WEBVTT 00:00:05.796 --> 00:00:09.170 SRINI DEVADAS: We're going to do something quite different 00:00:09.170 --> 00:00:10.860 in terms of a puzzle. 00:00:10.860 --> 00:00:16.550 And while it's a recursive puzzle, 00:00:16.550 --> 00:00:21.810 it's going to play out to be a recursive solution. 00:00:21.810 --> 00:00:27.559 It looks quite different from the N-queens puzzle. 00:00:27.559 --> 00:00:29.350 And so this is going to be a tiling puzzle. 00:00:34.570 --> 00:00:36.320 And so I set up the puzzle, and then we're 00:00:36.320 --> 00:00:42.440 going to take a little segue off into a canonical recursive 00:00:42.440 --> 00:00:45.310 algorithm called merge sort because I think 00:00:45.310 --> 00:00:49.790 it's good for you to see that algorithm before you dive 00:00:49.790 --> 00:00:51.470 into solving this particular puzzle 00:00:51.470 --> 00:00:53.630 because it kind of shows you the way. 00:00:53.630 --> 00:00:56.950 All right, and but I'll set up the puzzle. 00:00:56.950 --> 00:00:59.660 I guess I didn't have to erase this square. 00:00:59.660 --> 00:01:05.900 It's a courtyard that has a bunch-- 00:01:05.900 --> 00:01:06.470 of tiles. 00:01:06.470 --> 00:01:11.960 You're supposed to tile it, and it's all square tiles. 00:01:11.960 --> 00:01:18.438 And I'll just make it eight by eight like a chessboard. 00:01:21.782 --> 00:01:23.240 And in general, we are going to say 00:01:23.240 --> 00:01:31.177 that this is a 2 raised to n by 2 raised to n courtyard. 00:01:35.000 --> 00:01:40.210 And I want to tile this courtyard. 00:01:47.790 --> 00:01:49.800 So this is a three tile-- 00:01:49.800 --> 00:01:53.130 some people call it a tromino. 00:01:53.130 --> 00:01:56.250 You can think of it's really an L-shaped tile, 00:01:56.250 --> 00:01:59.550 and it can be oriented in any different way. 00:01:59.550 --> 00:02:03.080 It could be an inverted L, et cetera. 00:02:03.080 --> 00:02:09.020 And you're supposed to tile this courtyard using L-shaped tiles 00:02:09.020 --> 00:02:13.590 or trominos. 00:02:13.590 --> 00:02:15.530 And they're called trominos because there's 00:02:15.530 --> 00:02:18.320 three tiles in there-- three squares in there. 00:02:18.320 --> 00:02:22.080 And so that's your job. 00:02:22.080 --> 00:02:25.710 I guess kind of a weird job, but this is a puzzle. 00:02:25.710 --> 00:02:28.500 And so you're out there with 2 raised to n by 2 raised to n. 00:02:28.500 --> 00:02:30.840 And all you have are these L-shaped tiles, 00:02:30.840 --> 00:02:32.460 and you're not allowed to break them. 00:02:32.460 --> 00:02:36.300 If you break them, then that's not considered a valid tiling. 00:02:36.300 --> 00:02:39.480 That's bad, and that's your job. 00:02:42.840 --> 00:02:46.100 Oh, and obviously, it needs to look like this, 00:02:46.100 --> 00:02:48.180 and these look like the floor. 00:02:48.180 --> 00:02:52.200 Maybe you'd like to see the markings because these 00:02:52.200 --> 00:02:53.900 are tiles, after all. 00:02:53.900 --> 00:02:56.100 But you can't have something that is uneven. 00:02:56.100 --> 00:02:58.350 You can't have a tile on top of another tile. 00:02:58.350 --> 00:03:00.150 That's a definite no-no. 00:03:00.150 --> 00:03:04.710 And you can't not have a tile in any part of the courtyard. 00:03:04.710 --> 00:03:07.500 So I think that was kind of obvious, 00:03:07.500 --> 00:03:10.770 but it's worth explicitly stating just 00:03:10.770 --> 00:03:14.780 so you don't get confused here. 00:03:14.780 --> 00:03:21.750 So that's the question that we're going to try and answer. 00:03:21.750 --> 00:03:28.200 But as I said, I want to talk about something a little bit 00:03:28.200 --> 00:03:30.510 different for about 5 or 10 minutes 00:03:30.510 --> 00:03:32.910 and actually show you some code because I 00:03:32.910 --> 00:03:35.760 want to introduce this notion that's 00:03:35.760 --> 00:03:40.140 a little bit different from the N-queens recursive paradigm. 00:03:40.140 --> 00:03:44.310 It is recursive, but it's called recursive divide and conquer. 00:03:44.310 --> 00:03:48.450 And usually, when we use the term recursive divide 00:03:48.450 --> 00:03:52.200 and conquer, you might think that it's applicable, even 00:03:52.200 --> 00:03:54.250 to the N-queens recursive code. 00:03:54.250 --> 00:03:58.260 But from a technical term standpoint 00:03:58.260 --> 00:04:01.130 when we say recursive divide and conquer, 00:04:01.130 --> 00:04:04.950 we're really saying that we're taking the problem, 00:04:04.950 --> 00:04:12.200 and we are dividing it up into smaller problems. 00:04:12.200 --> 00:04:13.950 So you take the problem, and you divide it 00:04:13.950 --> 00:04:17.370 into two problems that are half the size or three problems that 00:04:17.370 --> 00:04:19.850 are a third of the size et cetera. 00:04:19.850 --> 00:04:22.470 Which is not what happened in the case of the N-queens 00:04:22.470 --> 00:04:23.640 problem. 00:04:23.640 --> 00:04:26.370 And in the case of the N-queens problem, you took a column, 00:04:26.370 --> 00:04:29.430 and you actually did iterations on the column. 00:04:29.430 --> 00:04:32.250 So it potentially exploded on you. 00:04:32.250 --> 00:04:35.030 And then you had a problem that is slightly smaller 00:04:35.030 --> 00:04:36.530 than the N-queens. 00:04:36.530 --> 00:04:39.810 It was effectively n minus 1 queens. 00:04:39.810 --> 00:04:44.880 And so it's not n over 2 or n over 2 or 3-- n over 3 00:04:44.880 --> 00:04:49.470 in the case of a three-way recursive divide and conquer. 00:04:49.470 --> 00:04:51.610 So there's a qualitative difference. 00:04:51.610 --> 00:04:55.050 And there's also a quantitative difference in the sense 00:04:55.050 --> 00:04:58.110 that these kinds of algorithms are 00:04:58.110 --> 00:05:01.140 going to be efficient in that they are going 00:05:01.140 --> 00:05:04.410 to run very quickly in polynomial time, 00:05:04.410 --> 00:05:05.820 perhaps quadratic time. 00:05:05.820 --> 00:05:10.440 We can do some analysis for both of the merge sort that we're 00:05:10.440 --> 00:05:12.300 going to do now and for the tiling 00:05:12.300 --> 00:05:14.190 puzzle, once we get to that. 00:05:14.190 --> 00:05:17.560 But that's something to keep in mind. 00:05:17.560 --> 00:05:23.700 So merge sort is the canonical recursive divide 00:05:23.700 --> 00:05:25.350 and conquer approach. 00:05:25.350 --> 00:05:27.950 And I'll put up code for it, but I'm not 00:05:27.950 --> 00:05:28.950 going to show it to you. 00:05:28.950 --> 00:05:30.900 You can look at it in the website. 00:05:30.900 --> 00:05:33.570 But I'm going to describe the algorithm 00:05:33.570 --> 00:05:38.360 to you using a trivial example. 00:05:38.360 --> 00:05:45.760 And so I want to sort these numbers in ascending order. 00:05:48.590 --> 00:05:50.290 So I have these four numbers that I want 00:05:50.290 --> 00:05:52.100 to sort in ascending order. 00:05:52.100 --> 00:05:54.730 And obviously, I could do this many different ways. 00:05:54.730 --> 00:05:56.750 We looked at selection sort. 00:05:56.750 --> 00:06:00.880 But this is a large array, and I want 00:06:00.880 --> 00:06:03.730 to do something more efficient than that quadratic algorithm 00:06:03.730 --> 00:06:04.990 and n squared algorithm. 00:06:04.990 --> 00:06:07.510 And merge sort, which is recursive divide and conquer, 00:06:07.510 --> 00:06:08.740 fits the bill. 00:06:08.740 --> 00:06:11.494 And what merge sort says is I'm going to split the array. 00:06:11.494 --> 00:06:12.910 I'm just going to take this array, 00:06:12.910 --> 00:06:16.352 and I'm going to split it up into two sub-arrays. 00:06:16.352 --> 00:06:18.310 And I could do this recursively, but let's just 00:06:18.310 --> 00:06:19.779 talk about the first step. 00:06:19.779 --> 00:06:21.820 So I'm going to go ahead and split this array up, 00:06:21.820 --> 00:06:23.650 and I have 76 and 32. 00:06:23.650 --> 00:06:29.830 And I'm going to call merge sort on this smaller array. 00:06:29.830 --> 00:06:32.080 And the first thing that merge sort does 00:06:32.080 --> 00:06:36.070 is split the array up into something smaller, 00:06:36.070 --> 00:06:38.740 and you end up getting that. 00:06:38.740 --> 00:06:43.570 So this is the first thing, and that's the first split. 00:06:43.570 --> 00:06:46.420 And then you end up splitting 48 and 32, 00:06:46.420 --> 00:06:52.520 and I'm sorry-- or 12, 76, 32, 48, 12. 00:06:52.520 --> 00:06:55.100 Yep, I got that right. 00:06:55.100 --> 00:06:57.437 And so far, you've done nothing. 00:06:57.437 --> 00:06:59.270 I mean, you've just sort of broken things up 00:06:59.270 --> 00:07:03.140 and turned this into four one arrays or one list 00:07:03.140 --> 00:07:04.610 or what have you. 00:07:04.610 --> 00:07:07.670 But then the fun starts where after you 00:07:07.670 --> 00:07:11.930 do the split, after you return, and so 76 as singleton 00:07:11.930 --> 00:07:14.087 is trivially sorted so there's no work to do. 00:07:14.087 --> 00:07:15.920 So there's no more splits that you could do. 00:07:15.920 --> 00:07:17.450 There's nothing that says you can-- 00:07:17.450 --> 00:07:20.520 I mean, 76 is a monolithic element in their array, 00:07:20.520 --> 00:07:22.020 so you can't do anything with it. 00:07:22.020 --> 00:07:23.550 And so you just leave it be. 00:07:23.550 --> 00:07:25.910 And the same thing with 32, and so on and so forth. 00:07:25.910 --> 00:07:29.420 But after your two calls return, you 00:07:29.420 --> 00:07:31.920 have to merge the two together. 00:07:31.920 --> 00:07:34.370 And the way you merge the two together-- 00:07:34.370 --> 00:07:37.520 the invariant here is that when you merge the two together, 00:07:37.520 --> 00:07:39.470 they have to be sorted in the order 00:07:39.470 --> 00:07:42.520 that you want the overall array to be sorted, 00:07:42.520 --> 00:07:44.450 which is ascending order. 00:07:44.450 --> 00:07:46.940 So in this case, you have down at the bottom, 00:07:46.940 --> 00:07:50.000 it's trivial code because you have two singleton lists. 00:07:50.000 --> 00:07:52.160 And you just decide which order they go into. 00:07:52.160 --> 00:07:56.140 OK, so the merge, in this case, would simply be produced, 00:07:56.140 --> 00:07:58.580 you would produce 32 and 76. 00:07:58.580 --> 00:08:03.140 OK, and in this case, the merger produced 12 and 48. 00:08:03.140 --> 00:08:08.600 So at this level, you're going to return 32 and 76 back. 00:08:08.600 --> 00:08:10.970 You're going to return 32 and 76 back. 00:08:10.970 --> 00:08:14.060 And here you're going to return 12 and 48 back. 00:08:14.060 --> 00:08:16.700 Now it gets a little more interesting. 00:08:16.700 --> 00:08:18.980 Now it gets a little more interesting because now you 00:08:18.980 --> 00:08:23.780 have 32 and 76 and 12 and 48, and those are both 00:08:23.780 --> 00:08:25.580 sorted in ascending order, and you 00:08:25.580 --> 00:08:27.770 need to merge them together. 00:08:27.770 --> 00:08:31.460 And so if you had a 128 element list, 00:08:31.460 --> 00:08:33.392 you would go, go, go all the way down, 00:08:33.392 --> 00:08:34.850 and then you would do some merging. 00:08:34.850 --> 00:08:39.950 Come up all the way and then the most time-consuming step really 00:08:39.950 --> 00:08:43.940 would be taking two 64 element lists 00:08:43.940 --> 00:08:46.520 that the left one is in ascending order, 00:08:46.520 --> 00:08:48.380 and the right one is in descending order. 00:08:48.380 --> 00:08:50.300 And then you need to merge them together 00:08:50.300 --> 00:08:53.180 so the result is in ascending order. 00:08:53.180 --> 00:08:55.190 That makes sense? 00:08:55.190 --> 00:08:57.726 And so in this case, the way we're 00:08:57.726 --> 00:08:59.780 going to do this is if you have-- 00:08:59.780 --> 00:09:06.380 and I can do this for this example or a different example. 00:09:06.380 --> 00:09:13.250 But I'll just stick with 32, 76, and this could be larger. 00:09:13.250 --> 00:09:15.210 This is 12 and 48. 00:09:15.210 --> 00:09:18.530 So I need to merge these two to produce in ascending order. 00:09:18.530 --> 00:09:21.440 And we use a straightforward algorithm 00:09:21.440 --> 00:09:24.140 that's called the two-finger algorithm, which 00:09:24.140 --> 00:09:26.840 says I'm going to start with my fingers pointing 00:09:26.840 --> 00:09:31.160 to the beginning of the two arrays, 00:09:31.160 --> 00:09:35.270 and I'm going to choose the element that's smaller. 00:09:35.270 --> 00:09:39.260 And I'm going to write it into my output array. 00:09:39.260 --> 00:09:41.300 So I'm going to compare 32 and 12. 00:09:41.300 --> 00:09:43.690 12 is smaller, so I'm going to write 12 down. 00:09:43.690 --> 00:09:45.980 And because I picked 12 from this array, 00:09:45.980 --> 00:09:48.710 and I didn't pick 32, I'm going to leave my top finger where 00:09:48.710 --> 00:09:51.380 it is, and I'm going to move my bottom finger-- 00:09:51.380 --> 00:09:54.860 my right-hand finger-- over to the next element. 00:09:54.860 --> 00:09:57.770 And then I'm going to compare 32 with 48 because this finger 00:09:57.770 --> 00:09:58.579 didn't move. 00:09:58.579 --> 00:10:00.620 And I'm going to figure out which one is smaller, 00:10:00.620 --> 00:10:01.410 and it's 32. 00:10:01.410 --> 00:10:03.180 And I'm going to put that over here. 00:10:03.180 --> 00:10:05.930 And then I'm going to move this finger because I took away 32. 00:10:05.930 --> 00:10:11.941 And then at this 48 is smaller, and then I'm off this way, 00:10:11.941 --> 00:10:13.190 assuming the array ended here. 00:10:13.190 --> 00:10:14.720 But if it didn't, I keep going. 00:10:14.720 --> 00:10:18.760 And then the last thing I do is put 76 down. 00:10:18.760 --> 00:10:22.120 So that is a recursive divide and conquer algorithm. 00:10:22.120 --> 00:10:25.140 It's the simplest algorithm I can think of, 00:10:25.140 --> 00:10:28.780 and that is still interesting. 00:10:28.780 --> 00:10:30.920 I'll show you code for it. 00:10:30.920 --> 00:10:35.470 I'm not going to explain it. 00:10:35.470 --> 00:10:37.570 It is worth looking at the overall code. 00:10:37.570 --> 00:10:40.240 And then the merge, which is this two-finger algorithm, 00:10:40.240 --> 00:10:42.220 is actually the most amount of code. 00:10:42.220 --> 00:10:45.700 But the split itself, as you can see, is you 00:10:45.700 --> 00:10:48.640 see as with our recursion, we say 00:10:48.640 --> 00:10:50.380 we want to have a base case. 00:10:50.380 --> 00:10:53.117 And so when the length of the list becomes 2, 00:10:53.117 --> 00:10:54.700 I'm just going to do that little flip. 00:10:54.700 --> 00:10:56.440 This is a sorting algorithm that only 00:10:56.440 --> 00:10:58.740 works for a list of length 2. 00:10:58.740 --> 00:11:01.300 But the beauty of this is that I can keep going down 00:11:01.300 --> 00:11:03.300 to get smaller and smaller lists, 00:11:03.300 --> 00:11:05.440 such that the sorting algorithm actually works. 00:11:05.440 --> 00:11:06.310 And this is it. 00:11:06.310 --> 00:11:09.790 This is a fine sorting algorithm for a base case. 00:11:09.790 --> 00:11:11.800 And we do have a base case here. 00:11:11.800 --> 00:11:16.250 We also have the other property that I'm actually calling. 00:11:16.250 --> 00:11:19.630 I'm making two recursive calls with significantly smaller 00:11:19.630 --> 00:11:20.470 problems. 00:11:20.470 --> 00:11:22.390 In this case, I'm using list splicing 00:11:22.390 --> 00:11:23.932 to get the first part of the list 00:11:23.932 --> 00:11:26.140 and the second part of the list, and I'm just calling 00:11:26.140 --> 00:11:29.380 merge sort on those two lists. 00:11:29.380 --> 00:11:33.340 And then finally, the workhorse in merge sort 00:11:33.340 --> 00:11:35.212 is the merge step-- this two finger thing 00:11:35.212 --> 00:11:36.420 that I just described to you. 00:11:36.420 --> 00:11:38.410 That is this merge thing. 00:11:38.410 --> 00:11:40.010 And this merge thing, as you can see-- 00:11:40.010 --> 00:11:41.830 there's not a lot of code. 00:11:41.830 --> 00:11:46.930 But it's doing this movement and the comparisons. 00:11:46.930 --> 00:11:50.670 And that's essentially what you have there. 00:11:50.670 --> 00:11:52.870 So given that context-- 00:11:52.870 --> 00:11:59.260 right given that context, let's go back to our tiling puzzle. 00:11:59.260 --> 00:12:03.070 And let's talk about this tiling puzzle and how we could 00:12:03.070 --> 00:12:05.920 we could solve this tiling puzzle. 00:12:05.920 --> 00:12:10.060 So you're going to break this courtyard up clearly 00:12:10.060 --> 00:12:11.650 because that's really what I've been 00:12:11.650 --> 00:12:14.470 trying to tell you in terms of recursive divide and conquer. 00:12:14.470 --> 00:12:19.660 First question-- look at this 2 raised to n, 00:12:19.660 --> 00:12:23.140 and 2 raised to n, and look at this three tile. 00:12:23.140 --> 00:12:26.290 I am going to ask you the question. 00:12:26.290 --> 00:12:30.340 Is there a solution that is a perfect tiling 00:12:30.340 --> 00:12:37.150 with no overlapping tiles and no squares with holes in them? 00:12:37.150 --> 00:12:38.110 Is someone? 00:12:41.860 --> 00:12:43.970 Fadi had his hand up first. 00:12:43.970 --> 00:12:45.029 Go for it. 00:12:45.029 --> 00:12:47.445 AUDIENCE: Well, if we look at the case of n is equal to 1, 00:12:47.445 --> 00:12:50.550 you have a two by two courtyard, then it's impossible, 00:12:50.550 --> 00:12:52.422 you can see that trivially. 00:12:52.422 --> 00:12:58.865 But in a general case, then we will have 2 to the 2n squares. 00:12:58.865 --> 00:13:01.568 And every time we're adding 3, 3, 3, the number of tiles 00:13:01.568 --> 00:13:04.452 is a multiple of 3. 00:13:04.452 --> 00:13:05.410 SRINI DEVADAS: Correct. 00:13:05.410 --> 00:13:09.290 So 2 to the 2n-- and Josh probably had the same answer-- 00:13:09.290 --> 00:13:12.020 2 to the 2n is not a multiple of 3. 00:13:12.020 --> 00:13:14.080 The only thing that divides it is 2. 00:13:20.940 --> 00:13:23.420 So OK so we're going to change the problem. 00:13:23.420 --> 00:13:25.350 We're going to say that there's a statue 00:13:25.350 --> 00:13:31.570 of your favorite person that is going to be out here. 00:13:31.570 --> 00:13:35.900 And so you don't need to tile that. 00:13:35.900 --> 00:13:44.790 So now you have 2 raised to 2n minus 1 00:13:44.790 --> 00:13:47.850 let me write that out properly. 00:13:47.850 --> 00:13:50.760 2 raised to 2n minus 1 squares. 00:13:56.560 --> 00:14:01.900 Now can you do this for all n? 00:14:01.900 --> 00:14:05.350 Clearly, it works for n equals 1. 00:14:05.350 --> 00:14:07.480 And then for n equals 2, you have what? 00:14:07.480 --> 00:14:13.300 2 raised to 4, which is 16 minus 115. 00:14:13.300 --> 00:14:15.050 That seems to work. 00:14:15.050 --> 00:14:18.250 And then you get up to, I think 80-- 00:14:18.250 --> 00:14:20.280 what is it? 00:14:20.280 --> 00:14:22.870 2 raised to 6 would be 64. 00:14:22.870 --> 00:14:24.550 And so that will be 63. 00:14:24.550 --> 00:14:25.469 That works too. 00:14:25.469 --> 00:14:26.510 In general, does it work? 00:14:30.150 --> 00:14:31.200 It does. 00:14:31.200 --> 00:14:34.260 So it turns out if you think about it, 00:14:34.260 --> 00:14:39.780 you can write this out as 2 raised to n minus 1 00:14:39.780 --> 00:14:42.840 times 2 raised to n plus 1. 00:14:42.840 --> 00:14:48.240 And in the middle there, pretend that there's 2 raised to n. 00:14:48.240 --> 00:14:50.050 So you have three consecutive numbers. 00:14:50.050 --> 00:14:53.460 And when you multiply three consecutive numbers, 00:14:53.460 --> 00:14:54.510 they're consecutive. 00:14:54.510 --> 00:14:56.250 Doesn't matter what the numbers are. 00:14:56.250 --> 00:14:59.760 The product is always going to be a multiple of 3 00:14:59.760 --> 00:15:02.670 because one of those things is going to be a multiple of 3. 00:15:02.670 --> 00:15:05.430 And you know that 2 raised to n is not a multiple of 3. 00:15:05.430 --> 00:15:07.680 This is not a product of three consecutive numbers. 00:15:07.680 --> 00:15:11.910 But if it were, and you pretended that 2 raised to n 00:15:11.910 --> 00:15:14.970 was in here, either this one is a multiple of 3, 00:15:14.970 --> 00:15:17.070 or that one is a multiple of 3. 00:15:17.070 --> 00:15:18.320 So this is good. 00:15:18.320 --> 00:15:24.090 So we know that at least from a standpoint of volume-- 00:15:24.090 --> 00:15:26.140 our surface area, if you will-- 00:15:26.140 --> 00:15:27.450 that this is going to work. 00:15:27.450 --> 00:15:30.210 Whereas the original problem didn't work from a surface area 00:15:30.210 --> 00:15:31.800 standpoint. 00:15:31.800 --> 00:15:34.920 So it comes down to how am I going 00:15:34.920 --> 00:15:38.910 to solve this problem using recursive divide and conquer? 00:15:38.910 --> 00:15:40.470 And so I can break this up. 00:15:40.470 --> 00:15:44.850 And you want to have a similar feel to the merge sort, 00:15:44.850 --> 00:15:48.670 where when you go down to the base case of merge sort, 00:15:48.670 --> 00:15:50.670 you had this trivial problem to solve. 00:15:50.670 --> 00:15:52.560 That's the beauty of divide and conquer. 00:15:52.560 --> 00:15:54.810 It becomes obvious when you have such a small problem 00:15:54.810 --> 00:15:56.820 like one queen. 00:15:56.820 --> 00:15:59.190 That's not technically not divide and conquer, 00:15:59.190 --> 00:16:01.020 but you do get to the smaller part. 00:16:01.020 --> 00:16:04.110 But clearly, in this case, this is a classic divide 00:16:04.110 --> 00:16:05.635 and conquer algorithm merge sort. 00:16:05.635 --> 00:16:08.010 And you can see that the first two or three lines of code 00:16:08.010 --> 00:16:12.150 are the trivial case of two elements that 00:16:12.150 --> 00:16:13.750 needed to be sorted. 00:16:13.750 --> 00:16:16.050 So let's go back and forget that. 00:16:16.050 --> 00:16:19.050 Let's look at this picture and think 00:16:19.050 --> 00:16:21.027 about how you'd break this up. 00:16:21.027 --> 00:16:22.110 How would I bring this up? 00:16:30.600 --> 00:16:33.096 So think two-dimensionally. 00:16:33.096 --> 00:16:35.950 And merge sort was a single dimension. 00:16:35.950 --> 00:16:38.361 OK, think two dimensional. 00:16:38.361 --> 00:16:39.360 This is two dimensional. 00:16:39.360 --> 00:16:40.420 This is a two-dimensional problem. 00:16:40.420 --> 00:16:42.380 It's in that sense, closer to N-queens. 00:16:42.380 --> 00:16:47.304 But merge sort was simply a single dimension. 00:16:47.304 --> 00:16:48.470 All right, I see Kevin had-- 00:16:48.470 --> 00:16:49.290 Fadi had his hand up. 00:16:49.290 --> 00:16:50.130 Kevin has a hand up. 00:16:50.130 --> 00:16:50.690 Someone else? 00:16:55.060 --> 00:16:56.226 All right, go ahead, Kevin. 00:16:56.226 --> 00:16:58.939 AUDIENCE: Do you keep cutting the square into quarters? 00:16:58.939 --> 00:17:00.230 SRINI DEVADAS: Quarters-- good. 00:17:00.230 --> 00:17:01.520 OK, yeah. 00:17:01.520 --> 00:17:04.660 So all right, so let's do that. 00:17:04.660 --> 00:17:05.319 Let's do that. 00:17:05.319 --> 00:17:06.402 You're on the right track. 00:17:10.380 --> 00:17:16.770 And by the way, this particular thing could be over there. 00:17:16.770 --> 00:17:20.329 And I'll just do this just to make sure. 00:17:20.329 --> 00:17:22.619 Your algorithm needs to work, regardless 00:17:22.619 --> 00:17:23.730 of where that statue is. 00:17:23.730 --> 00:17:25.980 So maybe the statue is over here. 00:17:25.980 --> 00:17:27.390 Maybe the statue is over there. 00:17:27.390 --> 00:17:29.907 There's only one statue that is covering one square, 00:17:29.907 --> 00:17:31.740 but it's not necessarily in near the center. 00:17:31.740 --> 00:17:33.330 It could even be anywhere. 00:17:33.330 --> 00:17:35.610 You need a general-purpose scheme 00:17:35.610 --> 00:17:39.210 that would work, regardless of where the statue is as long 00:17:39.210 --> 00:17:41.640 as it only occupies one square. 00:17:41.640 --> 00:17:43.200 That's the only constraint we have. 00:17:43.200 --> 00:17:46.380 So you're on the right track, Kevin. 00:17:46.380 --> 00:17:48.900 Someone else or, Fadi, you want to add to that? 00:17:51.894 --> 00:17:55.387 AUDIENCE: So in something that's similar to the merge sort 00:17:55.387 --> 00:17:58.443 so you divide it into quarters and then you 00:17:58.443 --> 00:18:00.568 reach the base case and then we tile that obviously 00:18:00.568 --> 00:18:01.567 with an L shaped tile. 00:18:01.567 --> 00:18:03.462 SRINI DEVADAS: Right, that's exactly right. 00:18:03.462 --> 00:18:03.962 So yeah. 00:18:03.962 --> 00:18:06.270 So that there's one thing that's missing in what 00:18:06.270 --> 00:18:09.120 both you and Kevin have put up. 00:18:09.120 --> 00:18:12.510 The thing with merge sort was I had exactly the same problem 00:18:12.510 --> 00:18:13.710 to solve. 00:18:13.710 --> 00:18:15.420 It was exactly the same problem. 00:18:15.420 --> 00:18:16.590 It was just smaller. 00:18:16.590 --> 00:18:19.830 I needed to sort the smaller array in ascending order. 00:18:19.830 --> 00:18:22.800 And I had two of them, and it was exactly the same problem. 00:18:22.800 --> 00:18:25.260 But when I split this up into four quarters like you 00:18:25.260 --> 00:18:28.200 guys are suggesting, do I have exactly the same four 00:18:28.200 --> 00:18:30.210 problems to solve? 00:18:30.210 --> 00:18:35.690 In fact, can I even solve this the 16 problem 00:18:35.690 --> 00:18:36.910 with L-shaped tiles? 00:18:36.910 --> 00:18:37.660 I can't. 00:18:37.660 --> 00:18:40.225 So at some level, I have three problems that I can't solve. 00:18:40.225 --> 00:18:42.100 And then a smaller problem that I don't quite 00:18:42.100 --> 00:18:44.266 know how to solve-- other than maybe splitting it up 00:18:44.266 --> 00:18:46.630 even more, which is kind of interesting, which is good. 00:18:46.630 --> 00:18:48.220 But then the next time, if I go ahead 00:18:48.220 --> 00:18:51.100 and split that, then wonderful. 00:18:51.100 --> 00:18:55.990 I will find one problem that's two by two that I can solve, 00:18:55.990 --> 00:18:58.550 but that doesn't solve my puzzle. 00:18:58.550 --> 00:19:02.350 So I have to do one more thing in order 00:19:02.350 --> 00:19:06.550 to make this look like merge sort or any recursive divide 00:19:06.550 --> 00:19:07.210 and conquer. 00:19:07.210 --> 00:19:09.700 One of the paradigms of recursive divide and conquer 00:19:09.700 --> 00:19:12.580 is you're not creating new types of problems. 00:19:12.580 --> 00:19:15.790 You're creating the same type of problem, except smaller. 00:19:15.790 --> 00:19:18.520 And therefore, eventually easier. 00:19:18.520 --> 00:19:20.590 So I have to do one more thing. 00:19:20.590 --> 00:19:24.700 And this is an aha moment here that I'm hoping-- 00:19:24.700 --> 00:19:26.950 I don't want to take an answer yet 00:19:26.950 --> 00:19:28.930 because I want you to think about it 00:19:28.930 --> 00:19:31.930 and perhaps arrive at it. 00:19:31.930 --> 00:19:36.280 But it's one of those things where you're going to go "oh!" 00:19:36.280 --> 00:19:38.380 if you don't know the answer. 00:19:38.380 --> 00:19:41.530 But it is worth spending another minute-- 00:19:41.530 --> 00:19:44.680 another minute for each of you thinking about it. 00:19:44.680 --> 00:19:47.680 How am I going to set it up so these quarters-- so all of this 00:19:47.680 --> 00:19:48.400 is good? 00:19:48.400 --> 00:19:50.200 I'm going to split it up into quarters, 00:19:50.200 --> 00:19:57.910 but I want all four problems to look exactly the same 00:19:57.910 --> 00:19:59.350 in the sense that they look-- 00:19:59.350 --> 00:20:01.210 this obviously looks different from these other ones. 00:20:01.210 --> 00:20:02.626 Well, there's a tile missing here, 00:20:02.626 --> 00:20:05.680 but there is no tile missing over in the other three. 00:20:05.680 --> 00:20:10.340 So how do I make all problems look the same? 00:20:10.340 --> 00:20:12.760 I know you know, but we need more. 00:20:17.610 --> 00:20:19.382 I'll give you 30 more seconds, and I'll 00:20:19.382 --> 00:20:20.340 give you one more hint. 00:20:27.150 --> 00:20:30.390 All! right, the hint is place one tile-- 00:20:30.390 --> 00:20:32.381 place one L-shaped tile. 00:20:32.381 --> 00:20:32.880 All right. 00:20:32.880 --> 00:20:34.214 Yeah, go for it. 00:20:34.214 --> 00:20:37.710 AUDIENCE: Yeah, you know that with those last four quarters-- 00:20:37.710 --> 00:20:39.948 you know that that L-shape has to go around? 00:20:39.948 --> 00:20:40.776 SRINI DEVADAS: Yes. 00:20:40.776 --> 00:20:41.567 Yeah, that's right. 00:20:41.567 --> 00:20:43.475 So you set it up so you tile. 00:20:46.680 --> 00:20:49.860 The reason I picked this L-shape this way 00:20:49.860 --> 00:20:54.050 is because this was the statue. 00:20:54.050 --> 00:21:01.170 So I wanted to remove a tile from each of the other 3/4. 00:21:01.170 --> 00:21:03.540 I'm sorry, remove a square. 00:21:03.540 --> 00:21:05.160 A tile is L-shaped. 00:21:05.160 --> 00:21:07.380 And so this was a missing square, 00:21:07.380 --> 00:21:10.232 which was part of the specification of the problem. 00:21:10.232 --> 00:21:11.690 It has nothing to do with the tile. 00:21:11.690 --> 00:21:14.370 It's just a missing square because there's a statue on it. 00:21:14.370 --> 00:21:18.390 But what I could do is if I oriented the tile in this way, 00:21:18.390 --> 00:21:25.140 now I have this being a smaller problem, which 00:21:25.140 --> 00:21:28.290 is one-quarter of the size of the original problem with one 00:21:28.290 --> 00:21:29.670 square missing. 00:21:29.670 --> 00:21:31.020 This one is one square missing. 00:21:31.020 --> 00:21:32.311 This one is one square missing. 00:21:32.311 --> 00:21:33.750 This one is one square missing. 00:21:33.750 --> 00:21:36.930 Now that makes all of the problems look the same. 00:21:36.930 --> 00:21:40.500 Now, at first, they're not exactly the same in the sense 00:21:40.500 --> 00:21:44.490 that the missing square is in a different position. 00:21:44.490 --> 00:21:47.940 But there's some generality that we need in our algorithm that 00:21:47.940 --> 00:21:50.970 says that eventually, if I keep breaking this up 00:21:50.970 --> 00:21:54.750 in this fashion, I just need to orient my L-shaped tile 00:21:54.750 --> 00:21:56.760 in the appropriate way. 00:21:56.760 --> 00:21:59.450 So there's only four different ways 00:21:59.450 --> 00:22:05.420 that an L-shaped tile is going to be oriented. 00:22:05.420 --> 00:22:08.420 It could be oriented like this. 00:22:08.420 --> 00:22:09.700 And I won't draw all of them. 00:22:09.700 --> 00:22:12.160 It could be oriented like that. 00:22:12.160 --> 00:22:17.910 And it could also be that's the same. 00:22:17.910 --> 00:22:18.870 So what else is there? 00:22:18.870 --> 00:22:22.030 Oh, it could be flipped this way. 00:22:22.030 --> 00:22:26.560 Yeah, like that and the fourth one. 00:22:26.560 --> 00:22:27.830 So that's what I did here. 00:22:27.830 --> 00:22:29.350 This one is what I put up there. 00:22:29.350 --> 00:22:33.760 And then as I break this up, let's 00:22:33.760 --> 00:22:36.370 just say that now I'm going to make four recursive calls. 00:22:36.370 --> 00:22:38.200 I'm going to place one tile-- 00:22:38.200 --> 00:22:40.360 one L-shaped tile-- and I'm going 00:22:40.360 --> 00:22:41.650 to mark off of these things. 00:22:41.650 --> 00:22:43.750 And I'm going to make four recursive calls 00:22:43.750 --> 00:22:47.350 to basically myself, except that the arguments are different. 00:22:47.350 --> 00:22:49.000 It's the same code, obviously. 00:22:49.000 --> 00:22:51.486 So when I go up here, I go like that, 00:22:51.486 --> 00:22:53.110 and hopefully, this will become clearer 00:22:53.110 --> 00:22:54.610 if you're going to do a tiling here. 00:22:54.610 --> 00:22:58.054 I'm going to go focus on the top left, and I'm going to do this. 00:22:58.054 --> 00:22:59.470 What am I going to do with respect 00:22:59.470 --> 00:23:05.110 to this at this particular quarter of the courtyard? 00:23:05.110 --> 00:23:09.600 How am I going to orient the L-shaped tile for this one? 00:23:09.600 --> 00:23:12.060 Point with your fingers. 00:23:12.060 --> 00:23:12.690 Exactly right. 00:23:12.690 --> 00:23:14.130 So this is the same as this. 00:23:14.130 --> 00:23:15.546 I'm going to go ahead and do that. 00:23:18.280 --> 00:23:21.480 So now I have this problem. 00:23:21.480 --> 00:23:26.590 So I'm going to mark that with a different color. 00:23:26.590 --> 00:23:29.740 So I'm focused on this problem, and I've done that break. 00:23:29.740 --> 00:23:34.840 And now I'm good because I have four two by two courtyards, 00:23:34.840 --> 00:23:39.670 and each of these four has one square 00:23:39.670 --> 00:23:43.420 missing in each of these four little two by two courtyards. 00:23:43.420 --> 00:23:45.190 So clearly, I can solve that. 00:23:45.190 --> 00:23:47.680 And then once I solve that, I move on to this one, 00:23:47.680 --> 00:23:50.570 and I move onto this one-- whatever order you want. 00:23:50.570 --> 00:23:51.340 That's it. 00:23:51.340 --> 00:23:54.940 And so the code for this is-- 00:23:54.940 --> 00:23:58.300 from a control flow standpoint, from a standpoint 00:23:58.300 --> 00:24:02.860 of the overall algorithm, it's extremely straightforward. 00:24:02.860 --> 00:24:04.720 It's almost exactly like merge sort, 00:24:04.720 --> 00:24:08.150 except that you have four-way recursion 00:24:08.150 --> 00:24:10.660 because it's two dimensional, and you end up 00:24:10.660 --> 00:24:11.620 getting quarters. 00:24:11.620 --> 00:24:13.740 And the only thing that's annoying about this code 00:24:13.740 --> 00:24:17.590 to write and also to understand, which you guys will-- 00:24:17.590 --> 00:24:23.630 I'll put this up, and I'll show you what to look for-- 00:24:23.630 --> 00:24:27.970 is how you represent this as such 00:24:27.970 --> 00:24:32.140 that you get all the values right. 00:24:32.140 --> 00:24:37.540 So you get an arbitrary square, to begin with, that is missing. 00:24:37.540 --> 00:24:41.320 And then you have to go to the middle. 00:24:41.320 --> 00:24:42.136 That's clear. 00:24:42.136 --> 00:24:44.260 You're going to go to the middle of this courtyard, 00:24:44.260 --> 00:24:45.500 and you're going to split it. 00:24:45.500 --> 00:24:47.920 And you're going to decide which way this L-shaped tile is 00:24:47.920 --> 00:24:51.790 oriented, and then you got to do the splitting again 00:24:51.790 --> 00:24:53.320 and so it's just bookkeeping. 00:24:53.320 --> 00:24:55.120 That's really what this is all about. 00:24:55.120 --> 00:24:57.220 It's not algorithmically interesting. 00:24:57.220 --> 00:24:59.800 It's just bookkeeping in terms of the orientation 00:24:59.800 --> 00:25:01.757 of these L-shaped tiles. 00:25:01.757 --> 00:25:03.340 So if you look at the code, and if you 00:25:03.340 --> 00:25:11.140 ignore a bunch of these indices of these different variables-- 00:25:11.140 --> 00:25:13.100 let me show you. 00:25:13.100 --> 00:25:18.120 Well, first, I'll show you what happens when you run the code. 00:25:18.120 --> 00:25:21.530 I'm running it for two different kinds of courtyards. 00:25:21.530 --> 00:25:25.780 And I've used A through, I guess, something like S, 00:25:25.780 --> 00:25:27.430 in this case, to signify a tile. 00:25:27.430 --> 00:25:28.810 So you see AAA here. 00:25:28.810 --> 00:25:31.120 AAA is an L-shaped tile. 00:25:31.120 --> 00:25:32.590 BBB is an L-shaped tile. 00:25:32.590 --> 00:25:34.840 FFF is an L-shaped tile, et cetera. 00:25:34.840 --> 00:25:38.020 The original specification of the problem-- 00:25:38.020 --> 00:25:40.120 this is just simply an eight by eight problem-- 00:25:40.120 --> 00:25:41.560 had this tile that you see here. 00:25:41.560 --> 00:25:45.130 That's a blank that I've covered up that was missing. 00:25:45.130 --> 00:25:47.170 I'm sorry, this square missing. 00:25:47.170 --> 00:25:52.750 And then I broke this up into two or four 4 by 4's. 00:25:52.750 --> 00:25:55.480 And then I went all the way up and then the first tile 00:25:55.480 --> 00:26:01.270 that I placed, I pretended to place this U tile over here. 00:26:01.270 --> 00:26:04.270 I could've placed it and then called it the A tile. 00:26:04.270 --> 00:26:09.430 But this is essentially what you think of the one in the center 00:26:09.430 --> 00:26:11.300 exactly the way this is oriented. 00:26:11.300 --> 00:26:12.646 That's your U tile over here. 00:26:12.646 --> 00:26:14.020 And then you go all the way down. 00:26:14.020 --> 00:26:17.440 And explicitly, it's just the A tile that displays first, 00:26:17.440 --> 00:26:19.880 and then the B tile, and so on and so forth. 00:26:19.880 --> 00:26:22.600 And if you take a look at the code, what's 00:26:22.600 --> 00:26:26.210 the best place to start? 00:26:26.210 --> 00:26:28.430 So this is the base case-- 00:26:28.430 --> 00:26:30.970 same thing as we've talked about before. 00:26:30.970 --> 00:26:32.830 You always need to have a base case. 00:26:32.830 --> 00:26:34.660 The base cases is a two by two. 00:26:34.660 --> 00:26:38.800 And this code places one L-shaped tile on a two 00:26:38.800 --> 00:26:41.170 by two yard. 00:26:41.170 --> 00:26:43.660 This is going to make four recursive calls, as you 00:26:43.660 --> 00:26:47.650 can see from range four, and there's 00:26:47.650 --> 00:26:50.710 two different kinds of recursive calls in the sense 00:26:50.710 --> 00:26:53.770 that they're both calling recursive tile, 00:26:53.770 --> 00:26:55.600 but the arguments of the recursive style 00:26:55.600 --> 00:26:59.890 are different for this recursive call 00:26:59.890 --> 00:27:03.790 because it already had the missing square in it. 00:27:03.790 --> 00:27:05.800 And these three recursive calls that are 00:27:05.800 --> 00:27:10.220 going to have the missing square in one of their corners. 00:27:10.220 --> 00:27:14.160 So that's why you see the if and the else here. 00:27:14.160 --> 00:27:17.830 And if the original for the recursive 00:27:17.830 --> 00:27:20.230 call that already had the missing square 00:27:20.230 --> 00:27:23.920 because it had the statue on it, that's the if statement. 00:27:23.920 --> 00:27:28.720 And the other ones are the ones that are in the corners. 00:27:28.720 --> 00:27:31.420 So the first things it would be a corner for this one. 00:27:31.420 --> 00:27:33.170 This is the corner of that, one and that's 00:27:33.170 --> 00:27:35.480 the corner of that one. 00:27:35.480 --> 00:27:37.570 So that's pretty much all I had. 00:27:37.570 --> 00:27:41.440 This code-- you need to gaze at it for a few minutes 00:27:41.440 --> 00:27:44.230 to figure out that I got all the arguments correct. 00:27:44.230 --> 00:27:46.240 But from a standpoint of, as I said, 00:27:46.240 --> 00:27:47.950 the control flow and the algorithm, 00:27:47.950 --> 00:27:51.310 it is hopefully completely clear to you at this point. 00:27:51.310 --> 00:27:52.920 If not, ask me questions. 00:27:52.920 --> 00:27:54.160 Come to office hours. 00:27:54.160 --> 00:27:56.250 Have a good long weekend.