WEBVTT 00:00:08.516 --> 00:00:09.890 SRINI DEVADAS: Let's get started. 00:00:09.890 --> 00:00:13.240 So today, we're going to look at one of my favorite puzzles. 00:00:13.240 --> 00:00:17.260 I'll say right at the beginning, that the coding associated 00:00:17.260 --> 00:00:20.440 with the puzzle is fairly straightforward. 00:00:20.440 --> 00:00:24.940 But the analysis associated with this puzzle is quite intricate. 00:00:24.940 --> 00:00:27.100 And in particular, we're going to have 00:00:27.100 --> 00:00:30.970 to do worst-case analysis, which is really a good thing for you 00:00:30.970 --> 00:00:36.160 to learn about, because asymptotic analysis 00:00:36.160 --> 00:00:39.140 in algorithms is all about worst case. 00:00:39.140 --> 00:00:39.640 All right. 00:00:39.640 --> 00:00:40.990 So keep that in mind. 00:00:40.990 --> 00:00:45.130 When you try to come up with the answers to my questions, 00:00:45.130 --> 00:00:47.440 always keep in mind that I'm going 00:00:47.440 --> 00:00:52.120 to be asking for what happens in the worst case. 00:00:52.120 --> 00:00:55.070 In the context of our puzzle-- 00:00:55.070 --> 00:00:58.000 which is called the crystal ball puzzle, 00:00:58.000 --> 00:01:00.640 and it's fairly popular in literature, 00:01:00.640 --> 00:01:03.550 though I've generalized it a little bit-- 00:01:03.550 --> 00:01:06.550 you have a situation where you're 00:01:06.550 --> 00:01:10.690 supposed to figure out the hardness coefficient 00:01:10.690 --> 00:01:12.920 of a crystal ball. 00:01:12.920 --> 00:01:14.860 And in the generalized version, you 00:01:14.860 --> 00:01:17.770 have a set of identical crystal balls. 00:01:17.770 --> 00:01:21.380 So how do you find this hardness coefficient? 00:01:21.380 --> 00:01:28.410 Well, you go up to the Shanghai Tower that has 128 floors-- 00:01:28.410 --> 00:01:37.419 1 through 128-- and you drop this ball from say, floor 1. 00:01:37.419 --> 00:01:39.210 So just you're standing here and let's say, 00:01:39.210 --> 00:01:40.600 you just drop it from there. 00:01:40.600 --> 00:01:43.750 Or you go up to floor 2, and you drop the ball. 00:01:43.750 --> 00:01:47.560 And the hardness coefficient is defined 00:01:47.560 --> 00:01:53.080 as the highest floor from which you can drop the ball 00:01:53.080 --> 00:01:56.200 and the ball does not break. 00:01:56.200 --> 00:02:04.240 So what this means is if at floor f, 00:02:04.240 --> 00:02:15.490 the ball doesn't break, and it breaks at f plus 1, 00:02:15.490 --> 00:02:19.480 then the hardness coefficient is f. 00:02:19.480 --> 00:02:24.190 And you can assume a monotonicity property 00:02:24.190 --> 00:02:30.130 that says that if it did not break at floor f, then 00:02:30.130 --> 00:02:32.200 it won't break-- 00:02:32.200 --> 00:02:39.250 this implies that it won't break at any floor less than 00:02:39.250 --> 00:02:40.780 or equal to f. 00:02:40.780 --> 00:02:42.430 Which makes sense, right? 00:02:42.430 --> 00:02:44.660 And otherwise, it would be kind of complicated. 00:02:44.660 --> 00:02:45.680 It would be random. 00:02:45.680 --> 00:02:49.060 And we won't be able to do any interesting analysis. 00:02:49.060 --> 00:02:53.770 So the easy version of the puzzle, 00:02:53.770 --> 00:02:56.470 of course, which is not particularly interesting, 00:02:56.470 --> 00:03:02.090 is if you needed to get this exactly-- 00:03:02.090 --> 00:03:05.270 so your job depends on getting that number exactly right-- 00:03:05.270 --> 00:03:06.400 so there's no shortcuts. 00:03:06.400 --> 00:03:07.524 You've got to get it right. 00:03:07.524 --> 00:03:09.850 You can't be off by one either way. 00:03:09.850 --> 00:03:15.670 And you have only one ball. 00:03:15.670 --> 00:03:17.270 So you only have one ball. 00:03:17.270 --> 00:03:19.319 So what can you do if you only have one ball 00:03:19.319 --> 00:03:20.860 and you have to get it exactly right? 00:03:20.860 --> 00:03:22.969 AUDIENCE: Just try every floor. 00:03:22.969 --> 00:03:24.260 SRINI DEVADAS: Try every floor. 00:03:24.260 --> 00:03:24.760 Right. 00:03:24.760 --> 00:03:29.510 And but more important, try it from lower floors 00:03:29.510 --> 00:03:30.920 to higher floors, right? 00:03:30.920 --> 00:03:34.670 So you drop it at floor 1. 00:03:34.670 --> 00:03:42.900 So one ball, you drop from floor 1. 00:03:42.900 --> 00:03:46.460 And if it breaks, then your hardness coefficient 00:03:46.460 --> 00:03:50.760 is, you return 0. 00:03:50.760 --> 00:03:56.130 But then if you keep going and let's say, you drop off dot, 00:03:56.130 --> 00:03:57.420 dot, dot-- 00:03:57.420 --> 00:04:05.310 of from floor 65 and it breaks, and you've gone in order, 00:04:05.310 --> 00:04:09.630 then the hardness coefficient is 64. 00:04:09.630 --> 00:04:10.620 So that's easy enough. 00:04:10.620 --> 00:04:13.320 There's no optimization here. 00:04:13.320 --> 00:04:15.160 It's the only thing you can do. 00:04:15.160 --> 00:04:18.810 So that's why it's not particularly interesting. 00:04:18.810 --> 00:04:21.149 Now if you have identical crystal balls that 00:04:21.149 --> 00:04:23.940 are guaranteed, they were manufactured 00:04:23.940 --> 00:04:27.885 in exactly the same way, et cetera, et cetera, then 00:04:27.885 --> 00:04:29.760 obviously, things get a lot more interesting, 00:04:29.760 --> 00:04:32.460 because you can try to minimize something. 00:04:32.460 --> 00:04:34.230 And I'm going to write it over here, 00:04:34.230 --> 00:04:38.940 because I want you to keep this in mind throughout the lecture. 00:04:38.940 --> 00:04:48.870 Our goal here is to minimize the worst-case number 00:04:48.870 --> 00:04:52.980 of drops for our problem. 00:04:52.980 --> 00:04:58.440 And we want to solve the problem in the general case 00:04:58.440 --> 00:05:00.420 with d balls. 00:05:00.420 --> 00:05:02.190 And I should say, d identical balls. 00:05:05.650 --> 00:05:07.930 But more important, the reason I'm writing this up 00:05:07.930 --> 00:05:14.550 is, as I said before, we need to focus on the worst case. 00:05:14.550 --> 00:05:19.010 So it may be the case that for a given set of balls, 00:05:19.010 --> 00:05:23.520 your algorithm does very well, in the sense that the hardness 00:05:23.520 --> 00:05:25.820 coefficient happened to be 42. 00:05:25.820 --> 00:05:34.480 And you picked the floor as of 41 to begin with. 00:05:34.480 --> 00:05:36.727 And so you dropped from 41, and it doesn't break. 00:05:36.727 --> 00:05:38.560 And then you go to 42, and it doesn't break. 00:05:38.560 --> 00:05:41.170 And you go to 43, and it breaks. 00:05:41.170 --> 00:05:42.820 And you go, oh, I only needed two drops 00:05:42.820 --> 00:05:46.570 for this particular problem. 00:05:46.570 --> 00:05:50.920 But if the hardness coefficient happened to be 65 00:05:50.920 --> 00:05:53.410 and you started from floor 41, I mean 00:05:53.410 --> 00:05:56.500 you'd need 20-odd number of drops. 00:05:56.500 --> 00:05:59.590 So you have to be careful in that you 00:05:59.590 --> 00:06:02.200 have to analyze whatever strategy you come up 00:06:02.200 --> 00:06:07.030 with in the case where the hardness coefficient could 00:06:07.030 --> 00:06:08.590 be pretty much anything. 00:06:08.590 --> 00:06:12.640 I mean it could be 0 or it could be 128 in our Shanghai Tower 00:06:12.640 --> 00:06:14.560 example. 00:06:14.560 --> 00:06:17.770 So the first interesting question 00:06:17.770 --> 00:06:23.930 is, what happens when I have two identical balls? 00:06:23.930 --> 00:06:30.010 So, oh, before we do that, according 00:06:30.010 --> 00:06:37.330 to my metric here, what is the worst-case number of drops 00:06:37.330 --> 00:06:39.530 for one ball? 00:06:39.530 --> 00:06:42.180 What is the worst-case number of drops for one ball? 00:06:44.750 --> 00:06:46.690 Give me a number. 00:06:46.690 --> 00:06:48.762 Someone else. 00:06:48.762 --> 00:06:49.470 Yeah, back there. 00:06:49.470 --> 00:06:51.820 AUDIENCE: 128. 00:06:51.820 --> 00:06:54.670 SRINI DEVADAS: 128, exactly right. 00:06:54.670 --> 00:06:56.230 So it's 128, because I could have 00:06:56.230 --> 00:06:59.590 had a ball that doesn't break from the 128 floor. 00:06:59.590 --> 00:07:01.150 I don't know that to begin with. 00:07:01.150 --> 00:07:02.770 I have to start with floor 1. 00:07:02.770 --> 00:07:06.100 And I got to go 1, 2, 3, 4, et cetera, right? 00:07:06.100 --> 00:07:08.080 So the worst case for one ball is 128. 00:07:08.080 --> 00:07:10.870 And you can't do better than that. 00:07:10.870 --> 00:07:15.070 But of course, it gets more interesting when d is large. 00:07:15.070 --> 00:07:19.030 So our goal now is to discover an algorithm that 00:07:19.030 --> 00:07:22.640 is going to minimize the worst-case number of drops 00:07:22.640 --> 00:07:25.060 for d equals 2. 00:07:25.060 --> 00:07:28.210 That's our current goal, right? 00:07:28.210 --> 00:07:32.470 And then we'll look at more, even more general things. 00:07:32.470 --> 00:07:36.120 So how would you do things if you had d equals 2? 00:07:39.400 --> 00:07:43.510 Just, it doesn't matter if you don't get our algorithm right. 00:07:43.510 --> 00:07:44.330 We'll analyze it. 00:07:44.330 --> 00:07:45.560 You propose something. 00:07:45.560 --> 00:07:46.509 We'll analyze it. 00:07:46.509 --> 00:07:48.550 And then we'll decide if we can do better or not. 00:07:48.550 --> 00:07:48.700 Yeah. 00:07:48.700 --> 00:07:49.525 Go ahead. 00:07:49.525 --> 00:07:52.500 AUDIENCE: Drop the ball from the halfway point, say 64. 00:07:52.500 --> 00:07:54.855 If it breaks there, then you can use the bottom half 00:07:54.855 --> 00:07:55.730 of what we are using. 00:07:55.730 --> 00:07:56.640 SRINI DEVADAS: Use the bottom half. 00:07:56.640 --> 00:07:56.970 Excellent. 00:07:56.970 --> 00:07:57.678 What's your name? 00:07:57.678 --> 00:07:58.810 AUDIENCE: Dhaman. 00:07:58.810 --> 00:07:59.120 SRINI DEVADAS: Sorry. 00:07:59.120 --> 00:07:59.850 AUDIENCE: Dhaman. 00:07:59.850 --> 00:08:00.910 SRINI DEVADAS: Dhaman. 00:08:00.910 --> 00:08:06.570 So Dhaman says that we should do binary search, which 00:08:06.570 --> 00:08:11.890 is very intuitive and kind of makes sense 00:08:11.890 --> 00:08:15.040 in a lot of context, including this one. 00:08:15.040 --> 00:08:21.970 And so let's drop from 64. 00:08:21.970 --> 00:08:25.450 And the reason she picked that is because 128 divided by 2 00:08:25.450 --> 00:08:26.710 is 64. 00:08:26.710 --> 00:08:39.340 And if it breaks, then you need to look at the interval, 1, 63 00:08:39.340 --> 00:08:41.890 because if it-- 00:08:41.890 --> 00:08:42.439 I'm sorry. 00:08:42.439 --> 00:08:43.230 Yeah, that's right. 00:08:43.230 --> 00:08:46.690 If it breaks, you need to look at the interval 1, 63 00:08:46.690 --> 00:08:47.790 with a second ball. 00:08:47.790 --> 00:08:50.350 I mean that first ball is gone. 00:08:50.350 --> 00:08:53.510 It's shards of crystal or whatever you want to call it. 00:08:53.510 --> 00:08:55.120 So you just have the second ball. 00:08:55.120 --> 00:08:58.860 And so now at this point, you have no choice. 00:08:58.860 --> 00:09:00.940 You only have one ball left. 00:09:00.940 --> 00:09:02.860 As I mentioned, your job depends on getting 00:09:02.860 --> 00:09:05.350 that hardness coefficient right, so you can only 00:09:05.350 --> 00:09:09.910 use that first algorithm in terms of starting from 1 00:09:09.910 --> 00:09:11.770 and going all the way to 63. 00:09:11.770 --> 00:09:13.840 And so what is the worst-case number 00:09:13.840 --> 00:09:17.560 of drops given that Dhaman picked the midpoint 00:09:17.560 --> 00:09:19.210 for this algorithm? 00:09:19.210 --> 00:09:21.880 Someone else. 00:09:21.880 --> 00:09:22.380 Yeah. 00:09:22.380 --> 00:09:22.880 Go ahead. 00:09:23.380 --> 00:09:25.860 AUDIENCE: It's 63. 00:09:25.860 --> 00:09:29.190 SRINI DEVADAS: It's 63. 00:09:29.190 --> 00:09:31.770 No actually, you're off by 1. 00:09:31.770 --> 00:09:36.780 Because I did drop the ball here, so close. 00:09:36.780 --> 00:09:42.400 So the first ball was dropped once, one drop. 00:09:42.400 --> 00:09:48.090 And this one, if in fact, the hardness coefficient was 64, 00:09:48.090 --> 00:09:52.200 you drop the second ball 63 times. 00:09:52.200 --> 00:09:56.960 And that would be the worst case in this instance. 00:09:56.960 --> 00:10:00.300 Now, I've skipped over one little thing. 00:10:00.300 --> 00:10:02.010 The fact that this is the midpoint 00:10:02.010 --> 00:10:05.160 implies that this is, in fact, the correct answer. 00:10:05.160 --> 00:10:06.960 But we do have to analyze. 00:10:06.960 --> 00:10:09.000 I mean remember, you know, when you do analysis, 00:10:09.000 --> 00:10:11.920 you do have to cover the entire spectrum here. 00:10:11.920 --> 00:10:18.030 So you do have to look at the case where the drop from 64 00:10:18.030 --> 00:10:18.960 does not break. 00:10:22.680 --> 00:10:30.150 And in that case, you're looking at 65 through 128. 00:10:30.150 --> 00:10:33.330 But you do have two balls for this. 00:10:33.330 --> 00:10:34.410 It didn't break. 00:10:34.410 --> 00:10:37.110 So you still have two balls for that. 00:10:37.110 --> 00:10:40.440 And so you can imagine that, now that you have two balls 00:10:40.440 --> 00:10:43.860 and you have a small interval, that you're 00:10:43.860 --> 00:10:46.230 going to be able to do something like 32 for that. 00:10:46.230 --> 00:10:48.162 The bottom line is, you don't even need to. 00:10:48.162 --> 00:10:49.620 You can get a little bit lazy here. 00:10:49.620 --> 00:10:51.786 You don't really need to analyze what the number is, 00:10:51.786 --> 00:10:53.870 because you know that this number, 63 plus 1, 00:10:53.870 --> 00:10:55.230 is larger than that-- 00:10:55.230 --> 00:10:56.984 because you wanted worst case, right? 00:10:56.984 --> 00:10:58.650 So this is the kind of thing that you'll 00:10:58.650 --> 00:11:01.440 have to do when you do algorithmic analysis. 00:11:01.440 --> 00:11:05.820 Maybe not exactly in a puzzle situation like this, 00:11:05.820 --> 00:11:08.730 but you have to look at all the cases. 00:11:08.730 --> 00:11:11.879 And you have to find kind of the max of all the cases. 00:11:11.879 --> 00:11:13.170 So is this the best you can do? 00:11:15.510 --> 00:11:16.010 Yeah. 00:11:16.010 --> 00:11:16.809 Go ahead. 00:11:16.809 --> 00:11:21.549 AUDIENCE: You could break the 128 into smaller sections using 00:11:21.549 --> 00:11:22.298 the first ball. 00:11:22.298 --> 00:11:24.294 So instead of breaking it in half, 00:11:24.294 --> 00:11:26.496 you could do it every 16 floors. 00:11:26.496 --> 00:11:27.787 SRINI DEVADAS: Every 16 floors. 00:11:27.787 --> 00:11:28.287 Yeah. 00:11:28.287 --> 00:11:29.783 AUDIENCE: Starting from the bottom, 00:11:29.783 --> 00:11:32.200 and then once the first ball breaks, 00:11:32.200 --> 00:11:35.490 then you have a smaller interval to test with the second ball. 00:11:35.490 --> 00:11:40.240 So let's say it breaks at the 32nd. 00:11:40.240 --> 00:11:44.730 Then you have to only test for 16 to 32, instead of 1 to 32. 00:11:44.730 --> 00:11:45.980 SRINI DEVADAS: That's correct. 00:11:45.980 --> 00:11:46.500 Yeah. 00:11:46.500 --> 00:11:47.550 So it shows how you can do it-- 00:11:47.550 --> 00:11:49.200 I mean, this is, obviously there's not 00:11:49.200 --> 00:11:51.450 going to be a whole lot of optimization 00:11:51.450 --> 00:11:53.640 here with d equals 2. 00:11:53.640 --> 00:11:56.670 But you'd be surprised as to how much better you 00:11:56.670 --> 00:12:00.030 can do than 64, which is obviously a big step 00:12:00.030 --> 00:12:03.930 forward from 128. 00:12:03.930 --> 00:12:07.690 But you can go a few more steps as well. 00:12:07.690 --> 00:12:09.594 And so what was your name? 00:12:09.594 --> 00:12:10.380 AUDIENCE: Lucy. 00:12:10.380 --> 00:12:11.213 SRINI DEVADAS: Lucy. 00:12:11.213 --> 00:12:13.460 So Lucy has it right. 00:12:13.460 --> 00:12:18.420 It's a similar idea, except you don't necessarily 00:12:18.420 --> 00:12:22.280 need to do the intuitive binary search. 00:12:22.280 --> 00:12:26.220 It's actually kind of different from that because of the way 00:12:26.220 --> 00:12:28.020 this problem is structured. 00:12:28.020 --> 00:12:34.860 And what her idea is is that we'll look at something 00:12:34.860 --> 00:12:36.730 smaller, a smaller interval. 00:12:41.450 --> 00:12:47.520 And she picked 16, so let's go 16, 32 is our algorithm. 00:12:47.520 --> 00:12:52.940 It's something that goes 16, 32, 48, 64, dot, dot, dot. 00:12:52.940 --> 00:12:57.500 And maybe gets up all the way to I guess, yeah 128 and 16. 00:13:00.582 --> 00:13:03.270 So 128 is a multiple of 16, right? 00:13:03.270 --> 00:13:05.990 Some math person make sure I'm right. 00:13:09.290 --> 00:13:10.290 So what happens here? 00:13:10.290 --> 00:13:12.140 So let's analyze this to make sure 00:13:12.140 --> 00:13:15.320 that we're good in terms of an improvement. 00:13:15.320 --> 00:13:16.980 We have to be careful. 00:13:16.980 --> 00:13:20.990 So let's just arbitrarily say that it 00:13:20.990 --> 00:13:25.310 broke at 64, because that's kind of what happened over 00:13:25.310 --> 00:13:27.110 in that previous algorithm too. 00:13:27.110 --> 00:13:31.280 So that means I've done four drops without a break. 00:13:34.750 --> 00:13:36.190 I'm sorry, four drops. 00:13:36.190 --> 00:13:41.650 Three drops without a break and four drops total. 00:13:41.650 --> 00:13:42.970 So I've done four drops. 00:13:42.970 --> 00:13:47.060 But now the cool thing is I don't have to start from 16. 00:13:47.060 --> 00:13:49.990 I know it didn't break at 48. 00:13:49.990 --> 00:13:58.200 So I can now look at four drops till break of first ball. 00:13:58.200 --> 00:14:00.560 That is what I wanted to say. 00:14:00.560 --> 00:14:05.470 And now, I can look at the interval 49. 00:14:05.470 --> 00:14:08.110 Because 48, it did not break-- 00:14:08.110 --> 00:14:12.100 49 to 63, inclusive. 00:14:12.100 --> 00:14:14.860 And so if you do that, that's 15. 00:14:14.860 --> 00:14:16.280 It's 15. 00:14:16.280 --> 00:14:19.430 The difference is 14, but it's inclusive, so it's 15. 00:14:19.430 --> 00:14:24.540 So 15 plus 4, that is 19 drops. 00:14:24.540 --> 00:14:25.180 Yeah. 00:14:25.180 --> 00:14:25.800 Back there. 00:14:25.800 --> 00:14:27.175 AUDIENCE: Wouldn't the worst case 00:14:27.175 --> 00:14:28.800 though be when it's at 127. 00:14:28.800 --> 00:14:32.890 SRINI DEVADAS: Yes, that's absolutely correct. 00:14:32.890 --> 00:14:35.670 So 19 is not the worst case. 00:14:35.670 --> 00:14:38.430 I mean it's looking good, but this was exactly 00:14:38.430 --> 00:14:41.490 the point I was trying to make right from the beginning. 00:14:41.490 --> 00:14:42.630 You are not done yet. 00:14:42.630 --> 00:14:45.630 I mean, you cannot just go off and say that 19 is the worst 00:14:45.630 --> 00:14:46.872 case for this algorithm. 00:14:46.872 --> 00:14:48.330 The worst case for this algorithm-- 00:14:48.330 --> 00:14:51.810 this particular algorithm occurs at a different point 00:14:51.810 --> 00:14:55.380 from the original algorithm that went midway. 00:14:55.380 --> 00:14:56.960 Every algorithm is different. 00:14:56.960 --> 00:15:00.210 The parameters are different here. 00:15:00.210 --> 00:15:02.960 And so I'm just calling a different algorithm. 00:15:02.960 --> 00:15:06.390 But the case that would happen is 00:15:06.390 --> 00:15:08.490 I guess you would have 112 here. 00:15:08.490 --> 00:15:11.760 And then you'd go 1, 2, so how many? 00:15:11.760 --> 00:15:14.537 So let's say it gets to 128. 00:15:14.537 --> 00:15:16.620 And if it doesn't break, then you're clearly done. 00:15:16.620 --> 00:15:18.600 So that's not the worst case. 00:15:18.600 --> 00:15:24.990 So if it gets to 128 and it breaks, then so how many drops 00:15:24.990 --> 00:15:25.650 have we done? 00:15:25.650 --> 00:15:28.530 128 divided 16 is what? 00:15:28.530 --> 00:15:29.400 Is 8. 00:15:29.400 --> 00:15:30.720 Are you sure? 00:15:30.720 --> 00:15:31.740 You're all sure? 00:15:31.740 --> 00:15:32.240 OK. 00:15:32.240 --> 00:15:33.780 Then I'm sure. 00:15:33.780 --> 00:15:36.600 So that's eight drops. 00:15:36.600 --> 00:15:42.720 And now you need to do 113 to 127. 00:15:42.720 --> 00:15:45.210 And that we know is 15. 00:15:45.210 --> 00:15:47.420 So that's 23. 00:15:47.420 --> 00:15:48.850 All right. 00:15:48.850 --> 00:15:53.320 So the best you have done here is 23. 00:15:53.320 --> 00:15:56.470 But are we satisfied? 00:15:56.470 --> 00:15:57.970 We're never satisfied. 00:15:57.970 --> 00:16:00.690 We always want to do a little bit better. 00:16:00.690 --> 00:16:03.640 So how did she pick 16? 00:16:03.640 --> 00:16:05.220 Why did you come up with 16, Lucy? 00:16:05.220 --> 00:16:06.686 It looked like a nice number? 00:16:06.686 --> 00:16:08.630 AUDIENCE: I just figured it divides 128. 00:16:08.630 --> 00:16:11.970 SRINI DEVADAS: You just figured, oh, nice. 00:16:11.970 --> 00:16:15.280 So if you'd had 17, then we would have had real trouble 00:16:15.280 --> 00:16:16.640 with all these numbers. 00:16:16.640 --> 00:16:17.140 Yeah. 00:16:17.140 --> 00:16:17.470 Go ahead. 00:16:17.470 --> 00:16:17.970 Josh. 00:16:17.970 --> 00:16:20.620 AUDIENCE: Think you can do a little bit better 00:16:20.620 --> 00:16:23.460 if you get closer to the square root of 128? 00:16:23.460 --> 00:16:25.190 SRINI DEVADAS: Ah, beautiful, beautiful. 00:16:25.190 --> 00:16:26.450 I love you guys. 00:16:26.450 --> 00:16:29.140 You guys make it so easy. 00:16:29.140 --> 00:16:30.640 So that's the next thing I was going 00:16:30.640 --> 00:16:34.300 to try and pry out of you, but you got it out 00:16:34.300 --> 00:16:37.720 without much effort from me. 00:16:37.720 --> 00:16:42.580 So you can kind of get analytical here as opposed 00:16:42.580 --> 00:16:47.770 to doing things empirically, which was try a number 00:16:47.770 --> 00:16:50.420 and figure out what happened, which is kind of what we did. 00:16:50.420 --> 00:16:52.570 And you can be a little more analytical about this 00:16:52.570 --> 00:16:55.960 and you could say, hey, suppose I 00:16:55.960 --> 00:16:59.980 want to find this number k, which is kind of the 64-- 00:16:59.980 --> 00:17:01.900 started with 64, went to 16. 00:17:01.900 --> 00:17:04.810 And now, we're thinking we can do better than 16 even. 00:17:04.810 --> 00:17:10.810 And we start dropping things at k, 2k, 3k. 00:17:10.810 --> 00:17:15.099 And I want to get this exactly right. 00:17:15.099 --> 00:17:17.069 So let me look at my notes. 00:17:17.069 --> 00:17:18.930 And you go dot, dot, dot-- 00:17:18.930 --> 00:17:26.250 n divided by k minus 1 times k and n divided by k times 00:17:26.250 --> 00:17:28.140 k, which is obviously m. 00:17:28.140 --> 00:17:30.630 And so I'm trying to, I'm trying to write it-- 00:17:30.630 --> 00:17:32.490 I'm trying to get this k. 00:17:32.490 --> 00:17:33.810 I want to write something. 00:17:33.810 --> 00:17:36.120 I want to minimize at some quantity 00:17:36.120 --> 00:17:38.670 and discover the k that is going to give me my best 00:17:38.670 --> 00:17:41.460 case in the worst case-- 00:17:41.460 --> 00:17:43.500 minimize the worst case, right? 00:17:43.500 --> 00:17:46.050 So that's really what I want. 00:17:46.050 --> 00:17:49.930 So if you look at what happens here, 00:17:49.930 --> 00:17:54.420 you're in a situation where you have to now, 00:17:54.420 --> 00:17:55.530 you have a count, right? 00:17:55.530 --> 00:17:57.160 I mean, well, what do you want to-- 00:17:57.160 --> 00:18:03.420 we've decided that k is this general strategy. 00:18:03.420 --> 00:18:05.280 And we don't quite know what k is. 00:18:05.280 --> 00:18:09.900 And we have to do some analysis of the worst-case situation 00:18:09.900 --> 00:18:13.740 in order to find the k that would work well for us. 00:18:13.740 --> 00:18:18.930 And so the way you do this is by, essentially, the analysis 00:18:18.930 --> 00:18:20.430 we've done so far. 00:18:20.430 --> 00:18:28.380 And kind of we have a nice, the point that was made here, 00:18:28.380 --> 00:18:32.100 I think, it was Kevin who said we have to go to the last drop, 00:18:32.100 --> 00:18:36.370 because that's kind of the case where it's the worst case. 00:18:36.370 --> 00:18:37.860 And that's similar here as well. 00:18:37.860 --> 00:18:40.950 Because if you go up all the way here 00:18:40.950 --> 00:18:43.560 and then it breaks, if it doesn't break here, 00:18:43.560 --> 00:18:44.520 you're done. 00:18:44.520 --> 00:18:48.960 If it breaks here, then you've got, you've done a lot of drops 00:18:48.960 --> 00:18:51.180 before it broke that first ball. 00:18:51.180 --> 00:18:54.750 And so that's the situation which you can pull in 00:18:54.750 --> 00:18:56.850 as your worst-case situation. 00:18:56.850 --> 00:18:59.880 So that's exactly what we did before. 00:18:59.880 --> 00:19:05.110 And so if you really think about it, 00:19:05.110 --> 00:19:20.290 assume first ball doesn't break until the last drop is 00:19:20.290 --> 00:19:22.390 the worst-case situation. 00:19:22.390 --> 00:19:26.530 And so what does that mean from a symbolic standpoint 00:19:26.530 --> 00:19:29.815 of with respect to the number of drops of-- 00:19:29.815 --> 00:19:34.240 so the number of drops in this case of the first ball? 00:19:34.240 --> 00:19:37.781 Can someone give me an equation for it? 00:19:37.781 --> 00:19:38.280 Yeah. 00:19:38.280 --> 00:19:39.204 Go ahead. 00:19:39.204 --> 00:19:40.128 AUDIENCE: n over k. 00:19:40.128 --> 00:19:41.128 SRINI DEVADAS: n over k. 00:19:41.128 --> 00:19:41.930 You a Kevin too? 00:19:41.930 --> 00:19:42.230 Yeah. 00:19:42.230 --> 00:19:42.730 OK. 00:19:44.860 --> 00:19:45.981 So n over k. 00:19:45.981 --> 00:19:46.480 Right. 00:19:46.480 --> 00:19:48.470 So that's what happened there. 00:19:48.470 --> 00:19:51.010 Now, well, it broke, right? 00:19:51.010 --> 00:19:52.610 That's what we decided here. 00:19:52.610 --> 00:20:08.242 And so now, you need to search n k plus 1 to-- 00:20:08.242 --> 00:20:10.480 well, you got to go all the way there. 00:20:10.480 --> 00:20:14.820 I'm sorry, it broke there, so you got to go n minus 1. 00:20:20.310 --> 00:20:22.535 This is the same as what I wrote before, 00:20:22.535 --> 00:20:23.910 but it's just that I had numbers. 00:20:23.910 --> 00:20:25.660 And it was kind of easier to write numbers 00:20:25.660 --> 00:20:26.940 than write these symbols. 00:20:26.940 --> 00:20:31.740 But we have to do this because we don't know what k is. 00:20:31.740 --> 00:20:35.720 So if you look at how many drops are-- 00:20:35.720 --> 00:20:39.720 and if you're just going to do this, you get-- 00:20:39.720 --> 00:20:45.750 if you notice, this gets simplified to n. 00:20:45.750 --> 00:20:48.710 Yeah, so this is n, and then you have 00:20:48.710 --> 00:20:55.570 minus k plus 1, all the way to n minus 1. 00:20:55.570 --> 00:20:59.630 So if you subtract, and this is a plus 1 here. 00:20:59.630 --> 00:21:00.860 Boy, this is confusing. 00:21:00.860 --> 00:21:02.300 I'm not good at math. 00:21:02.300 --> 00:21:03.800 I'm definitely not good at math. 00:21:03.800 --> 00:21:12.746 So this is k minus 1 drops in the worst case. 00:21:12.746 --> 00:21:14.640 All right. 00:21:14.640 --> 00:21:18.940 So you now need to add these two things together. 00:21:18.940 --> 00:21:26.810 So what we'd like to do is minimize n divide it 00:21:26.810 --> 00:21:29.600 by k, which is this plus that. 00:21:36.030 --> 00:21:38.220 We want to minimize this function. 00:21:38.220 --> 00:21:39.710 And it's a constant. 00:21:39.710 --> 00:21:41.610 k is a variable. 00:21:41.610 --> 00:21:44.580 And for a given n, we want to discover the k that 00:21:44.580 --> 00:21:45.960 minimizes this quantity. 00:21:45.960 --> 00:21:48.390 And it goes back to the intuition 00:21:48.390 --> 00:21:52.110 that I think Josh had about the square root of n. 00:21:52.110 --> 00:21:54.571 So again, high school calculus here, which I've forgotten, 00:21:54.571 --> 00:21:56.820 which hopefully you haven't because you still need it. 00:21:56.820 --> 00:21:57.320 I don't. 00:22:00.780 --> 00:22:03.930 And you end up essentially saying 00:22:03.930 --> 00:22:06.450 you can turn this into two raised 00:22:06.450 --> 00:22:13.710 to the square root of n minus 1 drops in the worst case. 00:22:13.710 --> 00:22:16.420 All right. 00:22:16.420 --> 00:22:23.120 So you can take the floor of that, and that's 11. 00:22:23.120 --> 00:22:24.860 So that's that square root of n. 00:22:24.860 --> 00:22:28.250 So 2 times 11 minus 1 is 21. 00:22:28.250 --> 00:22:29.530 So you can do 21. 00:22:29.530 --> 00:22:32.944 So you think 21 is the optimum? 00:22:32.944 --> 00:22:33.600 Yeah, Kevin. 00:22:33.600 --> 00:22:35.058 AUDIENCE: Can we do a little better 00:22:35.058 --> 00:22:36.360 if we make it go smaller? 00:22:36.360 --> 00:22:37.300 SRINI DEVADAS: Yes. 00:22:37.300 --> 00:22:39.340 You can do a little bit better. 00:22:39.340 --> 00:22:40.740 I don't want to do this just yet. 00:22:40.740 --> 00:22:41.070 Hopefully. 00:22:41.070 --> 00:22:41.740 We'll have time. 00:22:41.740 --> 00:22:44.970 We'll get to that because I want to go to the d greater than 2 00:22:44.970 --> 00:22:47.200 case, but you're absolutely right. 00:22:47.200 --> 00:22:51.360 It turns out that you can do, this would give you 21, 00:22:51.360 --> 00:22:54.490 as I mentioned, 22 minus 1. 00:22:54.490 --> 00:22:57.060 It turns out, you can do better but by a few drops. 00:22:57.060 --> 00:23:01.740 And the assumption we made here, what is the assumption that we 00:23:01.740 --> 00:23:03.600 made here? 00:23:03.600 --> 00:23:05.689 When I wrote this, what assumption did I make, 00:23:05.689 --> 00:23:07.730 which is kind of similar to what Kevin just said, 00:23:07.730 --> 00:23:10.250 but not exactly? 00:23:10.250 --> 00:23:14.830 Would someone point out the assumption that was made here? 00:23:14.830 --> 00:23:16.189 Not you. 00:23:16.189 --> 00:23:16.730 Someone else. 00:23:20.380 --> 00:23:20.880 All right. 00:23:20.880 --> 00:23:21.330 Ganatra. 00:23:21.330 --> 00:23:21.829 Go for it. 00:23:21.829 --> 00:23:23.400 AUDIENCE: The intervals are the same. 00:23:23.400 --> 00:23:24.233 SRINI DEVADAS: Yeah. 00:23:24.233 --> 00:23:26.220 The intervals are equal. 00:23:26.220 --> 00:23:30.150 So this, look, it's a different problem. 00:23:30.150 --> 00:23:35.120 If it didn't break here, then I got a smaller-- 00:23:35.120 --> 00:23:36.950 I have two balls. 00:23:36.950 --> 00:23:38.780 And I had a smaller problem. 00:23:38.780 --> 00:23:41.764 So why would I use the same k? 00:23:41.764 --> 00:23:43.430 I mean, that kind of makes sense, right? 00:23:43.430 --> 00:23:46.550 So now, it's more complicated and hopefully, we'll 00:23:46.550 --> 00:23:47.530 get to that. 00:23:47.530 --> 00:23:49.896 But I want to do other things first. 00:23:49.896 --> 00:23:50.770 So that's the reason. 00:23:50.770 --> 00:23:54.720 So at least you know why 21 isn't optimal, right? 00:23:54.720 --> 00:23:58.370 It's because of this constraint that we placed on ourselves 00:23:58.370 --> 00:24:01.310 in order to do this analysis to get to the point 00:24:01.310 --> 00:24:06.500 where we had 2 times square root of n minus 1 that corresponds 00:24:06.500 --> 00:24:09.740 to the equal interval constraint. 00:24:09.740 --> 00:24:10.290 So good. 00:24:10.290 --> 00:24:10.790 All right. 00:24:10.790 --> 00:24:12.590 Any questions about this? 00:24:12.590 --> 00:24:14.090 So as you can imagine, I'm not going 00:24:14.090 --> 00:24:18.410 to show you code that computes 2 times square root of n minus 1. 00:24:18.410 --> 00:24:20.720 So that doesn't make sense, but we 00:24:20.720 --> 00:24:24.230 do have code to look at that corresponds to the case 00:24:24.230 --> 00:24:27.320 where d could be 3, d could be 4. 00:24:27.320 --> 00:24:30.050 And then obviously, it gets more interesting in terms 00:24:30.050 --> 00:24:31.670 of what your strategy is. 00:24:31.670 --> 00:24:34.490 And we won't go with optimizing. 00:24:34.490 --> 00:24:38.690 In fact, there's an open problem as to what the optimum strategy 00:24:38.690 --> 00:24:40.530 would be for arbitrary d-- 00:24:40.530 --> 00:24:42.440 d meaning the number of balls-- 00:24:42.440 --> 00:24:44.780 if it's 5, 6, whatever-- 00:24:44.780 --> 00:24:47.180 partly because of what I just talked about with respect 00:24:47.180 --> 00:24:50.480 to you need to utilize unequal intervals. 00:24:50.480 --> 00:24:52.850 But the natural thing to do now is 00:24:52.850 --> 00:24:57.350 to move to d equals a arbitrary number of balls 00:24:57.350 --> 00:25:03.230 and stick with kind of the same algorithmic approach of having 00:25:03.230 --> 00:25:05.090 equal intervals. 00:25:05.090 --> 00:25:07.190 But the question is, what are these intervals? 00:25:07.190 --> 00:25:10.640 What and how would we find the size of these intervals? 00:25:10.640 --> 00:25:12.630 Because it's kind of not immediately obvious 00:25:12.630 --> 00:25:14.132 if I told you d was 5. 00:25:14.132 --> 00:25:16.340 And you don't want to just keep the three balls away, 00:25:16.340 --> 00:25:18.800 and sell them later, and just use the two, right? 00:25:18.800 --> 00:25:21.270 That doesn't give you your minimum. 00:25:21.270 --> 00:25:23.370 It doesn't minimize your worst case. 00:25:23.370 --> 00:25:27.460 You want to use these balls. 00:25:27.460 --> 00:25:27.990 Cool. 00:25:27.990 --> 00:25:28.590 All right. 00:25:28.590 --> 00:25:38.930 So let's talk a little bit about if I had d equals 3, 00:25:38.930 --> 00:25:41.690 I mean again, you can use your intuition. 00:25:41.690 --> 00:25:45.740 What do you think we should do? 00:25:45.740 --> 00:25:48.260 What happened for d equals 2 with respect to that interval 00:25:48.260 --> 00:25:51.590 and how is it related to n? 00:25:51.590 --> 00:25:55.550 So what do you think, just wild guess, maybe even? 00:25:55.550 --> 00:25:57.290 What do you think we need to do if we 00:25:57.290 --> 00:26:00.200 had d equals 3 for that very first ball in terms 00:26:00.200 --> 00:26:05.075 of what we need to look at in relation to the interval size? 00:26:07.791 --> 00:26:08.290 Yeah. 00:26:08.290 --> 00:26:08.790 Back there. 00:26:08.790 --> 00:26:10.340 AUDIENCE: We need a cube root of 128. 00:26:10.340 --> 00:26:11.610 SRINI DEVADAS: Ah, cube root, cube root. 00:26:11.610 --> 00:26:12.580 What's your name? 00:26:12.580 --> 00:26:12.840 AUDIENCE: Ryan. 00:26:12.840 --> 00:26:13.680 SRINI DEVADAS: Ryan. 00:26:13.680 --> 00:26:15.150 So Ryan says, you know, cube root. 00:26:15.150 --> 00:26:17.260 And yeah, that's the correct intuition. 00:26:17.260 --> 00:26:19.756 It's a little bit unclear, you know, what happens. 00:26:19.756 --> 00:26:21.630 You know, so you do cube roots, and maybe you 00:26:21.630 --> 00:26:24.560 have what's the cube root of 128? 00:26:24.560 --> 00:26:27.030 5, yeah, that's right. 00:26:27.030 --> 00:26:27.940 I think it's 5. 00:26:27.940 --> 00:26:28.440 OK. 00:26:28.440 --> 00:26:29.670 It's close to 5, you know. 00:26:29.670 --> 00:26:31.950 It's 6 or 5 or 6, right? 00:26:31.950 --> 00:26:35.760 And so maybe I do 6, and then I do 12. 00:26:35.760 --> 00:26:39.190 And it's a little bit strange. 00:26:39.190 --> 00:26:42.270 So it turns out, it's actually the 6 and 12, 00:26:42.270 --> 00:26:45.690 and you start with 128 when you have three balls, 00:26:45.690 --> 00:26:48.690 doesn't seem like it's going to give you the worst case, right? 00:26:48.690 --> 00:26:50.890 Because, you know, like in the worst case, 00:26:50.890 --> 00:26:54.100 you might be dropping the ball 6, 12, 18, 00:26:54.100 --> 00:26:56.640 24, et cetera, et cetera. 00:26:56.640 --> 00:26:59.160 So cube root is right, but it's something a little bit 00:26:59.160 --> 00:27:01.530 different, because it gave us this interval that 00:27:01.530 --> 00:27:03.640 was kind of too small. 00:27:03.640 --> 00:27:05.820 So how would I get something that's bigger? 00:27:05.820 --> 00:27:07.830 You'd want things that are a little bit bigger. 00:27:07.830 --> 00:27:10.900 I mean, I think you're going in the right direction, 00:27:10.900 --> 00:27:12.840 but I don't want 6 and 12 and 18. 00:27:12.840 --> 00:27:15.720 I mean, clearly, that's going to give me like 25 drops 00:27:15.720 --> 00:27:19.456 and that's even worse than the two ball case, which really 00:27:19.456 --> 00:27:21.830 means you should lose your job and never get another one, 00:27:21.830 --> 00:27:22.330 right? 00:27:22.330 --> 00:27:25.680 So if you do something that bad, right? 00:27:25.680 --> 00:27:29.700 So that's-- what would you do that's kind of different from 00:27:29.700 --> 00:27:31.330 cube root, but related to cube root? 00:27:34.020 --> 00:27:34.970 Yeah, go ahead Fadi. 00:27:34.970 --> 00:27:36.730 AUDIENCE: 128 raised to 2/3 maybe? 00:27:36.730 --> 00:27:37.046 SRINI DEVADAS: Yeah, yeah. 00:27:37.046 --> 00:27:37.546 Two thirds. 00:27:37.546 --> 00:27:38.150 Exactly right. 00:27:38.150 --> 00:27:40.370 So what you want to do is, you want to start-- 00:27:40.370 --> 00:27:42.880 it makes sense that when you have lots of balls, 00:27:42.880 --> 00:27:44.640 in terms of being able to use them, 00:27:44.640 --> 00:27:47.117 that you start with larger intervals and you shrink them. 00:27:47.117 --> 00:27:48.950 That's exactly what happened with two balls. 00:27:48.950 --> 00:27:52.410 You started with k in the case where we had two balls. 00:27:52.410 --> 00:27:55.050 And then we are down to one when we have one ball. 00:27:55.050 --> 00:27:56.707 So that's really what you're doing. 00:27:56.707 --> 00:27:58.290 When you have one ball, all you can do 00:27:58.290 --> 00:28:01.520 is go one floor at a time. 00:28:01.520 --> 00:28:03.580 When you have two balls, you go k, 00:28:03.580 --> 00:28:05.504 where k happened to be square root of n. 00:28:05.504 --> 00:28:07.170 And then we had three balls, you kind of 00:28:07.170 --> 00:28:10.620 want to start with like n raised to 2/3 00:28:10.620 --> 00:28:13.110 in terms of the interval. 00:28:13.110 --> 00:28:16.050 And then if I had four balls, the argument 00:28:16.050 --> 00:28:18.870 would be that I would start with n raised to 3/4. 00:28:18.870 --> 00:28:23.010 And then I would go to square root of n. 00:28:23.010 --> 00:28:25.090 And then I would go so on and so forth. 00:28:25.090 --> 00:28:26.850 And what is that reminding you of? 00:28:26.850 --> 00:28:29.880 What is this, in terms of these powers 00:28:29.880 --> 00:28:35.520 and getting up to not quite n, but 5, 6th n, so on. 00:28:35.520 --> 00:28:38.960 What does that remind you of from number theory 00:28:38.960 --> 00:28:43.460 or just sort of regular arithmetic? 00:28:43.460 --> 00:28:44.269 AUDIENCE: Bases. 00:28:44.269 --> 00:28:45.810 SRINI DEVADAS: Yeah, different bases. 00:28:45.810 --> 00:28:48.676 It's different radix arithmetic, right? 00:28:48.676 --> 00:28:51.050 So it turns out, the way you want to solve this problem-- 00:28:51.050 --> 00:28:55.200 and this is in keeping with the intuition you 00:28:55.200 --> 00:28:59.430 have with respect to the cube roots, and so on, and so forth. 00:28:59.430 --> 00:29:01.890 The way we want to do this is, we want 00:29:01.890 --> 00:29:05.175 to think about d digit numbers. 00:29:12.790 --> 00:29:17.040 And the numbers themselves, and d is the number of balls. 00:29:20.020 --> 00:29:21.290 And so that's the d digits. 00:29:21.290 --> 00:29:24.890 But what is the radix of these numbers? 00:29:24.890 --> 00:29:30.360 The radix of these numbers are going to be related to the k, 00:29:30.360 --> 00:29:31.920 to the intervals. 00:29:31.920 --> 00:29:34.340 So you essentially want to say, you know, 00:29:34.340 --> 00:29:36.470 I'm counting in multiples of k-- 00:29:36.470 --> 00:29:38.990 1k, 2k, 3k, et cetera. 00:29:38.990 --> 00:29:41.150 So if I had like radix k, that means 00:29:41.150 --> 00:29:44.720 I'm just incrementing that particular digit. 00:29:44.720 --> 00:29:48.920 You know, if I go, if I had radix 12, 00:29:48.920 --> 00:29:57.270 and I had 1 0 in radix 12, then this would be-- 00:30:02.120 --> 00:30:07.310 so when I go 2 0, then the most significant digit 00:30:07.310 --> 00:30:08.120 got incremented. 00:30:08.120 --> 00:30:11.540 And so this is plus 12. 00:30:11.540 --> 00:30:12.980 This is 12 times larger-- 00:30:12.980 --> 00:30:17.510 I'm sorry-- 12 larger than two 0's, 12 larger than 1 0 00:30:17.510 --> 00:30:19.300 in radix 12. 00:30:19.300 --> 00:30:20.240 Make sense? 00:30:20.240 --> 00:30:29.110 Just like 20 is 10 larger than 10 in decimal. 00:30:29.110 --> 00:30:31.610 So that's the way you want to think about this. 00:30:31.610 --> 00:30:37.790 And so you're going to do r-ary numbers. 00:30:37.790 --> 00:30:39.290 And the first thing that you need 00:30:39.290 --> 00:30:41.270 to do is, you need to figure out, 00:30:41.270 --> 00:30:44.080 based on the number of floors that you have, figure out, 00:30:44.080 --> 00:30:45.260 you've given d. 00:30:45.260 --> 00:30:47.799 You're given n, the number of floors. 00:30:47.799 --> 00:30:49.340 And you're given the number of balls. 00:30:49.340 --> 00:30:51.800 And you got to figure out what radix you're working in. 00:30:51.800 --> 00:30:53.810 And this is going to be a generalization 00:30:53.810 --> 00:30:57.820 of the square root of n, as you'll see. 00:30:57.820 --> 00:31:02.930 We want to choose a radix such that r 00:31:02.930 --> 00:31:08.840 raised to d is greater than n. 00:31:08.840 --> 00:31:16.800 Choose r such that r raised to d is greater than n. 00:31:16.800 --> 00:31:19.080 And so that comes back to, obviously, 00:31:19.080 --> 00:31:22.350 if you needed, if d were 2, then it makes sense 00:31:22.350 --> 00:31:24.150 that r would be square root of n, 00:31:24.150 --> 00:31:26.490 which is in keeping with what we talked about. 00:31:26.490 --> 00:31:30.027 But if d is 3, then that radix-- 00:31:30.027 --> 00:31:31.860 not necessarily the interval that we choose, 00:31:31.860 --> 00:31:34.777 because the intervals are going to be different-- 00:31:34.777 --> 00:31:36.610 is going to be more like the cube root of n, 00:31:36.610 --> 00:31:41.080 which is also proposed. 00:31:41.080 --> 00:31:42.810 Ryan mentioned that. 00:31:42.810 --> 00:31:50.310 And so in this case, let's say that n were 128 and d equals 4. 00:31:50.310 --> 00:31:56.430 Then we would choose r equals 4, because 4 raised to 4 00:31:56.430 --> 00:31:59.550 is 256, which is greater than 128. 00:31:59.550 --> 00:32:05.610 And we can't choose 3, because 3 raised to 4 is only 81. 00:32:05.610 --> 00:32:08.940 And 81 isn't greater than 128. 00:32:08.940 --> 00:32:11.840 So you'd have to choose 4. 00:32:11.840 --> 00:32:16.600 So what we're now saying is, we want to somehow figure out-- 00:32:16.600 --> 00:32:19.300 we're not quite there in an algorithm yet. 00:32:19.300 --> 00:32:22.830 But we have a kind of a path forward 00:32:22.830 --> 00:32:28.470 in that we want to look at that representation, 00:32:28.470 --> 00:32:33.750 that r-ary representation of these d digit numbers. 00:32:33.750 --> 00:32:37.710 And we're going to essentially increment. 00:32:37.710 --> 00:32:41.400 And so with our goal, roughly speaking, 00:32:41.400 --> 00:32:45.030 we're going to go with this representation. 00:32:45.030 --> 00:32:48.360 And we're going to start incrementing digits. 00:32:48.360 --> 00:32:51.430 So we just go 0, 1, 2 in terms of the digits. 00:32:51.430 --> 00:32:53.995 But because the representation is r-ary 00:32:53.995 --> 00:32:55.620 that means something different in terms 00:32:55.620 --> 00:32:58.230 of the number of floors, depending on what r is. 00:32:58.230 --> 00:33:01.410 I mean, just like I said, if we're 00:33:01.410 --> 00:33:06.512 this were ternary representation then in terms of floors, 00:33:06.512 --> 00:33:07.095 this is only-- 00:33:09.841 --> 00:33:13.620 this would be, for ternary, this would be three floors. 00:33:13.620 --> 00:33:19.230 2 0 would be three floors away from 1 0. 00:33:19.230 --> 00:33:22.040 In decimal, 2 0 is 10 floors away from 1 0, 00:33:22.040 --> 00:33:23.040 and so on, and so forth. 00:33:23.040 --> 00:33:24.251 That makes sense? 00:33:24.251 --> 00:33:24.750 Yep. 00:33:24.750 --> 00:33:28.689 Ask me questions if you're confused. 00:33:28.689 --> 00:33:29.730 So let's take an example. 00:33:29.730 --> 00:33:34.120 So and then we'll look at code that actually builds 00:33:34.120 --> 00:33:38.520 this algorithm from scratch. 00:33:38.520 --> 00:33:40.350 So we'll take our example of d equals 00:33:40.350 --> 00:33:44.460 4 for our canonical 128 floor Shanghai Tower, 00:33:44.460 --> 00:33:48.240 because you do want to get a sense of how things improve. 00:33:48.240 --> 00:33:50.760 We did improve from obviously d equals 1 00:33:50.760 --> 00:33:55.320 through d equals 2 with the equal interval assumption. 00:33:55.320 --> 00:33:59.550 But with this radix idea, we'd like 00:33:59.550 --> 00:34:03.060 to see how much better we can do with d equals 4. 00:34:03.060 --> 00:34:06.780 And we want to do substantially better than 21 drops 00:34:06.780 --> 00:34:10.429 or whatever we had back there. 00:34:10.429 --> 00:34:12.870 So r is 4. 00:34:12.870 --> 00:34:16.230 d is 4-- oh, I'm sorry-- d is 4. 00:34:16.230 --> 00:34:18.570 n equals 128. 00:34:18.570 --> 00:34:24.338 And what we're going to do is, the smallest number is 0000. 00:34:24.338 --> 00:34:30.460 And in brackets, I'm going write what the decimal representation 00:34:30.460 --> 00:34:33.719 is, because I think that does give you some intuition. 00:34:33.719 --> 00:34:37.104 You get a sense of how this translates into floors. 00:34:37.104 --> 00:34:38.520 And after a while, if you're going 00:34:38.520 --> 00:34:40.770 to tell someone to do this, you want to give them 00:34:40.770 --> 00:34:42.150 floor numbers in decimal. 00:34:42.150 --> 00:34:43.710 Because the Shanghai Tower doesn't 00:34:43.710 --> 00:34:48.179 have some weird representation for the floors 00:34:48.179 --> 00:34:51.150 when you get into an elevator or anything like that. 00:34:51.150 --> 00:34:53.070 So this is floor 0. 00:34:53.070 --> 00:35:00.000 And now, the highest number is 3333, which happens to be 255. 00:35:00.000 --> 00:35:02.250 I'm not going to write floor each time. 00:35:02.250 --> 00:35:04.230 But when I put things in brackets here, 00:35:04.230 --> 00:35:05.801 that's the decimal representation. 00:35:05.801 --> 00:35:06.300 OK. 00:35:06.300 --> 00:35:07.230 So that's 255. 00:35:07.230 --> 00:35:08.970 So that's obviously too big. 00:35:08.970 --> 00:35:09.940 We don't need that. 00:35:09.940 --> 00:35:12.570 But we're stuck with having to choose 00:35:12.570 --> 00:35:16.220 r equals 4 for the reasons I described to you before. 00:35:16.220 --> 00:35:19.500 So there's somewhere in here there's a 128. 00:35:19.500 --> 00:35:24.150 And I don't have it right up in front of me 00:35:24.150 --> 00:35:28.950 as to what 128 would correspond to. 00:35:28.950 --> 00:35:36.920 But 1233, for example, would be floor 111. 00:35:36.920 --> 00:35:38.380 So the way this works, and you all 00:35:38.380 --> 00:35:43.060 know you're radix arithmetic, it's 00:35:43.060 --> 00:35:51.310 3 times 1, plus 3 times 4, plus 2 times 4 square, plus 1 times 00:35:51.310 --> 00:35:52.960 4 cubed. 00:35:52.960 --> 00:35:53.950 That's what you have. 00:35:53.950 --> 00:35:55.630 And that's how you get 111. 00:35:55.630 --> 00:35:59.350 So what you have to do in order to do this right 00:35:59.350 --> 00:36:04.390 is to start with the most significant digit, which 00:36:04.390 --> 00:36:05.644 is the left-most digit. 00:36:05.644 --> 00:36:07.810 And that makes sense, because we want our intervals, 00:36:07.810 --> 00:36:09.910 when we have lots of balls, to be large. 00:36:09.910 --> 00:36:13.250 And that as you move and the balls break, 00:36:13.250 --> 00:36:17.410 you're going to start shifting rightward. 00:36:17.410 --> 00:36:19.960 And you're going to be incrementing different things. 00:36:19.960 --> 00:36:25.169 And that's essentially going to give us give us our algorithm. 00:36:25.169 --> 00:36:26.710 And so we'll talk about-- let me just 00:36:26.710 --> 00:36:29.110 give you an example of an algorithm. 00:36:29.110 --> 00:36:34.360 And we'll look at how the algorithm is implemented. 00:36:34.360 --> 00:36:37.120 As I mentioned, it turns into 10 lines of code, 00:36:37.120 --> 00:36:40.180 because all you're doing is translating 00:36:40.180 --> 00:36:42.400 into this representation. 00:36:42.400 --> 00:36:44.920 And you're just incrementing bits of the representation. 00:36:44.920 --> 00:36:46.120 And it's interactive. 00:36:46.120 --> 00:36:50.550 You can play with it later today or whenever. 00:36:50.550 --> 00:36:52.720 And you can sort of think of a hardness coefficient 00:36:52.720 --> 00:36:54.970 in your head then see how things work out. 00:36:54.970 --> 00:36:56.905 But before we end, we obviously will 00:36:56.905 --> 00:37:00.670 have to do a worst-case analysis of this algorithm 00:37:00.670 --> 00:37:04.930 because that's the whole point of this exercise. 00:37:04.930 --> 00:37:08.290 So in this case, where do you think I should-- 00:37:08.290 --> 00:37:11.560 so I don't want to drop, obviously, at floor 0. 00:37:11.560 --> 00:37:13.480 I mean, that's there's no notion of that. 00:37:13.480 --> 00:37:16.500 So where do I drop at, at the beginning? 00:37:16.500 --> 00:37:21.020 The first drop is at, given what I told you? 00:37:21.020 --> 00:37:23.580 You can tell me in any representation. 00:37:23.580 --> 00:37:25.220 Where would I drop the first ball 00:37:25.220 --> 00:37:27.350 if I have four balls and 128 floors? 00:37:32.480 --> 00:37:36.590 I said that I want to start with something that, obviously, 00:37:36.590 --> 00:37:38.450 there's some larger interval here. 00:37:38.450 --> 00:37:41.090 And I want to start on the left-hand side. 00:37:41.090 --> 00:37:43.520 And I want to increment the leftmost bit, which 00:37:43.520 --> 00:37:45.260 is the most significant bit. 00:37:45.260 --> 00:37:50.090 So I would start with 1000. 00:37:50.090 --> 00:38:00.560 So the first drop is at 1000, which is at floor 64. 00:38:00.560 --> 00:38:01.970 That's the first drop. 00:38:01.970 --> 00:38:09.374 Now, if in fact this does not break-- 00:38:09.374 --> 00:38:11.540 all right, so let's assume that this does not break. 00:38:14.970 --> 00:38:18.090 Then what do I do next? 00:38:18.090 --> 00:38:20.470 2000, right? 00:38:20.470 --> 00:38:25.030 And this would be 128. 00:38:25.030 --> 00:38:26.770 Let's assume that this breaks here. 00:38:29.400 --> 00:38:30.240 All right. 00:38:30.240 --> 00:38:32.370 So now, what do I do? 00:38:32.370 --> 00:38:33.000 It broke. 00:38:33.000 --> 00:38:35.190 I lost a ball. 00:38:35.190 --> 00:38:37.045 Now I have three balls left still. 00:38:37.045 --> 00:38:37.920 That's the good news. 00:38:37.920 --> 00:38:40.584 And how many-- so what do I need, where do I need to move? 00:38:40.584 --> 00:38:42.000 If I have three balls left, I need 00:38:42.000 --> 00:38:45.220 to be in a representation that has how many digits? 00:38:45.220 --> 00:38:46.420 Three digits. 00:38:46.420 --> 00:38:47.500 So I need to move. 00:38:47.500 --> 00:38:51.550 I a broke at 2000, so I know that it's less than 2000. 00:38:51.550 --> 00:38:54.820 So I need to go back to this, because this is the highest 00:38:54.820 --> 00:38:55.810 floor. 00:38:55.810 --> 00:38:58.710 In general, whenever things break, 00:38:58.710 --> 00:39:02.110 you go back to the highest floor that you know 00:39:02.110 --> 00:39:03.940 that the ball didn't break. 00:39:03.940 --> 00:39:05.740 I mean, that's the canonical strategy 00:39:05.740 --> 00:39:07.240 that we've followed all through. 00:39:07.240 --> 00:39:09.970 When the ball breaks, you go back 00:39:09.970 --> 00:39:11.560 and say, oh, but I do know that there 00:39:11.560 --> 00:39:15.130 was a highest floor in which the ball did not break. 00:39:15.130 --> 00:39:19.770 And I have to start with 1 plus that, right? 00:39:19.770 --> 00:39:22.820 So what I'm going to do here is I need to go up 00:39:22.820 --> 00:39:32.530 and because it broke, I need to look at 1001. 00:39:32.530 --> 00:39:34.900 The interval that I need to look at 00:39:34.900 --> 00:39:38.920 is 1001, because it's one more than that. 00:39:38.920 --> 00:39:44.410 And all the way to 1333, which is one less than that. 00:39:47.010 --> 00:39:51.750 So and this, you may feel a little bit better 00:39:51.750 --> 00:39:54.510 looking at this in decimal. 00:39:54.510 --> 00:39:55.210 I certainly do. 00:39:57.880 --> 00:40:01.560 It's 65, 127, which makes perfect sense. 00:40:01.560 --> 00:40:06.210 So far, we've played around with the balls, the first two balls. 00:40:06.210 --> 00:40:09.030 And so we haven't really seen anything very dramatically 00:40:09.030 --> 00:40:13.800 different from the two-ball case, but more is coming. 00:40:13.800 --> 00:40:15.690 Let's say this does not break. 00:40:15.690 --> 00:40:24.240 If this does not break, then what would I do now? 00:40:24.240 --> 00:40:25.830 If it does not break-- 00:40:25.830 --> 00:40:31.590 I'm sorry, my bad, my bad. 00:40:31.590 --> 00:40:33.990 I haven't even told you. 00:40:33.990 --> 00:40:35.820 I've just talked about the interval. 00:40:35.820 --> 00:40:38.570 And I haven't said what drop you need to make, right? 00:40:38.570 --> 00:40:41.040 So ignore what I said just now. 00:40:41.040 --> 00:40:44.190 I think that the next question is, the ball broke. 00:40:44.190 --> 00:40:45.360 So now I have a second ball. 00:40:48.010 --> 00:40:50.350 And I want the first drop of the second ball. 00:40:53.260 --> 00:40:55.227 And I want to look at this. 00:40:55.227 --> 00:40:56.560 I want to look at this interval. 00:40:56.560 --> 00:40:58.820 And it has to be somewhere inside this interval. 00:40:58.820 --> 00:40:59.903 But it's not going to be-- 00:40:59.903 --> 00:41:06.880 I'm not going to drop this at this floor, 65 or 1333. 00:41:06.880 --> 00:41:09.250 I want to pick something in between, right? 00:41:09.250 --> 00:41:13.390 And so now looking at this, tell me-- 00:41:13.390 --> 00:41:15.630 according to what you know so far-- 00:41:15.630 --> 00:41:18.400 where you think I should drop the second ball 00:41:18.400 --> 00:41:19.914 for the first time? 00:41:19.914 --> 00:41:20.798 Go ahead. 00:41:20.798 --> 00:41:21.950 AUDIENCE: 96. 00:41:21.950 --> 00:41:22.700 SRINI DEVADAS: 96. 00:41:22.700 --> 00:41:23.995 And how did you get that? 00:41:23.995 --> 00:41:26.690 AUDIENCE: Just like I did the half point. 00:41:26.690 --> 00:41:28.135 SRINI DEVADAS: And you put 1100? 00:41:28.135 --> 00:41:28.760 AUDIENCE: Yeah. 00:41:28.760 --> 00:41:29.460 SRINI DEVADAS: Perfect. 00:41:29.460 --> 00:41:30.196 What's your name? 00:41:30.196 --> 00:41:31.090 AUDIENCE: Evan. 00:41:31.090 --> 00:41:31.923 SRINI DEVADAS: Evan. 00:41:31.923 --> 00:41:33.560 So Evan says 96. 00:41:33.560 --> 00:41:37.880 I guess he thinks in decimal naturally, like most of us. 00:41:37.880 --> 00:41:41.540 But the way you would code this, by the way, 00:41:41.540 --> 00:41:45.830 is by incrementing this digit. 00:41:45.830 --> 00:41:49.040 And you'll get 1100. 00:41:49.040 --> 00:41:51.920 And in this case, let's say it does not break. 00:41:55.110 --> 00:41:58.770 Then where do you go, Evan? 00:41:58.770 --> 00:41:59.977 AUDIENCE: Above floor 96. 00:41:59.977 --> 00:42:00.810 SRINI DEVADAS: Yeah. 00:42:00.810 --> 00:42:02.150 After 96, it doesn't break. 00:42:02.150 --> 00:42:03.732 What would you do? 00:42:03.732 --> 00:42:06.930 AUDIENCE: Then you would go to what is that? 00:42:06.930 --> 00:42:08.590 SRINI DEVADAS: Well, it's- 00:42:08.590 --> 00:42:10.050 AUDIENCE: In decimal? 00:42:10.050 --> 00:42:12.400 SRINI DEVADAS: No, in this representation-- 00:42:14.806 --> 00:42:15.930 whatever is easier for you. 00:42:15.930 --> 00:42:17.680 AUDIENCE: It would be 1200. 00:42:17.680 --> 00:42:19.260 SRINI DEVADAS: Perfect, 1200, which 00:42:19.260 --> 00:42:22.080 happens to be I think 112. 00:42:22.080 --> 00:42:23.490 So you go to floor 112. 00:42:23.490 --> 00:42:29.216 This was 96, and then you needed to add another 16 to that-- 00:42:29.216 --> 00:42:33.030 yep, 112 and so on. 00:42:33.030 --> 00:42:35.060 So I won't bore you with this. 00:42:35.060 --> 00:42:37.600 You get a sense of what's going on here. 00:42:37.600 --> 00:42:41.790 And obviously with these things, you can wave your hands 00:42:41.790 --> 00:42:44.100 and you get a sense of what's going on. 00:42:44.100 --> 00:42:48.780 But ultimately, you know, what's great about computer science 00:42:48.780 --> 00:42:53.040 and programming is you've got to teach a dumb computer to do 00:42:53.040 --> 00:42:54.720 exactly the right thing. 00:42:54.720 --> 00:42:57.270 And there's no two things about it in terms of you 00:42:57.270 --> 00:42:58.810 understand this algorithm. 00:42:58.810 --> 00:43:00.270 I didn't understand this algorithm 00:43:00.270 --> 00:43:02.630 until I coded it, fully. 00:43:02.630 --> 00:43:04.380 I mean, maybe I still don't understand it, 00:43:04.380 --> 00:43:07.950 considering the mistakes I'm making. 00:43:07.950 --> 00:43:12.960 But that's just because I've lost a bunch of gray cells. 00:43:12.960 --> 00:43:15.300 So let me show you what the code looks like. 00:43:15.300 --> 00:43:18.270 It's 15 lines of code. 00:43:18.270 --> 00:43:19.290 It's not that much. 00:43:19.290 --> 00:43:23.850 And as you can imagine, the representation is-- 00:43:23.850 --> 00:43:25.100 I could have used many things. 00:43:25.100 --> 00:43:27.910 I just used a list for each of these digits. 00:43:27.910 --> 00:43:30.360 I'm incrementing these digits. 00:43:30.360 --> 00:43:34.630 And I won't spend a whole lot of time on the code, 00:43:34.630 --> 00:43:39.780 but what I want to do before we end here is talk about-- 00:43:39.780 --> 00:43:43.230 actually, let me do that first, before we go to the code. 00:43:43.230 --> 00:43:45.930 I want to talk about the worst case here, 00:43:45.930 --> 00:43:48.000 because that's what this is all about. 00:43:48.000 --> 00:43:51.440 I want to get an equation, just like we did for 2 times 00:43:51.440 --> 00:43:54.540 square root of n minus 1 and all of these different things, I 00:43:54.540 --> 00:43:57.420 want to get an equation for the worst case in terms 00:43:57.420 --> 00:44:01.920 of the number of drops in relation to r and d, 00:44:01.920 --> 00:44:03.720 which are obviously related to n. 00:44:03.720 --> 00:44:06.567 Because r raised to d has to be greater than n. 00:44:06.567 --> 00:44:08.400 So if you kind of look at this, and it turns 00:44:08.400 --> 00:44:11.580 out this is not a difficult analysis-- 00:44:11.580 --> 00:44:13.134 if you think about it, and you start 00:44:13.134 --> 00:44:14.550 thinking about worst case, and you 00:44:14.550 --> 00:44:17.940 think about how we're incrementing these things, 00:44:17.940 --> 00:44:19.170 there's phases. 00:44:19.170 --> 00:44:23.267 And if you can think about a phase as when a ball breaks, 00:44:23.267 --> 00:44:24.850 you know, that's when a phase is done. 00:44:24.850 --> 00:44:28.050 And you move to the second phase. 00:44:28.050 --> 00:44:32.450 Tell me how many drops I have? 00:44:32.450 --> 00:44:34.894 You break it up in terms of the number of phases 00:44:34.894 --> 00:44:36.810 or the number of balls, whichever way you want 00:44:36.810 --> 00:44:39.360 to think about it, and use the same kind 00:44:39.360 --> 00:44:41.700 of worst-case analysis we've done with respect 00:44:41.700 --> 00:44:48.520 to what happens in the worst case. 00:44:48.520 --> 00:44:50.480 You have to keep doing a bunch of drops. 00:44:50.480 --> 00:44:54.720 And really, for these algorithms, the floor 127, 00:44:54.720 --> 00:44:56.910 the hardness coefficient of-- 00:44:56.910 --> 00:45:01.340 actually, the hardness coefficient of 127 00:45:01.340 --> 00:45:02.490 is the hard one, right? 00:45:02.490 --> 00:45:04.740 It's the difficult one, because you kind of go through 00:45:04.740 --> 00:45:06.360 and you go all the way there. 00:45:06.360 --> 00:45:08.220 And you do all of these things before-- you 00:45:08.220 --> 00:45:11.700 drop the ball a lot of times before it breaks. 00:45:11.700 --> 00:45:13.650 And you do that for every ball. 00:45:13.650 --> 00:45:15.540 That would be the worst case, right? 00:45:15.540 --> 00:45:17.520 So that's kind of where things are at. 00:45:17.520 --> 00:45:20.760 So from a symbolic standpoint, what 00:45:20.760 --> 00:45:23.760 is the worst case in the first phase 00:45:23.760 --> 00:45:27.870 in terms of the number of drops of the first ball, 00:45:27.870 --> 00:45:31.546 from a symbolic standpoint? 00:45:31.546 --> 00:45:33.170 It's the first digit I'm talking about. 00:45:33.170 --> 00:45:35.110 It's the first ball I'm talking about. 00:45:35.110 --> 00:45:39.180 And I'm dropping this ball, as you can see, 00:45:39.180 --> 00:45:40.560 at these different points. 00:45:40.560 --> 00:45:46.800 So in general, how many drops would I have before I'm either, 00:45:46.800 --> 00:45:50.320 I may be done completely because I know it doesn't break a floor 00:45:50.320 --> 00:45:52.220 128 or what have you. 00:45:52.220 --> 00:45:55.480 But what is the worst case when that ball breaks? 00:45:55.480 --> 00:45:57.554 Give me an equation. 00:45:57.554 --> 00:45:58.054 Someone. 00:46:01.350 --> 00:46:02.220 So I drop it once. 00:46:02.220 --> 00:46:03.430 I drop it twice. 00:46:03.430 --> 00:46:06.200 Am I going to drop it like, if r is 5, 00:46:06.200 --> 00:46:09.436 am I going to drop it 10 times? 00:46:09.436 --> 00:46:10.810 What am I doing in terms of the-- 00:46:10.810 --> 00:46:13.985 what is the upper limit in terms of this digit? 00:46:13.985 --> 00:46:14.859 AUDIENCE: r minus 1. 00:46:14.859 --> 00:46:15.900 SRINI DEVADAS: r minus 1. 00:46:15.900 --> 00:46:18.150 And where am I starting from? 00:46:18.150 --> 00:46:19.500 1, so there you go. 00:46:19.500 --> 00:46:20.530 What's the answer? 00:46:20.530 --> 00:46:21.377 AUDIENCE: r minus 1. 00:46:21.377 --> 00:46:22.710 SRINI DEVADAS: r minus 1, right? 00:46:22.710 --> 00:46:23.740 So that's it. 00:46:23.740 --> 00:46:30.870 So it turns out that I have r minus 1 drops in the worst 00:46:30.870 --> 00:46:38.020 case for the first ball. 00:46:38.020 --> 00:46:41.520 And, I mean, it's the same representation. 00:46:41.520 --> 00:46:44.920 Every digit is the same radix, obviously. 00:46:44.920 --> 00:46:48.220 And now, that ball broke and I moved to the second digit 00:46:48.220 --> 00:46:49.660 with the second ball. 00:46:49.660 --> 00:46:51.850 And I'm going to start again. 00:46:51.850 --> 00:46:56.170 And in the worst case, I could have r minus 1 drops for that 00:46:56.170 --> 00:46:59.519 as well, right, because I'm going to go increment from 0. 00:46:59.519 --> 00:47:01.060 The first one I'm going to do is a 1. 00:47:01.060 --> 00:47:04.370 And then I'm going to do a 2 and all the way to r minus 1. 00:47:04.370 --> 00:47:10.790 So r minus 1 drops for the second ball. 00:47:10.790 --> 00:47:14.550 And then I go all the way to d balls. 00:47:14.550 --> 00:47:21.726 So my answer-- yeah, go ahead. 00:47:21.726 --> 00:47:22.400 Over here. 00:47:22.400 --> 00:47:25.686 Ganatra. 00:47:25.686 --> 00:47:27.630 AUDIENCE: So r is 4 in this case. 00:47:27.630 --> 00:47:29.880 And we say the worse case is 3. 00:47:29.880 --> 00:47:31.543 But how do we get 3? 00:47:31.543 --> 00:47:34.210 Because if it doesn't break on either of those, 00:47:34.210 --> 00:47:38.631 on the 1000 and 2000, aren't we just done 00:47:38.631 --> 00:47:40.287 and that's only two drops? 00:47:40.287 --> 00:47:41.120 SRINI DEVADAS: Yeah. 00:47:41.120 --> 00:47:43.540 So it gets a little-- 00:47:43.540 --> 00:47:44.650 you're absolutely right. 00:47:44.650 --> 00:47:47.671 But if the number were 255-- 00:47:47.671 --> 00:47:49.420 so one of the things that's happening here 00:47:49.420 --> 00:47:53.440 is we've lost some precision because r raised to d needed 00:47:53.440 --> 00:47:55.780 to be this large number of 255. 00:47:55.780 --> 00:47:58.895 And this algorithm is actually going to work for 255 floors. 00:47:58.895 --> 00:48:00.320 AUDIENCE: Oh, it's because of n. 00:48:00.320 --> 00:48:01.800 SRINI DEVADAS: Yeah, because of n. 00:48:01.800 --> 00:48:04.719 So it's not, yeah, so that's kind of-- you know, 00:48:04.719 --> 00:48:06.010 it's a good question you asked. 00:48:06.010 --> 00:48:09.670 But unfortunately, you kind of want 00:48:09.670 --> 00:48:14.110 to choose a case where you have perfect precision in terms 00:48:14.110 --> 00:48:17.090 of r raised to d is exactly what you need it to be. 00:48:17.090 --> 00:48:18.700 And in that case, this analysis works. 00:48:18.700 --> 00:48:20.117 AUDIENCE: So it works the max end. 00:48:20.117 --> 00:48:21.699 SRINI DEVADAS: That's it, the max end. 00:48:21.699 --> 00:48:22.760 That's exactly right. 00:48:22.760 --> 00:48:25.110 So I mean, 255 is substantially larger than 128. 00:48:25.110 --> 00:48:27.160 So it kind of makes sense that, as you said, 00:48:27.160 --> 00:48:30.014 that you would be able to shave off certain things. 00:48:30.014 --> 00:48:31.680 And so you can kind of do that, but it's 00:48:31.680 --> 00:48:33.010 a little more involved. 00:48:33.010 --> 00:48:36.730 But I think the intuition is important. 00:48:36.730 --> 00:48:40.060 And the intuition is simply that your worst-case number 00:48:40.060 --> 00:48:46.240 of drops, assuming you're being a little bit imprecise, 00:48:46.240 --> 00:48:52.000 would simply be r minus 1 times d in the worst case. 00:48:52.000 --> 00:48:57.760 And this would be true for all n which is less than r 00:48:57.760 --> 00:49:01.370 raised to d, for all n. 00:49:01.370 --> 00:49:04.390 So now, just in terms of numbers, 00:49:04.390 --> 00:49:07.990 I had r equals 4 and d equals 4. 00:49:07.990 --> 00:49:12.730 And so you end up getting 12 drops for the 128 case, all 00:49:12.730 --> 00:49:15.710 the way to 255 as well. 00:49:15.710 --> 00:49:18.040 You need 12 drops in the worst case, 00:49:18.040 --> 00:49:23.260 even for 255 floors, which is really what you were asking. 00:49:23.260 --> 00:49:23.990 So good. 00:49:23.990 --> 00:49:24.490 Right. 00:49:24.490 --> 00:49:26.330 That makes sense? 00:49:26.330 --> 00:49:30.100 So I'm not going to spend a whole lot of time on this code, 00:49:30.100 --> 00:49:31.630 but that's it. 00:49:31.630 --> 00:49:33.940 It's converting. 00:49:33.940 --> 00:49:37.540 I could've used a Python library function for that, 00:49:37.540 --> 00:49:39.866 but that's simply a conversion because I 00:49:39.866 --> 00:49:40.990 wanted to print things out. 00:49:40.990 --> 00:49:45.472 I don't even need that, this particular subroutine, 00:49:45.472 --> 00:49:46.930 which is essentially something that 00:49:46.930 --> 00:49:52.870 is doing conversion from decimal to r-ary arithmetic. 00:49:52.870 --> 00:49:57.130 And as you can see, you see I subtract and adding. 00:49:57.130 --> 00:49:59.770 The adding this is the incrementation that I just 00:49:59.770 --> 00:50:03.490 erased, but you do see it up there from 1 0 to 2 0. 00:50:03.490 --> 00:50:04.960 That's why you add. 00:50:04.960 --> 00:50:08.350 Sometimes you need to subtract, because things break. 00:50:08.350 --> 00:50:12.220 And when things break, you need to go back to the highest floor 00:50:12.220 --> 00:50:14.140 where the ball didn't break. 00:50:14.140 --> 00:50:17.350 So that's where you do the subtraction. 00:50:17.350 --> 00:50:22.780 And so this is how, if you ran this, 00:50:22.780 --> 00:50:26.200 it's not the greatest user interface, I'll admit. 00:50:26.200 --> 00:50:27.580 But that's not the point. 00:50:33.560 --> 00:50:34.420 The Crystal. 00:50:42.110 --> 00:50:47.950 So I'm going to do at 128 and 4, which is our example. 00:50:47.950 --> 00:50:50.250 And it says radix shows it as ball 4. 00:50:50.250 --> 00:50:52.090 Did the ball break? 00:50:52.090 --> 00:50:53.740 I say no. 00:50:53.740 --> 00:50:56.410 And then drop the ball. 00:50:56.410 --> 00:50:59.390 Did the ball Break if I say no here, then I'm done. 00:50:59.390 --> 00:51:01.480 So I'm going to go ahead and say yes. 00:51:01.480 --> 00:51:06.370 And then I'm going to say no and then no. 00:51:06.370 --> 00:51:07.720 I'm going to keep going-- 00:51:07.720 --> 00:51:13.030 no, no, no, no-- because this is the worst case. 00:51:20.770 --> 00:51:23.110 And I go to 11 in this case. 00:51:23.110 --> 00:51:26.891 So you do get close in the sense that Ganatra was asking, 00:51:26.891 --> 00:51:28.140 you know, could you shave off? 00:51:28.140 --> 00:51:31.090 You only shaved off one because of the imprecision. 00:51:31.090 --> 00:51:34.510 So there is a case where even though you would only 00:51:34.510 --> 00:51:38.460 get 12 for 255, but you do get 11 for 128. 00:51:38.460 --> 00:51:39.680 OK. 00:51:39.680 --> 00:51:40.180 Cool. 00:51:40.180 --> 00:51:40.680 Good. 00:51:40.680 --> 00:51:43.000 So I'll let you look at the code. 00:51:43.000 --> 00:51:47.894 If you guys can leave any time you want. 00:51:47.894 --> 00:51:49.310 I just wanted to mention, I wanted 00:51:49.310 --> 00:51:50.726 to go back to the one thing that I 00:51:50.726 --> 00:51:53.860 didn't get that I mentioned, but I haven't gotten to do, 00:51:53.860 --> 00:51:55.930 which is this is still-- 00:51:55.930 --> 00:51:57.750 you could still do better. 00:51:57.750 --> 00:52:00.760 You can still do better not because of what Ganatra 00:52:00.760 --> 00:52:03.480 mentioned-- you know, that's just that imprecision thing, 00:52:03.480 --> 00:52:05.860 but assuming even that was taken care 00:52:05.860 --> 00:52:07.930 of because the numbers worked out, 00:52:07.930 --> 00:52:11.200 You can do better because k can be made a different. 00:52:11.200 --> 00:52:12.970 You could do unequal intervals. 00:52:12.970 --> 00:52:15.820 So let me put that in the notes and you 00:52:15.820 --> 00:52:16.930 can read it by yourself. 00:52:16.930 --> 00:52:18.320 You had a question, Evan. 00:52:18.320 --> 00:52:19.170 AUDIENCE: Yeah. 00:52:19.170 --> 00:52:20.920 When you were going through the ball drops 00:52:20.920 --> 00:52:23.770 and you kept answering no, that the ball did not break, 00:52:23.770 --> 00:52:27.550 why did it start asking you or why did it start telling you 00:52:27.550 --> 00:52:29.400 where the next ball dropped? 00:52:29.400 --> 00:52:31.550 Because it went one, two, and then 00:52:31.550 --> 00:52:33.810 it went also to ball three and ball four. 00:52:33.810 --> 00:52:36.056 So will it just stay ball two the whole time it 00:52:36.056 --> 00:52:36.680 if never broke? 00:52:36.680 --> 00:52:38.180 SRINI DEVADAS: That's exactly right, 00:52:38.180 --> 00:52:40.150 but that's because I'm lazy. 00:52:42.740 --> 00:52:43.630 Yeah. 00:52:43.630 --> 00:52:45.580 So it'd be a fine exercise to fix that. 00:52:45.580 --> 00:52:47.437 I mean, that's not one of the exercises. 00:52:47.437 --> 00:52:49.770 That's how you turn, you know after 30 years of teaching 00:52:49.770 --> 00:52:51.269 whenever I get a question, I turn it 00:52:51.269 --> 00:52:52.940 into an opportunity for homework. 00:52:52.940 --> 00:52:53.631 OK. 00:52:53.631 --> 00:52:54.130 Right. 00:52:54.130 --> 00:52:54.820 That's how it works. 00:52:54.820 --> 00:52:55.060 All right. 00:52:55.060 --> 00:52:55.560 Good. 00:52:55.560 --> 00:52:57.360 See you guys next time.