:10.291 --> 00:32:14.628 So let's stop that, though, and check the most constrained 00:32:14.628 --> 00:32:17.965 first, assignments only. 00:32:17.965 --> 00:32:19.500 I don't know how long's this going to take. 00:32:22.069 --> 00:32:24.338 That's the dirty little secret. 00:32:24.338 --> 00:32:28.209 If we had arranged our states from most constrained to least 00:32:28.209 --> 00:32:30.644 constrained, ordinary depth first search with none of this 00:32:30.644 --> 00:32:34.048 stuff we talked about today would work just fine. 00:32:34.048 --> 00:32:35.282 All right. 00:32:38.919 --> 00:32:42.656 So it's a little bit like games. 00:32:42.656 --> 00:32:44.425 Do you use progressive deepening, or do 00:32:44.425 --> 00:32:45.693 you use alpha beta? 00:32:45.693 --> 00:32:47.061 And the answer is both. 00:32:47.061 --> 00:32:51.899 You use everything you've got to deal with the problems. 00:32:51.899 --> 00:32:56.604 And depending on the problem, one or another of the things 00:32:56.604 --> 00:32:59.440 you incorporate into your approach will work just great, 00:32:59.440 --> 00:33:02.242 if you're lucky. 00:33:02.242 --> 00:33:06.246 So now, I promise that this is useful not only for people who 00:33:06.246 --> 00:33:06.947 want to color maps. 00:33:06.947 --> 00:33:08.015 God, who wants to do that? 00:33:08.015 --> 00:33:10.217 We know it can be done with four colors. 00:33:10.217 --> 00:33:13.087 But it's also useful for doing all kinds of 00:33:13.087 --> 00:33:15.723 resource planning problems. 00:33:15.723 --> 00:33:18.092 So I want to show you a resource planning problem, and 00:33:18.092 --> 00:33:19.994 I want you think about-- while I'm doing it-- 00:33:19.994 --> 00:33:22.096 think about whether it's actually analogous to the map 00:33:22.096 --> 00:33:22.896 coloring problem. 00:33:22.896 --> 00:33:24.465 All right? 00:33:24.465 --> 00:33:25.366 So here's the deal. 00:33:25.366 --> 00:33:29.136 You have just landed a summer job with the Jet 00:33:29.136 --> 00:33:32.406 Green, a new airline. 00:33:32.406 --> 00:33:40.748 And Jet Green is a low cost, no frills, hardly any 00:33:40.748 --> 00:33:43.650 maintenance type of airline. 00:33:43.650 --> 00:33:47.921 And they want to fly mostly between Boston and New York. 00:33:47.921 --> 00:33:50.691 Occasionally they want to fly to Los Angeles. 00:33:50.691 --> 00:33:52.059 And they're trying to get by with the 00:33:52.059 --> 00:33:54.061 smallest number of airplanes. 00:33:54.061 --> 00:33:57.531 So that's why we have a kind of resource allocation problem 00:33:57.531 --> 00:33:59.700 with Jet Green. 00:33:59.700 --> 00:34:02.069 So I'm going to write down what their 00:34:02.069 --> 00:34:02.536 schedule looks like. 00:34:02.536 --> 00:34:08.775 They have one flight, F1, that goes from Boston 00:34:08.775 --> 00:34:14.614 to JFK, like so. 00:34:14.614 --> 00:34:17.951 It's an early in the day flight. 00:34:17.951 --> 00:34:21.221 Then they want to have another one, F2, that 00:34:21.221 --> 00:34:24.491 flies from JFK to Boston. 00:34:29.563 --> 00:34:32.666 And then they want to have another flight a little later 00:34:32.666 --> 00:34:36.803 in the day that flies from Boston to JFK. 00:34:39.606 --> 00:34:41.541 And a little later than that, they want to have another 00:34:41.541 --> 00:34:46.780 flight that goes from JFK to Boston. 00:34:46.780 --> 00:34:49.416 They're going to start off mostly as a shuttle airline in 00:34:49.416 --> 00:34:49.882 the beginning. 00:34:49.882 --> 00:34:55.621 So that's F1, F2, F3, and F4. 00:34:55.621 --> 00:35:00.627 And they have a fifth flight, F5, that goes from Boston to 00:35:00.627 --> 00:35:03.797 Los Angeles, that takes a long time. 00:35:03.797 --> 00:35:06.633 So it looks like this on the schedule. 00:35:06.633 --> 00:35:09.136 Of course we have time going that way. 00:35:09.136 --> 00:35:12.506 So that's Boston to LAX. 00:35:16.043 --> 00:35:22.082 Now your job is to determine if they can fly this schedule 00:35:22.082 --> 00:35:24.818 with four aircraft? 00:35:24.818 --> 00:35:27.254 And naturally you don't want to over use any one aircraft, 00:35:27.254 --> 00:35:30.290 because you would like to have even wear on them. 00:35:30.290 --> 00:35:31.058 Right? 00:35:31.058 --> 00:35:35.262 So as you make your choices, you'll rotate the aircraft. 00:35:35.262 --> 00:35:40.333 So you'll assign to this one to A1, aircraft number one. 00:35:40.333 --> 00:35:42.335 This one will be A2. 00:35:42.335 --> 00:35:44.571 This one will be A3. 00:35:44.571 --> 00:35:47.340 And this one will be A4. 00:35:47.340 --> 00:35:51.578 And, oops, there's no aircraft left for the flight to Los 00:35:51.578 --> 00:35:54.447 Angeles, because you only have four. 00:35:54.447 --> 00:35:55.615 So, it's obvious, right? 00:35:55.615 --> 00:36:00.654 This is 100%, exactly, the map coloring problem, even down to 00:36:00.654 --> 00:36:02.589 the four choices. 00:36:02.589 --> 00:36:06.226 Because, you have the constraint, the no single 00:36:06.226 --> 00:36:11.498 physical aircraft can fly two flights at the same time. 00:36:11.498 --> 00:36:13.099 Just like no two adjacent states can 00:36:13.099 --> 00:36:15.001 be colored the same. 00:36:15.001 --> 00:36:20.207 So there's a no same time constraint, like so. 00:36:27.380 --> 00:36:31.818 So if you were assigning aircraft to these flights, you 00:36:31.818 --> 00:36:35.255 would get down to F4, the fourth flight, and you would 00:36:35.255 --> 00:36:41.494 say, well, let's see, this guy down here can be 00:36:41.494 --> 00:36:42.762 A1, A2, A3, or A4. 00:36:42.762 --> 00:36:45.899 But if I choose A4 for that fourth flight, then there 00:36:45.899 --> 00:36:49.736 would be nothing left in its domain. 00:36:49.736 --> 00:36:56.843 So you've thus set the problem up to be identical to the map 00:36:56.843 --> 00:36:57.744 coloring problem. 00:36:57.744 --> 00:37:00.313 And, of course, you can enrich it with other kinds of 00:37:00.313 --> 00:37:01.581 constraints. 00:37:01.581 --> 00:37:03.383 So, for example, you might have-- 00:37:03.383 --> 00:37:05.185 this is a not same time constraint-- 00:37:13.226 --> 00:37:18.198 and these, I mean, this is at JFK. 00:37:18.198 --> 00:37:22.769 And it flies out of JFK, so maybe you can use the same 00:37:22.769 --> 00:37:23.870 aircraft for those. 00:37:23.870 --> 00:37:25.739 But not if they're right up against each other, because 00:37:25.739 --> 00:37:28.308 you have a minimum ground time rule. 00:37:28.308 --> 00:37:30.210 So there's a minimum ground time constraint here. 00:37:32.846 --> 00:37:34.681 And there's a minimum ground time constraint here. 00:37:37.617 --> 00:37:39.419 There's a minimum ground time constraint here. 00:37:44.224 --> 00:37:47.494 And if these are at the same city, then you've got to allow 00:37:47.494 --> 00:37:49.696 enough time for them to fly between the two cities that 00:37:49.696 --> 00:37:50.997 are involved. 00:37:50.997 --> 00:37:52.999 So the constraints can get a little bit more complicated, 00:37:52.999 --> 00:37:55.602 but the idea is the same. 00:37:55.602 --> 00:37:58.872 So you say to me, I don't trust you, show me. 00:37:58.872 --> 00:37:59.839 OK. 00:37:59.839 --> 00:38:02.809 So let me show you. 00:38:02.809 --> 00:38:04.878 Oh, by the way, there's one more way to make this map 00:38:04.878 --> 00:38:06.112 coloring problem easier, right? 00:38:09.616 --> 00:38:10.516 Let's see. 00:38:10.516 --> 00:38:13.987 The arrangement is going to be alphabetical, the type is 00:38:13.987 --> 00:38:19.192 going to be assignments only, and we know that's a loser. 00:38:19.192 --> 00:38:24.831 But not if we use a whole lot of colors. 00:38:24.831 --> 00:38:28.701 So it's the use of four colors that got us into trouble. 00:38:28.701 --> 00:38:30.737 Now that's an aside, but it'll be coming back 00:38:30.737 --> 00:38:31.704 in a moment or two. 00:38:31.704 --> 00:38:35.141 Scheduling, here's that problem. 00:38:35.141 --> 00:38:37.310 Boom. 00:38:37.310 --> 00:38:39.112 You can almost see it working just like the 00:38:39.112 --> 00:38:40.813 map coloring thing. 00:38:40.813 --> 00:38:44.150 But that's maybe too easy. 00:38:44.150 --> 00:38:45.385 Let's do this one. 00:38:49.055 --> 00:38:52.191 See this is just like [? simplicia. ?] 00:38:52.191 --> 00:38:55.695 We've kind of got a goofy arrangement here that's 00:38:55.695 --> 00:38:58.965 guaranteed to lose at the bottom because of choices made 00:38:58.965 --> 00:39:00.900 at the top. 00:39:00.900 --> 00:39:01.934 But that's OK. 00:39:01.934 --> 00:39:07.573 We can that we can stop this, and we can change to check 00:39:07.573 --> 00:39:09.642 neighbors only. 00:39:09.642 --> 00:39:10.877 And boom, there it goes. 00:39:14.013 --> 00:39:16.416 Or alternatively, let me see if I can do it this way. 00:39:16.416 --> 00:39:20.953 Most constrained first, type will be assignments only. 00:39:20.953 --> 00:39:23.423 Boom. 00:39:23.423 --> 00:39:24.657 That worked fine, too. 00:39:26.959 --> 00:39:29.428 So you might choose to have a slightly harder 00:39:29.428 --> 00:39:33.099 problem, like this. 00:39:33.099 --> 00:39:35.768 And we can search away. 00:39:40.573 --> 00:39:42.608 Actually I don't know if this will complete or not. 00:39:42.608 --> 00:39:46.179 This is a randomly generated example. 00:39:46.179 --> 00:39:49.248 But you're going to lose your summer job if you can't figure 00:39:49.248 --> 00:39:51.150 out whether you can do this, or not. 00:39:51.150 --> 00:39:52.418 So what are you going to do? 00:39:55.388 --> 00:39:57.256 [? Elliott, ?] you got any thoughts about how you're 00:39:57.256 --> 00:40:00.426 going to save your job? 00:40:00.426 --> 00:40:03.029 So here's the question you've been asked. 00:40:03.029 --> 00:40:08.568 How many airplanes does Jet Green need? 00:40:08.568 --> 00:40:12.204 And you decide, well, four seemed to work before, so 00:40:12.204 --> 00:40:14.206 we'll try four here. 00:40:14.206 --> 00:40:17.176 You're not sure if it's going to terminate or not, I mean, 00:40:17.176 --> 00:40:20.379 in your lifetime, let alone in your summer job. 00:40:24.951 --> 00:40:27.086 [? Elliott, ?] let me give you a hint. 00:40:27.086 --> 00:40:28.354 Look at the outline. 00:40:30.323 --> 00:40:33.826 The outline up here, on the board, the last item. 00:40:33.826 --> 00:40:34.794 [? ELLIOTT: [INAUDIBLE] ?] 00:40:34.794 --> 00:40:34.860 PROF. 00:40:34.860 --> 00:40:36.128 PATRICK WINSTON: Yeah, what's that mean? 00:40:39.198 --> 00:40:42.568 What's the maximum number of airplanes we're going to need? 00:40:42.568 --> 00:40:47.173 Suppose we've got five flights, what's the maximum 00:40:47.173 --> 00:40:49.241 number of airplanes we would ever need? 00:40:49.241 --> 00:40:50.543 Five. 00:40:50.543 --> 00:40:53.212 What's the minimum number of airplanes we'll need? 00:40:53.212 --> 00:40:54.947 One. 00:40:54.947 --> 00:40:59.719 So let's try it with a small number of airplanes and a 00:40:59.719 --> 00:41:00.953 large number of airplanes. 00:41:07.593 --> 00:41:11.363 So that showed us very fast that we can't 00:41:11.363 --> 00:41:12.631 do it with one airplane. 00:41:21.407 --> 00:41:26.045 That showed us very fast we can do it with ten airplanes. 00:41:26.045 --> 00:41:28.981 SPEAKER 1: [INAUDIBLE] 00:41:28.981 --> 00:41:33.886 amount of overlappage from up [INAUDIBLE]. 00:41:33.886 --> 00:41:33.953 PROF. 00:41:33.953 --> 00:41:35.187 PATRICK WINSTON: Volunteer? 00:41:37.323 --> 00:41:40.326 What are we going to do to find the actual number, as 00:41:40.326 --> 00:41:41.260 fast as possible. 00:41:41.260 --> 00:41:45.297 And at least give our boss a reasonable answer, even if not 00:41:45.297 --> 00:41:47.399 necessarily the exact number very fast? 00:41:50.135 --> 00:41:50.803 It's easy, right? 00:41:50.803 --> 00:41:54.039 We're going to start up here with one computer, and we're 00:41:54.039 --> 00:41:57.042 going to start down here with another computer, 00:41:57.042 --> 00:41:59.111 and see what happens. 00:41:59.111 --> 00:42:01.480 So let's see if we can do it with nine. 00:42:06.452 --> 00:42:09.421 Let's see if we can do it with eight. 00:42:13.292 --> 00:42:16.295 Let's see if we can do it with seven. 00:42:16.295 --> 00:42:17.796 These take almost zero time, right? 00:42:17.796 --> 00:42:19.031 Because they're under constraint. 00:42:22.901 --> 00:42:24.303 Wow, that's good, seven. 00:42:24.303 --> 00:42:26.005 Let's try six. 00:42:26.005 --> 00:42:27.239 Actually let's try two. 00:42:30.976 --> 00:42:32.778 It loses fast. 00:42:32.778 --> 00:42:34.013 Let's try three. 00:42:44.056 --> 00:42:44.323 I don't know. 00:42:44.323 --> 00:42:47.626 Maybe if you let it run one long enough three will work. 00:42:47.626 --> 00:42:49.294 I doubt it. 00:42:49.294 --> 00:42:51.296 While we're at it though, we might as well go back here and 00:42:51.296 --> 00:42:51.964 try it with six. 00:42:51.964 --> 00:42:53.232 Remember seven worked real fast. 00:42:57.202 --> 00:42:59.705 Gees, six, that was six, right? 00:42:59.705 --> 00:43:00.072 Yeah. 00:43:00.072 --> 00:43:01.306 So let's try it with five. 00:43:04.843 --> 00:43:05.144 OK. 00:43:05.144 --> 00:43:06.912 So it runs real fast with five. 00:43:06.912 --> 00:43:09.848 It terminates real quick with two, so we got 00:43:09.848 --> 00:43:11.917 three and four left. 00:43:11.917 --> 00:43:16.155 So we could tell our boss, a la any time algorithm, that 00:43:16.155 --> 00:43:18.090 you're not real sure, but you know it's going to be either 00:43:18.090 --> 00:43:19.458 three or four. 00:43:19.458 --> 00:43:21.460 And then, you got two computers. 00:43:21.460 --> 00:43:25.297 You can let both run and see if either one terminate. 00:43:25.297 --> 00:43:26.899 So you have three and four. 00:43:26.899 --> 00:43:31.336 My guess is that three will eventually give up. 00:43:31.336 --> 00:43:33.238 But of course, there's another little problem here. 00:43:33.238 --> 00:43:35.908 We haven't used the most constraint first. 00:43:35.908 --> 00:43:38.710 If we did that, we might be able to do it even faster. 00:43:38.710 --> 00:43:41.813 Actually, I don't think I can make that switch without 00:43:41.813 --> 00:43:44.016 getting another random assignment, but 00:43:44.016 --> 00:43:46.518 let's see what happens. 00:43:46.518 --> 00:43:48.153 Maybe so. 00:43:48.153 --> 00:43:49.421 SPEAKER 2: [INAUDIBLE] 00:43:52.391 --> 00:43:52.457 PROF. 00:43:52.457 --> 00:43:54.192 PATRICK WINSTON: Oh, I already had most constraint first? 00:43:54.192 --> 00:43:55.561 OK. 00:43:55.561 --> 00:43:57.829 So it didn't help to actually switch. 00:43:57.829 --> 00:44:00.933 And I think I've got a new schedule to work here. 00:44:00.933 --> 00:44:03.368 So that's the end of the story. 00:44:03.368 --> 00:44:07.439 You can do good resource allocation if you do several 00:44:07.439 --> 00:44:08.707 things are once. 00:44:08.707 --> 00:44:11.643 Number one, you always want to use most constraint first. 00:44:11.643 --> 00:44:13.545 Number two you want to propagate through domains 00:44:13.545 --> 00:44:15.247 produced to a single algorithm. 00:44:15.247 --> 00:44:17.582 And number three, if you really try to figure out what 00:44:17.582 --> 00:44:21.286 the minimum number of resources needed is, you do 00:44:21.286 --> 00:44:24.589 this under over business and you can quickly converge on a 00:44:24.589 --> 00:44:27.726 narrow range where the search is taking a long time, and be 00:44:27.726 --> 00:44:31.496 sure that it lies within that narrow range. 00:44:31.496 --> 00:44:34.499 Because when you over resource, it's fast to 00:44:34.499 --> 00:44:36.868 complete, and when you under resource, 00:44:36.868 --> 00:44:39.371 it's fast to terminate. 00:44:39.371 --> 00:44:41.373 So you can just squeeze it right down into 00:44:41.373 --> 00:44:42.808 a very small range. 00:44:42.808 --> 00:44:44.476 And that is the end of the story. 00:44:44.476 --> 00:44:45.911 Enjoy your holiday on Monday. 00:44:45.911 --> 00:44:50.849 We'll have two classes next week on Wednesday and Friday, 00:44:50.849 --> 00:44:52.117 as advertised.