WEBVTT 00:00:05.767 --> 00:00:08.100 SRINI DEVADAS: How many of you have attended a celebrity 00:00:08.100 --> 00:00:10.520 party? 00:00:10.520 --> 00:00:11.630 I haven't. 00:00:11.630 --> 00:00:13.460 Well, if you count MIT professors 00:00:13.460 --> 00:00:16.550 as being celebrities, I've been to a lot of them, 00:00:16.550 --> 00:00:20.870 but that's not the case, right? 00:00:20.870 --> 00:00:25.034 People don't really think of us as celebrities, 00:00:25.034 --> 00:00:25.950 which is fine with me. 00:00:31.940 --> 00:00:37.720 So this puzzle, I call it the best time to party. 00:00:37.720 --> 00:00:52.536 And-- I'm having a little trouble erasing the-- 00:00:52.536 --> 00:00:56.570 oh you're right. 00:00:56.570 --> 00:00:59.390 There's another one? 00:00:59.390 --> 00:01:00.190 All right. 00:01:00.190 --> 00:01:02.500 Well, I'm probably going to run out of this. 00:01:02.500 --> 00:01:03.460 But you know what? 00:01:03.460 --> 00:01:05.140 This time I'll use-- 00:01:05.140 --> 00:01:07.780 I don't think I need this super thick chalk. 00:01:07.780 --> 00:01:09.310 There's a slightly-- 00:01:09.310 --> 00:01:10.060 I'll use this one. 00:01:23.310 --> 00:01:26.150 So setup here-- with all of these things, 00:01:26.150 --> 00:01:28.490 there's always a little bit of setup-- 00:01:28.490 --> 00:01:31.920 is, you got a ticket to this party. 00:01:31.920 --> 00:01:34.950 There are going to be all these celebrities. 00:01:34.950 --> 00:01:39.720 And it's a timed ticket with some flexibility. 00:01:39.720 --> 00:01:42.155 You only get to stay for a certain amount of time 00:01:42.155 --> 00:01:44.030 before you have to leave and other people are 00:01:44.030 --> 00:01:47.180 going to come in and hobnob with these celebrities. 00:01:47.180 --> 00:01:51.300 But you get to pick the particular time that you go, 00:01:51.300 --> 00:01:52.010 right? 00:01:52.010 --> 00:01:53.960 And so, initially, we'll set things up 00:01:53.960 --> 00:01:56.020 in a fairly straightforward way. 00:01:56.020 --> 00:02:02.750 We're going to say that you have an hour, a particular hour, 00:02:02.750 --> 00:02:05.250 maybe it's 6:00 to 7:00, 7:00 to 8:00, what have you. 00:02:05.250 --> 00:02:06.320 You get to choose. 00:02:06.320 --> 00:02:07.650 That's part of the puzzle. 00:02:07.650 --> 00:02:10.520 How do you pick the best time to party? 00:02:10.520 --> 00:02:13.370 And you do have a schedule. 00:02:13.370 --> 00:02:16.010 And amazingly, these celebrities are going 00:02:16.010 --> 00:02:17.570 to stick to those schedules. 00:02:17.570 --> 00:02:20.380 But the schedule has been published with respect 00:02:20.380 --> 00:02:23.780 to when particular celebrities are going to be at this party. 00:02:23.780 --> 00:02:24.830 All right? 00:02:24.830 --> 00:02:30.500 And as you can imagine, as the title of the puzzle gives away, 00:02:30.500 --> 00:02:35.960 you want to choose the hour such that the maximum number 00:02:35.960 --> 00:02:41.990 of celebrities-- so you're not particularly interested 00:02:41.990 --> 00:02:47.900 in particular celebrities or one specific celebrity, 00:02:47.900 --> 00:02:50.150 you just wanted sort of max out number of selfies 00:02:50.150 --> 00:02:52.330 so you can put it on your Facebook page, Instagram, 00:02:52.330 --> 00:02:53.420 what have you. 00:02:53.420 --> 00:02:55.520 And you have that schedule and you're 00:02:55.520 --> 00:02:58.040 trying to optimize the number. 00:02:58.040 --> 00:03:00.860 So here you go. 00:03:00.860 --> 00:03:03.380 We're going to have a-- 00:03:03.380 --> 00:03:06.980 let's just solve this again manually first, 00:03:06.980 --> 00:03:09.680 and then we'll think of algorithms. 00:03:09.680 --> 00:03:11.300 We're going to have intervals again. 00:03:14.090 --> 00:03:22.190 And so think of these as hours. 00:03:22.190 --> 00:03:24.225 Beyonce comes in at 6:00, leaves at 7. 00:03:29.036 --> 00:03:30.285 Taylor from 7:00 through 9:00. 00:03:33.680 --> 00:03:36.886 Brad 10:00 to 11:00. 00:03:40.780 --> 00:03:44.982 Katie 10:00 to 12:00. 00:03:47.940 --> 00:03:52.367 Tom 8:00 to 10:00. 00:04:10.410 --> 00:04:11.850 Going to run out of room here. 00:04:11.850 --> 00:04:12.480 All right. 00:04:12.480 --> 00:04:15.210 So that's the schedule that's published, right? 00:04:15.210 --> 00:04:18.060 And we can think of these as hours-- 00:04:18.060 --> 00:04:19.019 comes and goes. 00:04:19.019 --> 00:04:22.079 There's one small subtlety, which 00:04:22.079 --> 00:04:24.930 is, I'm going to think of these as 00:04:24.930 --> 00:04:30.790 closed intervals on the left end and open at the right end. 00:04:30.790 --> 00:04:35.460 So all that means is, if this is 6:00 PM, 00:04:35.460 --> 00:04:39.375 and if you come in at 6:00 PM on the dot, well, 00:04:39.375 --> 00:04:40.750 you certainly get to see Beyonce. 00:04:40.750 --> 00:04:43.610 And if you come in at 6:59, you get to see Beyonce, 00:04:43.610 --> 00:04:46.640 but if you come in at 7:00, she's gone. 00:04:46.640 --> 00:04:47.910 Right? 00:04:47.910 --> 00:04:51.210 And the same thing with Taylor at 9:00. 00:04:51.210 --> 00:04:56.280 So you you have to choose your time properly. 00:04:56.280 --> 00:04:59.460 So what's the best time? 00:04:59.460 --> 00:05:02.120 You get to go for any hour that you choose. 00:05:02.120 --> 00:05:04.890 What's the best time to go to get 00:05:04.890 --> 00:05:07.190 the maximum number of selfies? 00:05:07.190 --> 00:05:08.910 Anyone? 00:05:08.910 --> 00:05:11.860 You can do this manually. 00:05:11.860 --> 00:05:16.960 Tell me what the time is and then I'll ask you 00:05:16.960 --> 00:05:18.940 what your algorithm was? 00:05:18.940 --> 00:05:19.690 Right? 00:05:19.690 --> 00:05:23.160 So-- someone who hasn't answered yet. 00:05:28.140 --> 00:05:29.634 Yep? 00:05:29.634 --> 00:05:31.740 AUDIENCE: Either 6:00 or 10:00. 00:05:31.740 --> 00:05:32.740 SRINI DEVADAS: Either 6:00 or-- what's your name? 00:05:32.740 --> 00:05:33.550 AUDIENCE: Kevin. 00:05:33.550 --> 00:05:33.860 SRINI DEVADAS: Oh! 00:05:33.860 --> 00:05:34.130 Kevin. 00:05:34.130 --> 00:05:34.670 Right. 00:05:34.670 --> 00:05:35.169 Sorry. 00:05:37.212 --> 00:05:38.170 You're the other Kevin. 00:05:38.170 --> 00:05:38.940 All right. 00:05:38.940 --> 00:05:41.472 So-- 6:00 o'clock or 10:00 o'clock, 00:05:41.472 --> 00:05:43.430 because at 6:00 o'clock-- so who are the people 00:05:43.430 --> 00:05:44.513 you'd see at 6:00 o'clock? 00:05:44.513 --> 00:05:46.770 AUDIENCE: Beyonce and Alicia Keys. 00:05:46.770 --> 00:05:48.966 SRINI DEVADAS: That's only two. 00:05:48.966 --> 00:05:49.850 AUDIENCE: Yeah. 00:05:49.850 --> 00:05:52.210 SRINI DEVADAS: So can you do better than two? 00:05:52.210 --> 00:05:54.340 Can someone do better than what Kevin did? 00:05:59.610 --> 00:06:00.300 Yeah, go ahead. 00:06:00.300 --> 00:06:01.743 Kanishka 00:06:01.743 --> 00:06:03.186 AUDIENCE: That will be Brad. 00:06:03.186 --> 00:06:05.591 So at 10:00, you'll get Brad. 00:06:05.591 --> 00:06:08.689 Katy, and Drake. 00:06:08.689 --> 00:06:09.480 SRINI DEVADAS: Yes. 00:06:09.480 --> 00:06:10.396 So you could do three. 00:06:10.396 --> 00:06:12.670 At 10:00, you would get Brad. 00:06:12.670 --> 00:06:16.360 You would get Katie, and then you'd also get Drake. 00:06:16.360 --> 00:06:17.210 OK? 00:06:17.210 --> 00:06:18.190 So that's three. 00:06:18.190 --> 00:06:18.890 OK? 00:06:18.890 --> 00:06:24.890 So that's the best you can do in this puzzle, right? 00:06:24.890 --> 00:06:27.080 But you may have hundreds of celebrities 00:06:27.080 --> 00:06:28.830 and you want to pick the right time. 00:06:28.830 --> 00:06:31.550 So obviously, we want to write a computer program here, right? 00:06:31.550 --> 00:06:35.180 What algorithm, Kevin or Kanishka 00:06:35.180 --> 00:06:37.790 did you use to do this? 00:06:37.790 --> 00:06:40.550 How do you do it in your head? 00:06:40.550 --> 00:06:44.966 AUDIENCE: I first of all saw which 00:06:44.966 --> 00:06:47.800 were the good candidates or shortlisting them essentially. 00:06:47.800 --> 00:06:49.840 SRINI DEVADAS: Did you go, either one of you, 00:06:49.840 --> 00:06:52.760 did you go hour by hour? 00:06:52.760 --> 00:06:54.140 Did you go 6:00 o'clock-- 00:06:54.140 --> 00:06:55.370 how many people are there? 00:06:55.370 --> 00:06:57.680 Or did you try and intersect, or what did you do? 00:07:02.440 --> 00:07:05.160 What's a reasonable way of proceeding here? 00:07:05.160 --> 00:07:05.810 Right? 00:07:05.810 --> 00:07:09.137 What's a reasonable way of proceeding? 00:07:09.137 --> 00:07:10.720 Someone other than-- yeah, back there. 00:07:10.720 --> 00:07:14.309 AUDIENCE: Look at the most frequent amount of time. 00:07:14.309 --> 00:07:16.100 SRINI DEVADAS: The frequent amount of time? 00:07:16.100 --> 00:07:18.440 So look at the hours. 00:07:18.440 --> 00:07:20.304 You say 6:00 o'clock and you try and see 00:07:20.304 --> 00:07:21.470 how many sixes are up there? 00:07:21.470 --> 00:07:22.722 Or-- right. 00:07:22.722 --> 00:07:23.430 What's your name? 00:07:23.430 --> 00:07:24.734 AUDIENCE: Nisha. 00:07:24.734 --> 00:07:26.150 SRINI DEVADAS: So Nisha says she's 00:07:26.150 --> 00:07:28.250 going to go look for 6:00 o'clock 00:07:28.250 --> 00:07:30.601 and see if many sixes are up there. 00:07:30.601 --> 00:07:31.100 Right? 00:07:31.100 --> 00:07:32.570 And there's two sixes up there. 00:07:32.570 --> 00:07:35.250 And maybe that's why Kevin said six, right? 00:07:35.250 --> 00:07:40.760 On the other hand, clearly that didn't quite work in this case, 00:07:40.760 --> 00:07:44.660 because what happened was, 10:00 o'clock-- there's only two 00:07:44.660 --> 00:07:45.240 of those. 00:07:45.240 --> 00:07:47.210 And so that looks good, too, 6:00 and 10:00, 00:07:47.210 --> 00:07:52.580 but you missed the fact that 10 exists in 9:00 and 11:00, 00:07:52.580 --> 00:07:53.210 correct? 00:07:53.210 --> 00:07:55.085 So you're not explicitly seeing these things. 00:07:57.890 --> 00:07:59.360 But that gives you an algorithm. 00:07:59.360 --> 00:08:00.943 I mean, it does give you an algorithm. 00:08:00.943 --> 00:08:03.920 The algorithm is-- let me go ahead 00:08:03.920 --> 00:08:06.750 and do a little bit better than what you described, 00:08:06.750 --> 00:08:10.040 a little more relaxed check, right? 00:08:10.040 --> 00:08:14.990 It's not just an identical, I want six to be in the table. 00:08:14.990 --> 00:08:17.600 But I want to see if six is contained 00:08:17.600 --> 00:08:21.470 in the interval associated with any particular celebrity. 00:08:21.470 --> 00:08:22.040 Correct? 00:08:22.040 --> 00:08:25.520 So if I did six, I'd say six is contained in this interval. 00:08:25.520 --> 00:08:27.020 It's not contained in this interval. 00:08:27.020 --> 00:08:30.000 Not, not, not, not, yes. 00:08:30.000 --> 00:08:30.500 Right? 00:08:30.500 --> 00:08:31.970 So I'd get two. 00:08:31.970 --> 00:08:34.039 And then when I've finally got to 10, 00:08:34.039 --> 00:08:38.460 I'd get contained, contained, contained, and I'd get three. 00:08:38.460 --> 00:08:39.020 Right? 00:08:39.020 --> 00:08:42.890 So there's clearly an algorithm here 00:08:42.890 --> 00:08:47.660 that depends on the number of hours that you're looking at. 00:08:47.660 --> 00:08:52.730 Depends, really, on the granularity of time, OK? 00:08:52.730 --> 00:08:57.560 So this particular algorithm would say, let me go ahead 00:08:57.560 --> 00:09:03.500 and enumerate the different times that I could possibly go. 00:09:03.500 --> 00:09:04.400 I could start 6:00. 00:09:04.400 --> 00:09:06.320 I could start at 7:00. 00:09:06.320 --> 00:09:07.940 But maybe we'll start at 6:30. 00:09:07.940 --> 00:09:10.240 I could go from 6:30 to 7:30. 00:09:10.240 --> 00:09:12.410 You're allowed to do that, right? 00:09:12.410 --> 00:09:15.830 The point is, if you stick with the original description, 00:09:15.830 --> 00:09:18.350 you could go ahead and enumerate 6:00 through 12:00, 00:09:18.350 --> 00:09:20.870 and then you could generate a bunch of different numbers 00:09:20.870 --> 00:09:22.880 associated with the number of celebrities 00:09:22.880 --> 00:09:27.570 and you could go ahead and pick what time you want to go. 00:09:27.570 --> 00:09:28.430 Right? 00:09:28.430 --> 00:09:32.930 So it's a little bit like our previous puzzle in the sense 00:09:32.930 --> 00:09:37.850 that there's a relatively straightforward strategy that 00:09:37.850 --> 00:09:42.230 enumerates different hours and goes through and does 00:09:42.230 --> 00:09:45.650 a bunch of computation and then picks the maximum, 00:09:45.650 --> 00:09:46.490 in this case. 00:09:46.490 --> 00:09:49.120 Previously, it was the minimum for, you will all conform. 00:09:49.120 --> 00:09:51.320 Here, it's the maximum, right? 00:09:51.320 --> 00:09:55.940 What is a disadvantage of this particular algorithm 00:09:55.940 --> 00:09:57.330 that we just described? 00:09:57.330 --> 00:10:01.777 What is one potential disadvantage? 00:10:01.777 --> 00:10:02.277 Yeah? 00:10:05.560 --> 00:10:06.320 Go ahead, Fadi. 00:10:06.320 --> 00:10:09.270 AUDIENCE: Well, it's not linear, so you have 00:10:09.270 --> 00:10:12.436 to make multiple passes on. 00:10:12.436 --> 00:10:14.269 First of all, you have to enumerate 00:10:14.269 --> 00:10:15.352 all of the possible times. 00:10:15.352 --> 00:10:16.972 And then for every time, you have 00:10:16.972 --> 00:10:20.220 to go through the list over and over again. 00:10:20.220 --> 00:10:22.050 So there's a lot of computational-- 00:10:24.787 --> 00:10:26.620 SRINI DEVADAS: There's a lot of computation. 00:10:26.620 --> 00:10:29.682 There's a lot of computational overhead. 00:10:29.682 --> 00:10:31.140 Let me show you what the code looks 00:10:31.140 --> 00:10:35.700 like for the straightforward algorithm, all right? 00:10:35.700 --> 00:10:40.530 And so it's going to, again, be intuitive. 00:10:40.530 --> 00:10:43.560 It's going to do exactly what we just described, right? 00:10:43.560 --> 00:10:45.960 And so if you look at what you have up there-- 00:10:45.960 --> 00:10:52.470 schedule up there is simply a set of intervals, right? 00:10:52.470 --> 00:10:54.330 And they're tuples. 00:10:54.330 --> 00:10:57.450 And so I have something that's bigger than this example here, 00:10:57.450 --> 00:10:58.540 but it includes that. 00:10:58.540 --> 00:11:00.570 And there's a whole bunch of things in there. 00:11:00.570 --> 00:11:04.020 And now I'm going to look at-- 00:11:04.020 --> 00:11:07.424 this is just a start time for a particular celebrity. 00:11:07.424 --> 00:11:09.090 This is the end time for that celebrity. 00:11:09.090 --> 00:11:10.350 I'm just setting that up. 00:11:10.350 --> 00:11:14.130 And I'm going to go ahead and-- 00:11:14.130 --> 00:11:19.807 I'm looking at for each c in schedule, 00:11:19.807 --> 00:11:20.890 I'm going to look at this. 00:11:20.890 --> 00:11:21.973 I'm going to look at that. 00:11:21.973 --> 00:11:23.400 I'm going to look at that. 00:11:23.400 --> 00:11:30.570 I'm going to figure out what the start time is and the end is. 00:11:30.570 --> 00:11:33.990 And I'm just doing this simply because I 00:11:33.990 --> 00:11:40.110 want to figure out what the earliest start time for any 00:11:40.110 --> 00:11:41.315 of these celebrities are. 00:11:41.315 --> 00:11:42.630 So in this case, it's six. 00:11:42.630 --> 00:11:44.985 And the latest time that a celebrity is around 00:11:44.985 --> 00:11:45.990 is 12, right? 00:11:45.990 --> 00:11:48.170 So that's all of this first thing does. 00:11:48.170 --> 00:11:51.540 It's just finding the range that I have to deal with. 00:11:51.540 --> 00:11:53.430 So there's not much there. 00:11:53.430 --> 00:11:55.290 I'm going to go ahead and compute what's 00:11:55.290 --> 00:11:57.000 called the celebrity density. 00:11:57.000 --> 00:11:58.530 And I'll show you the code for that. 00:11:58.530 --> 00:12:01.150 But that's essentially what we describe, 00:12:01.150 --> 00:12:04.590 which is for a particular hour between start and end, 00:12:04.590 --> 00:12:07.380 how many celebrities do I see at that particular hour, right? 00:12:07.380 --> 00:12:09.840 And then for the next hour, how many celebrities do I see? 00:12:09.840 --> 00:12:14.040 I'm just calling it the density, right? 00:12:14.040 --> 00:12:20.130 And then, the count is a list that 00:12:20.130 --> 00:12:23.490 is going to have celebrity densities for particular times 00:12:23.490 --> 00:12:26.400 that correspond to the indices of count. 00:12:26.400 --> 00:12:28.500 And I'm just going to go through that list 00:12:28.500 --> 00:12:31.590 and I'm going to figure out the time that has max count, 00:12:31.590 --> 00:12:34.380 or the maximum number of celebrities. 00:12:34.380 --> 00:12:36.840 And there's a couple of different ways of doing that. 00:12:36.840 --> 00:12:40.640 Don't worry about that, but I'm happy to answer questions 00:12:40.640 --> 00:12:43.810 if you have them. 00:12:43.810 --> 00:12:46.080 And then, I just say the best time 00:12:46.080 --> 00:12:48.810 to end the party is at this time. 00:12:48.810 --> 00:12:51.960 Time, which was discovered here, and the number 00:12:51.960 --> 00:12:55.050 of celebrities who are going to be attending is max count. 00:12:55.050 --> 00:12:55.900 OK. 00:12:55.900 --> 00:12:59.890 So this is essentially the code for the algorithm. 00:12:59.890 --> 00:13:08.050 And this thing here is doing the slightly more relaxed check 00:13:08.050 --> 00:13:11.460 that I alluded to, which was like when you have 6:00, 00:13:11.460 --> 00:13:14.820 you're not just looking for 6:00 here, 00:13:14.820 --> 00:13:18.600 you're looking for 6:00 within this interval 00:13:18.600 --> 00:13:19.870 that looks like this. 00:13:19.870 --> 00:13:23.100 So 6:00 is definitely within this, but it would not be-- 00:13:26.760 --> 00:13:34.230 6:00 is not inside 5:00 and 6:00 as I described to you before, 00:13:34.230 --> 00:13:34.890 right? 00:13:34.890 --> 00:13:36.690 So that's simply the check. 00:13:36.690 --> 00:13:39.450 That's why you have a less than equal to here 00:13:39.450 --> 00:13:41.670 and a strictly greater than over here, 00:13:41.670 --> 00:13:45.790 just to take care of that closed and open part of it. 00:13:45.790 --> 00:13:47.850 So again, if you don't understand 00:13:47.850 --> 00:13:50.160 every nuance in this code, it's not that big a deal. 00:13:50.160 --> 00:13:53.740 But hopefully you have the overall picture, right? 00:13:53.740 --> 00:13:55.590 Makes sense, right? 00:13:55.590 --> 00:13:59.040 So as you can imagine, there's a much better way. 00:13:59.040 --> 00:14:00.840 There's a much better way that isn't 00:14:00.840 --> 00:14:03.900 as exhaustive as this one. 00:14:03.900 --> 00:14:08.850 And do people have a sense of what a better way would be, 00:14:08.850 --> 00:14:10.330 or would you like a hint? 00:14:10.330 --> 00:14:14.640 And anyone want to conjecture a different way of solving this 00:14:14.640 --> 00:14:15.990 problem that's-- 00:14:15.990 --> 00:14:16.895 yeah, go ahead. 00:14:16.895 --> 00:14:19.120 AUDIENCE: The exhaustiveness of this algorithm? 00:14:19.120 --> 00:14:20.350 SRINI DEVADAS: Ah, so the exhaustiveness 00:14:20.350 --> 00:14:22.120 of this algorithm is as follows. 00:14:22.120 --> 00:14:25.290 So basically what I'm saying is-- 00:14:25.290 --> 00:14:29.130 what I'm saying is I'm going to go look at the range of times. 00:14:29.130 --> 00:14:31.320 And all of these are per hour, right? 00:14:31.320 --> 00:14:32.830 Everything is per hour. 00:14:32.830 --> 00:14:34.830 I'm going to look at the range of times, 00:14:34.830 --> 00:14:39.870 and I'm going to go ahead and include 6:00 00:14:39.870 --> 00:14:41.220 through 12:00 here. 00:14:41.220 --> 00:14:42.900 And I'm going to go 6:00, and I'm 00:14:42.900 --> 00:14:47.640 going to go figure out at 6:00 how many celebrities exist. 00:14:47.640 --> 00:14:49.290 And then at 7:00, how many celebrities 00:14:49.290 --> 00:14:52.380 exist, and then at 8:00 all the way to 12:00, right? 00:14:52.380 --> 00:14:56.400 And I'm going to get in count, which is that list there, I'm 00:14:56.400 --> 00:14:58.220 going to get for each-- 00:14:58.220 --> 00:15:02.730 for count 6:00, I'm going to say two. 00:15:02.730 --> 00:15:07.960 And for count 10:00, I'm going to get three, 00:15:07.960 --> 00:15:09.540 because 10 is the hour. 00:15:09.540 --> 00:15:11.290 And then I'm going to go through that list 00:15:11.290 --> 00:15:14.200 and figure out that 10 had the maximum number in it. 00:15:14.200 --> 00:15:16.400 And that's the exhaustiveness, right? 00:15:16.400 --> 00:15:18.550 That make sense? 00:15:18.550 --> 00:15:20.840 So if we're good with that, this algorithm works. 00:15:20.840 --> 00:15:24.330 But it's a little painful, right? 00:15:24.330 --> 00:15:27.880 So what-- any ideas as to how we could do things 00:15:27.880 --> 00:15:32.410 quite differently, which would be more efficient? 00:15:32.410 --> 00:15:34.030 And let me give out the word. 00:15:34.030 --> 00:15:37.150 I want an incremental way of doing this. 00:15:40.040 --> 00:15:41.310 How could I do this? 00:15:41.310 --> 00:15:45.380 I have a starting time, and I want 00:15:45.380 --> 00:15:49.430 to compute the same densities in an incremental way. 00:15:52.020 --> 00:15:56.300 Someone else other than Fadi? 00:15:56.300 --> 00:15:58.139 Yeah, back there. 00:15:58.139 --> 00:16:00.430 AUDIENCE: Well, you have to start the n time thing just 00:16:00.430 --> 00:16:06.470 for each list of whatever tuple or whatever you could 00:16:06.470 --> 00:16:09.040 keep like a count-- a different count, 00:16:09.040 --> 00:16:13.079 a dictionary of the start time, and then increment through each 00:16:13.079 --> 00:16:15.579 of the start times until the end and just add one each time. 00:16:15.579 --> 00:16:18.465 So you're basically iterating through each set of times 00:16:18.465 --> 00:16:21.050 and finding the count of each time 00:16:21.050 --> 00:16:22.560 and whichever time is the most. 00:16:22.560 --> 00:16:23.930 SRINI DEVADAS: You're absolutely on the right track. 00:16:23.930 --> 00:16:26.130 And it turns out you don't even need dictionaries. 00:16:26.130 --> 00:16:28.530 You can do this with just plain lists. 00:16:28.530 --> 00:16:31.680 But the idea here is actually a very compelling one. 00:16:31.680 --> 00:16:33.930 It's something that you'll see in other algorithms. 00:16:33.930 --> 00:16:37.950 Whenever you have something where you're doing repeated 00:16:37.950 --> 00:16:40.170 computation, there's always-- 00:16:40.170 --> 00:16:45.400 many a time, there's a way of removing redundancy and turning 00:16:45.400 --> 00:16:48.090 into incremental computation, right? 00:16:48.090 --> 00:16:51.190 And so the insight here is the following. 00:16:51.190 --> 00:16:55.290 The only time that celebrity density, which 00:16:55.290 --> 00:16:58.640 is the number of celebrities that are in the room, changes 00:16:58.640 --> 00:17:00.810 is obviously when a celebrity enters 00:17:00.810 --> 00:17:03.570 or when a celebrity leaves, right? 00:17:03.570 --> 00:17:04.380 I mean, that's it. 00:17:04.380 --> 00:17:06.799 I mean, we're all celebrities here. 00:17:06.799 --> 00:17:10.170 And I guess we did have someone leave, so there you go. 00:17:10.170 --> 00:17:12.520 And then maybe someone is going to come in, right? 00:17:12.520 --> 00:17:17.579 So I just need to monitor sort of input and output, 00:17:17.579 --> 00:17:20.369 or entry and exit. 00:17:20.369 --> 00:17:22.920 And I also know what the endpoints 00:17:22.920 --> 00:17:29.250 are with respect to the 6:00, 6:00 PM versus 12:00, correct? 00:17:29.250 --> 00:17:32.300 So what I can do here is the following. 00:17:37.440 --> 00:17:46.310 Suppose I had a chart that looks like this, right? 00:17:46.310 --> 00:17:53.700 And what I'm drawing here are these intervals. 00:17:53.700 --> 00:17:55.640 They aren't exactly what I had up there, 00:17:55.640 --> 00:17:57.850 but I'm drawing those intervals up, right? 00:17:57.850 --> 00:18:04.970 So you could think of this, if you'd like, as being 6:00. 00:18:04.970 --> 00:18:08.810 You can think of these two lined up 00:18:08.810 --> 00:18:11.150 as being 7:00, and so on and so forth. 00:18:14.150 --> 00:18:15.530 So I could draw these out. 00:18:15.530 --> 00:18:18.070 And I don't really have to draw these out 00:18:18.070 --> 00:18:21.900 in a computer program, but I can certainly draw them out 00:18:21.900 --> 00:18:25.550 in terms of giving you the intuition, right? 00:18:25.550 --> 00:18:28.510 So what I'm going to do is I'm going to start with zero. 00:18:28.510 --> 00:18:31.830 So before 6:00, there's absolutely no one in the room, 00:18:31.830 --> 00:18:34.130 so I'm at zero. 00:18:34.130 --> 00:18:36.170 And the moment I see-- 00:18:38.840 --> 00:18:42.770 I take one of these things, and if I see a celebrity, 00:18:42.770 --> 00:18:44.210 I can increment the count. 00:18:44.210 --> 00:18:49.580 So here at 5:00 I'm at zero, at 6:00 I get one. 00:18:49.580 --> 00:18:55.870 And at 7:00, no one has left, and two more people 00:18:55.870 --> 00:18:59.920 have come in, so I get three. 00:18:59.920 --> 00:19:03.620 And now, the next thing happens out here. 00:19:03.620 --> 00:19:08.740 Let's say this one here is 8:00, I would get four for that. 00:19:08.740 --> 00:19:11.770 And then out here it's 9:00. 00:19:11.770 --> 00:19:16.180 I go 4 minus 1 equals 3, because this celebrity left. 00:19:18.950 --> 00:19:21.580 So essentially, what I want to do 00:19:21.580 --> 00:19:27.820 is not have to compute this over and over for every time running 00:19:27.820 --> 00:19:29.950 through all of the list of celebrities 00:19:29.950 --> 00:19:31.457 like I did before, for 6:00, 7:00, 00:19:31.457 --> 00:19:33.040 through all the celebrities for 7:00-- 00:19:33.040 --> 00:19:34.540 I meant through all the celebrities. 00:19:34.540 --> 00:19:42.163 But if I sort the celebrities in increasing order of start 00:19:42.163 --> 00:19:45.970 times, essentially this picture is important 00:19:45.970 --> 00:19:51.820 and it requires sorting, because I went from left to right. 00:19:51.820 --> 00:19:54.340 And when I go from left to right, 00:19:54.340 --> 00:19:56.620 I'm essentially saying I want to see 00:19:56.620 --> 00:20:00.670 things that come earlier before the ones that come later, 00:20:00.670 --> 00:20:01.210 right? 00:20:01.210 --> 00:20:03.730 That's essentially what time goes to the right, right? 00:20:03.730 --> 00:20:05.420 So I'm going from left to right. 00:20:05.420 --> 00:20:13.600 So if I take these intervals and I sort them, and basically sort 00:20:13.600 --> 00:20:19.960 them by the start times, then all I need to do 00:20:19.960 --> 00:20:22.750 is, in the sorted list of intervals, 00:20:22.750 --> 00:20:24.160 I just showed you the computation 00:20:24.160 --> 00:20:26.406 that we have to perform. 00:20:26.406 --> 00:20:28.030 The computation that we have to perform 00:20:28.030 --> 00:20:30.760 is-- this was the first interval in the sorted list 00:20:30.760 --> 00:20:34.420 of intervals, because it was the leftmost interval. 00:20:34.420 --> 00:20:38.620 And I take that and I go ahead and increment it to one, right? 00:20:38.620 --> 00:20:42.070 Then, the next thing I do is I don't 00:20:42.070 --> 00:20:45.490 have to worry about any time. 00:20:45.490 --> 00:20:47.260 Even if there was a time, if this was 6:00 00:20:47.260 --> 00:20:50.860 and this was 8:00, for example, as long as there's 00:20:50.860 --> 00:20:53.830 no celebrity that came in at 7:00, 00:20:53.830 --> 00:20:56.470 there's no reason for me to change anything 00:20:56.470 --> 00:20:58.360 in my data structure, right? 00:20:58.360 --> 00:21:00.820 The only times that are interesting 00:21:00.820 --> 00:21:04.180 are when celebrities enter and leave. 00:21:04.180 --> 00:21:06.430 So the other nice thing about this algorithm 00:21:06.430 --> 00:21:09.580 is that it will work if Beyonce came 00:21:09.580 --> 00:21:13.660 in at 6:39:39, 39 seconds-- 00:21:13.660 --> 00:21:18.550 39 minutes, 39 seconds and left at 6:51 whatever, 00:21:18.550 --> 00:21:22.540 because I'm only concerned with the entry 00:21:22.540 --> 00:21:24.400 of a celebrity and the exit of a celebrity 00:21:24.400 --> 00:21:30.130 that correspond to these points that you see here, right? 00:21:30.130 --> 00:21:33.460 Does the algorithm make sense from an intuitive standpoint? 00:21:33.460 --> 00:21:34.960 Do you see what's happening here? 00:21:34.960 --> 00:21:37.420 Hopefully the picture gives you a sense 00:21:37.420 --> 00:21:40.480 for how you're doing the correct computation. 00:21:40.480 --> 00:21:42.520 And the only times that things change 00:21:42.520 --> 00:21:45.580 are when-- at the end points of these intervals, right? 00:21:45.580 --> 00:21:48.640 When you see someone, increment, and when someone leaves, 00:21:48.640 --> 00:21:49.960 decrement, right? 00:21:49.960 --> 00:21:54.940 So this set of numbers that you generate 00:21:54.940 --> 00:21:58.060 is essentially the density down at the bottom 00:21:58.060 --> 00:22:02.980 that corresponds to, at particular times, not 00:22:02.980 --> 00:22:05.830 necessarily on the hour, but whenever the celebrities come 00:22:05.830 --> 00:22:07.180 and go. 00:22:07.180 --> 00:22:10.630 And so that list that you get is all 00:22:10.630 --> 00:22:14.350 that you need to look at to figure out what the maximum is, 00:22:14.350 --> 00:22:14.980 right? 00:22:14.980 --> 00:22:17.650 So ultimately down below, in this particular example, 00:22:17.650 --> 00:22:20.860 perhaps four was the maximum. 00:22:20.860 --> 00:22:22.690 And it started with zero. 00:22:22.690 --> 00:22:26.121 It climbs up, maybe goes to four, maybe goes to five, 00:22:26.121 --> 00:22:28.120 and then goes down again, maybe climbs up again. 00:22:28.120 --> 00:22:30.400 Who cares? 00:22:30.400 --> 00:22:33.040 Once you get to the point where you've written the program, 00:22:33.040 --> 00:22:35.500 size doesn't matter anymore, right? 00:22:35.500 --> 00:22:37.120 So the last thing I'll do is I'll 00:22:37.120 --> 00:22:38.330 show you the code for this. 00:22:38.330 --> 00:22:39.910 It will be up on the website. 00:22:39.910 --> 00:22:44.170 And feel free to come to office hours 00:22:44.170 --> 00:22:46.750 and ask me questions about the code, 00:22:46.750 --> 00:22:49.790 or before lecture tomorrow. 00:22:49.790 --> 00:22:53.110 But this is the entirety of the code. 00:22:53.110 --> 00:22:58.140 And I'm not even using the built in sort functions in Python. 00:22:58.140 --> 00:22:59.890 I went ahead and wrote it just to give you 00:22:59.890 --> 00:23:02.560 a sense of how long the code would be. 00:23:02.560 --> 00:23:04.450 But it's a much more elegant algorithm. 00:23:04.450 --> 00:23:06.910 It would run faster on large examples. 00:23:06.910 --> 00:23:09.250 And I don't want to get into asymptotic complexity. 00:23:09.250 --> 00:23:12.440 That's really beyond this class. 00:23:12.440 --> 00:23:14.890 But as you can imagine, rather than going 00:23:14.890 --> 00:23:18.040 through the list of celebrities over and over, 00:23:18.040 --> 00:23:21.880 we're actually doing sorting exactly once, right? 00:23:21.880 --> 00:23:23.230 So that's better. 00:23:23.230 --> 00:23:26.420 But essentially, what happens here is simple. 00:23:26.420 --> 00:23:30.230 You go ahead and you create each of these times. 00:23:30.230 --> 00:23:32.590 So this is particular celebrity, particular time, 00:23:32.590 --> 00:23:34.510 and this is the start point of the celebrity. 00:23:34.510 --> 00:23:36.160 So you have to mark that. 00:23:36.160 --> 00:23:38.470 This is very different from that. 00:23:38.470 --> 00:23:42.970 What happens here is I am adding. 00:23:42.970 --> 00:23:45.850 What happens here is I'm subtracting. 00:23:45.850 --> 00:23:47.859 So I have to know whether this point is 00:23:47.859 --> 00:23:49.150 very different from that point. 00:23:49.150 --> 00:23:51.430 This is the same as that, same as that. 00:23:51.430 --> 00:23:52.990 And all of these are similar, but I 00:23:52.990 --> 00:23:56.290 have to differentiate between the two for obvious reasons. 00:23:56.290 --> 00:23:58.750 So I have to have start and end. 00:23:58.750 --> 00:24:01.270 And then I go ahead and take these times. 00:24:01.270 --> 00:24:03.810 Each of these times that you see with these black dots 00:24:03.810 --> 00:24:06.051 here, they all need to be sorted, 00:24:06.051 --> 00:24:08.050 because that's the only way I'm going to go from 00:24:08.050 --> 00:24:09.850 left to right, right? 00:24:09.850 --> 00:24:14.020 That's the only way, by going through that list of times. 00:24:14.020 --> 00:24:16.300 So I'm going to go ahead and do that sorting. 00:24:16.300 --> 00:24:18.310 And I could have done times dot sort by the way. 00:24:18.310 --> 00:24:22.750 I don't have to have sort list, but I just, as I said, 00:24:22.750 --> 00:24:25.360 I wanted to show you the code without using 00:24:25.360 --> 00:24:27.580 any Python library functions. 00:24:27.580 --> 00:24:33.400 And then choose time is simply looking through that list, 00:24:33.400 --> 00:24:36.460 finding that maximum number, be it four or five, 00:24:36.460 --> 00:24:38.890 and returning the index of that, because that tells you 00:24:38.890 --> 00:24:41.170 what the particular time is. 00:24:41.170 --> 00:24:44.050 And that time will always be-- 00:24:44.050 --> 00:24:44.920 can someone tell me? 00:24:44.920 --> 00:24:46.390 Last question and I'll let you go. 00:24:49.680 --> 00:24:53.170 What am I going to get-- on any given example, what 00:24:53.170 --> 00:24:59.610 am I going to get as the best time, any given example? 00:24:59.610 --> 00:25:02.580 I'll get you if Ganatra. 00:25:02.580 --> 00:25:03.550 Yeah? 00:25:03.550 --> 00:25:05.940 AUDIENCE: A time when more people come and leave. 00:25:05.940 --> 00:25:08.190 SRINI DEVADAS: A time when more people come and leave. 00:25:08.190 --> 00:25:10.212 That's correct, but I want something stronger. 00:25:10.212 --> 00:25:10.920 What's your name? 00:25:10.920 --> 00:25:11.730 AUDIENCE: Josh. 00:25:11.730 --> 00:25:12.210 SRINI DEVADAS: Josh? 00:25:12.210 --> 00:25:13.918 Yeah, Josh is right, but I want something 00:25:13.918 --> 00:25:18.630 even stronger than a time that more people come than leave. 00:25:18.630 --> 00:25:23.730 I mean, it's correct, but a variant observation. 00:25:23.730 --> 00:25:25.970 Someone else? 00:25:25.970 --> 00:25:28.160 I want to say something about-- 00:25:28.160 --> 00:25:31.640 there's two sets of points here, right? 00:25:31.640 --> 00:25:33.970 There's two sets of points here. 00:25:33.970 --> 00:25:36.290 Which set of points are going to be 00:25:36.290 --> 00:25:41.939 the ones that are in play, with respect to the maximum time. 00:25:41.939 --> 00:25:43.230 That's maybe a better question. 00:25:46.200 --> 00:25:47.217 Go ahead. 00:25:47.217 --> 00:25:48.800 AUDIENCE: The arrival time of someone. 00:25:48.800 --> 00:25:49.580 SRINI DEVADAS: That's right. 00:25:49.580 --> 00:25:51.360 That arrival time of a celebrity, right? 00:25:51.360 --> 00:25:54.630 And it's true that what you said, George is correct. 00:25:54.630 --> 00:25:56.250 Ultimately, the best time is going 00:25:56.250 --> 00:26:01.740 to be when the maximum number of arrivals happen. 00:26:01.740 --> 00:26:05.217 But the bottom line is, I'm never going to-- 00:26:05.217 --> 00:26:07.050 this is actually an interesting observation, 00:26:07.050 --> 00:26:09.522 and there's an exercise related to this observation 00:26:09.522 --> 00:26:10.980 for this puzzle that you might want 00:26:10.980 --> 00:26:14.100 to look at that I'll put up on the website 00:26:14.100 --> 00:26:15.650 as soon as we're done here. 00:26:15.650 --> 00:26:18.570 You're always going to pick one of these beginning times, 00:26:18.570 --> 00:26:20.370 because that's when the increment happens. 00:26:20.370 --> 00:26:22.453 You'd never pick one of these, because that's when 00:26:22.453 --> 00:26:24.390 a decrement happens, right? 00:26:24.390 --> 00:26:25.860 So that's all I wanted to say. 00:26:25.860 --> 00:26:28.170 And the reason I wanted to say that, as I said, 00:26:28.170 --> 00:26:31.470 there's an extra question on top of this, and yet 00:26:31.470 --> 00:26:33.750 another algorithm to solve this that 00:26:33.750 --> 00:26:37.020 is really based on that final observation that we made. 00:26:37.020 --> 00:26:37.650 All right. 00:26:37.650 --> 00:26:38.610 Well, thank you. 00:26:38.610 --> 00:26:41.880 Sorry for going over, but as I said, this is IAP. 00:26:41.880 --> 00:26:43.700 Forgive me.