WEBVTT 00:00:00.000 --> 00:00:01.940 align:middle line:90% [SQUEAKING] 00:00:01.940 --> 00:00:04.365 align:middle line:90% [RUSTLING] 00:00:04.365 --> 00:00:05.820 align:middle line:90% [CLICKING] 00:00:05.820 --> 00:00:11.650 align:middle line:90% 00:00:11.650 --> 00:00:13.270 align:middle line:84% PROFESSOR: In this video, we'll look 00:00:13.270 --> 00:00:16.810 align:middle line:84% at a powerful tool in the probabilistic method known 00:00:16.810 --> 00:00:20.470 align:middle line:84% as a bounded differences inequality, which 00:00:20.470 --> 00:00:24.460 align:middle line:84% also goes by the name Azuma-Hoeffding inequality. 00:00:24.460 --> 00:00:27.370 align:middle line:84% This inequality tells us that if we 00:00:27.370 --> 00:00:33.320 align:middle line:84% have a function of independent random variables and such 00:00:33.320 --> 00:00:36.280 align:middle line:84% that this function doesn't change very much if we just 00:00:36.280 --> 00:00:38.830 align:middle line:84% change one of the inputs of the function, 00:00:38.830 --> 00:00:41.420 align:middle line:84% then the output of this function, 00:00:41.420 --> 00:00:43.690 align:middle line:84% given the random input, is fairly 00:00:43.690 --> 00:00:46.240 align:middle line:90% concentrated around its mean. 00:00:46.240 --> 00:00:50.110 align:middle line:84% Let's look at the actual statement of this theorem. 00:00:50.110 --> 00:00:55.990 align:middle line:84% We have x1, a random variable taking values in the set 00:00:55.990 --> 00:01:00.830 align:middle line:84% or in the probability space omega 1, and so on. 00:01:00.830 --> 00:01:04.010 align:middle line:90% So x1, x2, to xn. 00:01:04.010 --> 00:01:06.400 align:middle line:84% So these are independent random variables. 00:01:06.400 --> 00:01:10.450 align:middle line:84% And it's very important that they're independent. 00:01:10.450 --> 00:01:17.140 align:middle line:84% We have a function, f, which takes as input n coordinates 00:01:17.140 --> 00:01:19.180 align:middle line:90% and outputs a real number. 00:01:19.180 --> 00:01:22.060 align:middle line:84% And the hypothesis of this theorem 00:01:22.060 --> 00:01:24.160 align:middle line:84% is that this function f satisfies 00:01:24.160 --> 00:01:26.020 align:middle line:90% the following property-- 00:01:26.020 --> 00:01:32.260 align:middle line:84% that if we look at two different outputs of f on inputs x 00:01:32.260 --> 00:01:37.840 align:middle line:84% and y such that x and y differ on exactly one coordinate. 00:01:37.840 --> 00:01:41.660 align:middle line:84% So by only changing a single coordinate of f, 00:01:41.660 --> 00:01:44.110 align:middle line:84% the assumption is that the value of f 00:01:44.110 --> 00:01:47.590 align:middle line:90% changes by no more than one. 00:01:47.590 --> 00:01:54.520 align:middle line:84% So f in some sense is fairly smooth to the fluctuations 00:01:54.520 --> 00:01:55.670 align:middle line:90% in the input. 00:01:55.670 --> 00:01:59.620 align:middle line:84% If you change only one input coordinate, then the output of f 00:01:59.620 --> 00:02:05.230 align:middle line:84% does not change too much, does not change by more than one. 00:02:05.230 --> 00:02:08.289 align:middle line:84% OK, what is the conclusion of the theorem? 00:02:08.289 --> 00:02:14.120 align:middle line:84% It says that the random variable obtained by evaluating f 00:02:14.120 --> 00:02:18.290 align:middle line:84% on these independent random input coordinates-- 00:02:18.290 --> 00:02:20.420 align:middle line:84% that's what we call this output, z-- 00:02:20.420 --> 00:02:23.330 align:middle line:84% and this random variable z satisfies 00:02:23.330 --> 00:02:26.210 align:middle line:84% the following concentration inequality. 00:02:26.210 --> 00:02:29.610 align:middle line:84% For every non-negative real lambda, 00:02:29.610 --> 00:02:34.700 align:middle line:84% the probability that z exceeds its expectation 00:02:34.700 --> 00:02:38.910 align:middle line:84% by at least lambda is at most this quantity here, 00:02:38.910 --> 00:02:45.000 align:middle line:84% which goes down rapidly as lambda gets large. 00:02:45.000 --> 00:02:49.830 align:middle line:84% And also, we also have a lower tail concentration bound, 00:02:49.830 --> 00:02:53.910 align:middle line:84% which says that the probability that z is significantly 00:02:53.910 --> 00:03:00.060 align:middle line:84% below its expectation by more than lambda, at least lambda, 00:03:00.060 --> 00:03:04.860 align:middle line:84% is this probability is upper bounded by this same quantity 00:03:04.860 --> 00:03:10.230 align:middle line:84% which, again, decays extremely quickly when lambda is large. 00:03:10.230 --> 00:03:14.610 align:middle line:84% So this is the statement of the bounded differences inequality. 00:03:14.610 --> 00:03:18.240 align:middle line:84% Again, the intuition here is that given independent inputs 00:03:18.240 --> 00:03:21.630 align:middle line:84% to a function which satisfies this property, that it does not 00:03:21.630 --> 00:03:25.920 align:middle line:84% change by more than one, upon changing any single coordinate, 00:03:25.920 --> 00:03:28.740 align:middle line:84% the output random variable is very 00:03:28.740 --> 00:03:31.780 align:middle line:90% concentrated around its mean. 00:03:31.780 --> 00:03:33.370 align:middle line:84% In the rest of this video, I want 00:03:33.370 --> 00:03:37.870 align:middle line:84% to present three applications of this inequality. 00:03:37.870 --> 00:03:40.930 align:middle line:84% The first application is meant to illustrate 00:03:40.930 --> 00:03:44.440 align:middle line:84% a very simple example of a function 00:03:44.440 --> 00:03:47.060 align:middle line:90% where we can apply the theorem. 00:03:47.060 --> 00:03:53.690 align:middle line:84% And this function is simply the function taking Boolean input, 00:03:53.690 --> 00:03:59.650 align:middle line:84% so n Boolean inputs, and outputs a real number obtained by simply 00:03:59.650 --> 00:04:01.305 align:middle line:90% adding up the input numbers. 00:04:01.305 --> 00:04:07.440 align:middle line:90% 00:04:07.440 --> 00:04:11.490 align:middle line:84% In other words, this is a sum of n different coin 00:04:11.490 --> 00:04:14.740 align:middle line:84% tosses, each coin toss resulting in a zero or a one. 00:04:14.740 --> 00:04:17.730 align:middle line:84% And you can check that this function has 00:04:17.730 --> 00:04:20.680 align:middle line:84% the property that we require in this theorem, 00:04:20.680 --> 00:04:24.580 align:middle line:84% namely that if you flip just one of the input coordinates, 00:04:24.580 --> 00:04:26.610 align:middle line:90% the output changes by-- 00:04:26.610 --> 00:04:31.200 align:middle line:84% well, in this case, exactly one, but certainly no more than one. 00:04:31.200 --> 00:04:34.440 align:middle line:84% Now we can apply the Azuma-Hoeffding inequality 00:04:34.440 --> 00:04:37.990 align:middle line:84% or the bounded differences inequality to this function, 00:04:37.990 --> 00:04:43.960 align:middle line:84% and that gets us a tail bound on the binomial distribution. 00:04:43.960 --> 00:04:53.810 align:middle line:90% 00:04:53.810 --> 00:05:00.200 align:middle line:84% This case, you know that the expectation of z is exactly n 00:05:00.200 --> 00:05:05.660 align:middle line:84% over 2, and it tells you that the deviation cannot exceed very 00:05:05.660 --> 00:05:11.760 align:middle line:84% much beyond something on the order of the square root of n. 00:05:11.760 --> 00:05:15.750 align:middle line:90% 00:05:15.750 --> 00:05:22.020 align:middle line:84% This bound is also known as Chernoff bound. 00:05:22.020 --> 00:05:28.800 align:middle line:84% And in fact, the proof of the bounded differences inequality 00:05:28.800 --> 00:05:32.970 align:middle line:84% is very similar to the proof of the Chernoff bound, which you 00:05:32.970 --> 00:05:34.325 align:middle line:90% can view in a different video. 00:05:34.325 --> 00:05:37.480 align:middle line:90% 00:05:37.480 --> 00:05:40.080 align:middle line:84% Let me now go on to the next example. 00:05:40.080 --> 00:05:45.380 align:middle line:90% 00:05:45.380 --> 00:05:49.820 align:middle line:84% The next example concerns a problem called the coupon 00:05:49.820 --> 00:05:50.635 align:middle line:90% collector problem. 00:05:50.635 --> 00:05:56.320 align:middle line:90% 00:05:56.320 --> 00:06:02.920 align:middle line:84% The setup is that we have independent random numbers 00:06:02.920 --> 00:06:07.180 align:middle line:84% s1 through sn, and they're each chosen uniformly 00:06:07.180 --> 00:06:09.250 align:middle line:90% from the numbers 1 through n. 00:06:09.250 --> 00:06:13.260 align:middle line:84% So these numbers are uniformly and independently chosen. 00:06:13.260 --> 00:06:20.990 align:middle line:90% 00:06:20.990 --> 00:06:25.280 align:middle line:84% So you can imagine a setup as having n different coupons. 00:06:25.280 --> 00:06:30.780 align:middle line:84% And each time you draw a random coupon, see what it is, 00:06:30.780 --> 00:06:32.790 align:middle line:84% return it to the box, and draw it again. 00:06:32.790 --> 00:06:36.650 align:middle line:84% And these s1 through sn's is a list of coupons 00:06:36.650 --> 00:06:40.430 align:middle line:90% that you draw from this box. 00:06:40.430 --> 00:06:43.610 align:middle line:84% And the random variable that we're interested in, z, 00:06:43.610 --> 00:06:47.510 align:middle line:84% is the number of missing coupons, the number of coupons 00:06:47.510 --> 00:06:50.185 align:middle line:84% that you have not seen through this process. 00:06:50.185 --> 00:06:53.360 align:middle line:90% 00:06:53.360 --> 00:06:59.660 align:middle line:84% In other words, this is the number of elements of 1 00:06:59.660 --> 00:07:09.380 align:middle line:84% through n that are not among the elements s1 through sn. 00:07:09.380 --> 00:07:11.260 align:middle line:84% So this is a number of missing coupons. 00:07:11.260 --> 00:07:14.680 align:middle line:84% It is a random variable, because s1 through sn are random. 00:07:14.680 --> 00:07:17.300 align:middle line:90% 00:07:17.300 --> 00:07:22.880 align:middle line:84% We wish to understand how z is concentrated around its mean. 00:07:22.880 --> 00:07:27.650 align:middle line:84% And for that, we can apply the bounded differences inequality. 00:07:27.650 --> 00:07:33.860 align:middle line:84% Note that this function, given viewing this quantity 00:07:33.860 --> 00:07:40.140 align:middle line:84% as a function from s1 through sn as inputs to the output number, 00:07:40.140 --> 00:07:43.460 align:middle line:84% this function satisfies the required hypothesis. 00:07:43.460 --> 00:07:48.170 align:middle line:84% Namely, if you change one of the si's, you 00:07:48.170 --> 00:07:51.620 align:middle line:84% do not change the number of missing coupons 00:07:51.620 --> 00:07:54.070 align:middle line:90% by more than one. 00:07:54.070 --> 00:07:57.570 align:middle line:84% Number of coupons, missing coupons might not change at all. 00:07:57.570 --> 00:08:00.125 align:middle line:84% But it cannot change by more than one. 00:08:00.125 --> 00:08:03.050 align:middle line:90% 00:08:03.050 --> 00:08:07.520 align:middle line:84% So this means that we can apply the bounded differences 00:08:07.520 --> 00:08:10.790 align:middle line:84% inequality to deduce the conclusion 00:08:10.790 --> 00:08:18.210 align:middle line:84% that the probability that z deviates from its expectation 00:08:18.210 --> 00:08:23.050 align:middle line:84% by more than lambda is at most 2, 00:08:23.050 --> 00:08:26.610 align:middle line:84% because here we're using upper and lower bounds simultaneously. 00:08:26.610 --> 00:08:32.890 align:middle line:84% Times 2 to the minus e 2 lambda squared over n. 00:08:32.890 --> 00:08:36.370 align:middle line:90% 00:08:36.370 --> 00:08:40.330 align:middle line:84% And the expectation is something that we 00:08:40.330 --> 00:08:43.240 align:middle line:84% can calculate pretty easily using linearity 00:08:43.240 --> 00:08:47.330 align:middle line:84% of expectations of these n different numbers, 00:08:47.330 --> 00:08:49.240 align:middle line:90% these n different coupons. 00:08:49.240 --> 00:08:54.250 align:middle line:84% Each single coupon is missing with probability 1 minus 1 00:08:54.250 --> 00:09:00.670 align:middle line:84% over n to the n, because this is the probability 00:09:00.670 --> 00:09:11.600 align:middle line:84% that a specific coupon, coupon i, is missing, 00:09:11.600 --> 00:09:16.280 align:middle line:84% which is the event that coupon i is not 00:09:16.280 --> 00:09:20.780 align:middle line:84% drawn in each of the n different random draws. 00:09:20.780 --> 00:09:27.230 align:middle line:84% And this quantity is very close to n over e. 00:09:27.230 --> 00:09:27.730 align:middle line:90% OK. 00:09:27.730 --> 00:09:31.040 align:middle line:84% So this is an application of the bounded differences 00:09:31.040 --> 00:09:36.450 align:middle line:84% inequality to a function which is not linear, like before. 00:09:36.450 --> 00:09:38.840 align:middle line:84% So the first example is a much simpler example 00:09:38.840 --> 00:09:43.110 align:middle line:84% because the function f is simply a sum of its inputs. 00:09:43.110 --> 00:09:46.610 align:middle line:84% And here, there's a more complicated function. 00:09:46.610 --> 00:09:53.870 align:middle line:84% Our last example involves an even subtler function. 00:09:53.870 --> 00:10:00.170 align:middle line:84% And here, the result is a classic theorem in probabilistic 00:10:00.170 --> 00:10:05.625 align:middle line:84% combinatorics due to Shamir and Spencer from the '80s. 00:10:05.625 --> 00:10:12.570 align:middle line:90% 00:10:12.570 --> 00:10:16.080 align:middle line:84% The theorem concerns the chromatic number 00:10:16.080 --> 00:10:17.620 align:middle line:90% of a random graph. 00:10:17.620 --> 00:10:38.880 align:middle line:84% So let z be the chromatic number of the random graph, G n, p OK, 00:10:38.880 --> 00:10:42.890 align:middle line:84% so G n, p is the Erdos-Rényi random graph obtained by taking 00:10:42.890 --> 00:10:46.550 align:middle line:84% n vertices and putting an edge between every pair of vertices 00:10:46.550 --> 00:10:48.410 align:middle line:90% with probability p. 00:10:48.410 --> 00:10:53.570 align:middle line:84% So throw a probability p coin for each possible edge 00:10:53.570 --> 00:10:57.350 align:middle line:84% independently and construct a random graph this way. 00:10:57.350 --> 00:11:00.440 align:middle line:84% And then z is the chromatic number of this random graph. 00:11:00.440 --> 00:11:03.140 align:middle line:84% So it is the minimum number of colors required 00:11:03.140 --> 00:11:07.310 align:middle line:84% to color all the vertices so that no two adjacent vertices 00:11:07.310 --> 00:11:08.620 align:middle line:90% receive the same color. 00:11:08.620 --> 00:11:11.470 align:middle line:90% 00:11:11.470 --> 00:11:13.510 align:middle line:90% This is some random variable. 00:11:13.510 --> 00:11:16.900 align:middle line:84% And this random variable is pretty complicated. 00:11:16.900 --> 00:11:19.630 align:middle line:90% It's pretty hard to analyze. 00:11:19.630 --> 00:11:23.630 align:middle line:84% But nevertheless, using the bounded differences inequality, 00:11:23.630 --> 00:11:27.280 align:middle line:84% let us deduce the following concentration bound, 00:11:27.280 --> 00:11:32.320 align:middle line:84% showing that z typically is not too far 00:11:32.320 --> 00:11:35.620 align:middle line:90% away from its expectation. 00:11:35.620 --> 00:11:41.340 align:middle line:84% And specifically, we have the bound 00:11:41.340 --> 00:11:45.300 align:middle line:84% saying that z deviates from expectation by more 00:11:45.300 --> 00:11:48.270 align:middle line:90% than this quantity here. 00:11:48.270 --> 00:11:51.990 align:middle line:84% And this event has probability at most 2 times 00:11:51.990 --> 00:11:54.980 align:middle line:90% e to the minus 2 lambda squared. 00:11:54.980 --> 00:11:58.540 align:middle line:90% 00:11:58.540 --> 00:12:00.165 align:middle line:90% OK, so let's prove this theorem. 00:12:00.165 --> 00:12:04.060 align:middle line:90% 00:12:04.060 --> 00:12:07.270 align:middle line:84% The interesting part of this proof 00:12:07.270 --> 00:12:10.660 align:middle line:84% is how to set up the function, f, 00:12:10.660 --> 00:12:15.190 align:middle line:84% so that we can apply the bounded differences inequality. 00:12:15.190 --> 00:12:20.060 align:middle line:84% So right now, z is some quantity which seems kind of complicated, 00:12:20.060 --> 00:12:22.420 align:middle line:84% and it's based on something that is random. 00:12:22.420 --> 00:12:26.980 align:middle line:84% So how can we phrase it in terms of independent random variables 00:12:26.980 --> 00:12:29.590 align:middle line:84% in a way so that we can apply the bounded differences 00:12:29.590 --> 00:12:31.360 align:middle line:90% inequality? 00:12:31.360 --> 00:12:37.540 align:middle line:84% Well, one way to do it, and it's a natural first attempt, 00:12:37.540 --> 00:12:44.590 align:middle line:84% is to view z as a function with n, choose two inputs, 00:12:44.590 --> 00:12:48.330 align:middle line:84% one input for each possible edge of the random graph. 00:12:48.330 --> 00:12:51.110 align:middle line:90% 00:12:51.110 --> 00:12:52.970 align:middle line:84% That is a valid choice in the sense 00:12:52.970 --> 00:12:57.180 align:middle line:84% that it satisfies the bounded differences condition, 00:12:57.180 --> 00:12:59.390 align:middle line:84% but it will not give the correct bound 00:12:59.390 --> 00:13:02.630 align:middle line:84% because it will turn out to have way too many variables 00:13:02.630 --> 00:13:04.680 align:middle line:90% in this function. 00:13:04.680 --> 00:13:06.390 align:middle line:84% We'll take a look at a different method 00:13:06.390 --> 00:13:10.290 align:middle line:84% that will be able to provide us with the desired bound. 00:13:10.290 --> 00:13:15.300 align:middle line:84% And there is some neat idea here on how to cluster 00:13:15.300 --> 00:13:18.310 align:middle line:90% the random variables together. 00:13:18.310 --> 00:13:27.640 align:middle line:84% So we will represent our graph on n vertices labeled 1 00:13:27.640 --> 00:13:30.430 align:middle line:90% through n. 00:13:30.430 --> 00:13:33.970 align:middle line:84% And this graph, which has some edges and we'll 00:13:33.970 --> 00:13:36.670 align:middle line:84% need to represent these edges, we'll 00:13:36.670 --> 00:13:50.640 align:middle line:84% represent them as an element of the following product set-- 00:13:50.640 --> 00:13:56.480 align:middle line:90% 00:13:56.480 --> 00:14:00.890 align:middle line:84% omega 1 times omega 2-- so these are Cartesian products-- 00:14:00.890 --> 00:14:02.990 align:middle line:90% times omega 3. 00:14:02.990 --> 00:14:08.300 align:middle line:90% So 1, 2, omega n minus one. 00:14:08.300 --> 00:14:12.920 align:middle line:84% And how we're going to encode the graph 00:14:12.920 --> 00:14:17.480 align:middle line:84% using the element of this product set is as follows-- 00:14:17.480 --> 00:14:27.825 align:middle line:84% omega 1 will be the set 0 comma 1, and this set will record-- 00:14:27.825 --> 00:14:31.810 align:middle line:90% 00:14:31.810 --> 00:14:34.480 align:middle line:84% either 0 or 1, which one gets chosen-- 00:14:34.480 --> 00:14:38.800 align:middle line:84% will record whether there is an edge between the first vertex 00:14:38.800 --> 00:14:42.130 align:middle line:90% and the second vertex. 00:14:42.130 --> 00:14:47.470 align:middle line:84% So if we choose 0, then there's no edge between 1 and 2. 00:14:47.470 --> 00:14:52.290 align:middle line:84% If we choose 1, then there is an edge between 1 and 2. 00:14:52.290 --> 00:15:02.970 align:middle line:84% Omega 2 is the product set of 0, 1 with itself. 00:15:02.970 --> 00:15:09.750 align:middle line:84% And the two bits encodes whether there 00:15:09.750 --> 00:15:12.120 align:middle line:84% is an edge between 1 and 3, and also 00:15:12.120 --> 00:15:14.700 align:middle line:84% whether there is an edge between 2 and 3. 00:15:14.700 --> 00:15:20.400 align:middle line:84% That information is encoded in the element of omega 2 00:15:20.400 --> 00:15:23.180 align:middle line:90% that we choose. 00:15:23.180 --> 00:15:33.320 align:middle line:84% And omega 3 is now 0 comma 1, raise to the third power. 00:15:33.320 --> 00:15:39.820 align:middle line:84% And here, these three bits record 00:15:39.820 --> 00:15:45.700 align:middle line:84% whether the three edges from 1, 2, and 3, to the fourth vertex 00:15:45.700 --> 00:15:50.510 align:middle line:84% are included or not included in this graph and so on. 00:15:50.510 --> 00:16:00.570 align:middle line:90% 00:16:00.570 --> 00:16:05.070 align:middle line:84% OK, so every graph on n-labeled vertices 00:16:05.070 --> 00:16:09.930 align:middle line:84% can be represented as an element of this product set and vice 00:16:09.930 --> 00:16:11.850 align:middle line:90% versa. 00:16:11.850 --> 00:16:16.320 align:middle line:84% Basically, we are clustering the edges together 00:16:16.320 --> 00:16:22.540 align:middle line:84% according to the right end point of its edge 00:16:22.540 --> 00:16:24.860 align:middle line:84% according to this vertex order of the vertices. 00:16:24.860 --> 00:16:35.290 align:middle line:90% 00:16:35.290 --> 00:16:38.470 align:middle line:90% So why is this useful? 00:16:38.470 --> 00:16:41.710 align:middle line:84% If we have two graphs, g and g prime, 00:16:41.710 --> 00:16:54.225 align:middle line:84% that differ on edges around only one vertex-- 00:16:54.225 --> 00:16:58.580 align:middle line:90% 00:16:58.580 --> 00:17:03.450 align:middle line:84% so if g and g prime differ on edges around only one 00:17:03.450 --> 00:17:10.760 align:middle line:84% vertex, then the chromatic number, or chi 00:17:10.760 --> 00:17:18.060 align:middle line:84% of g minus chi of g prime, this, in absolute value, 00:17:18.060 --> 00:17:21.099 align:middle line:90% is at most one. 00:17:21.099 --> 00:17:26.200 align:middle line:84% So if you have two graphs and one can be obtained from 00:17:26.200 --> 00:17:29.210 align:middle line:84% the other by modifying edges around a single vertex, 00:17:29.210 --> 00:17:33.020 align:middle line:84% then their chromatic numbers cannot differ by more than one. 00:17:33.020 --> 00:17:35.950 align:middle line:84% And the reason is that we could have just 00:17:35.950 --> 00:17:42.040 align:middle line:84% chosen a new color for the vertex that is involved, 00:17:42.040 --> 00:17:47.470 align:middle line:84% so we would not need to use more than one color change to go 00:17:47.470 --> 00:17:51.710 align:middle line:90% from one graph to another. 00:17:51.710 --> 00:17:55.280 align:middle line:84% OK, so this is a important fact that turns out 00:17:55.280 --> 00:17:57.380 align:middle line:90% to be incredibly useful. 00:17:57.380 --> 00:18:02.720 align:middle line:84% And it's useful because if we have a graph represented 00:18:02.720 --> 00:18:06.080 align:middle line:84% as an element of this product set, 00:18:06.080 --> 00:18:12.920 align:middle line:84% then having another graph that's obtained by just changing one 00:18:12.920 --> 00:18:17.810 align:middle line:84% coordinate results in a graph where 00:18:17.810 --> 00:18:20.270 align:middle line:84% the only changes to the graph itself are 00:18:20.270 --> 00:18:24.240 align:middle line:90% edges around a single vertex. 00:18:24.240 --> 00:18:30.980 align:middle line:84% And therefore, the function, f, that sends 00:18:30.980 --> 00:18:35.710 align:middle line:84% an element of this to an element of the reals-- 00:18:35.710 --> 00:18:38.270 align:middle line:84% basically, the chromatic number of the graph-- 00:18:38.270 --> 00:18:42.846 align:middle line:84% satisfies the bounded differences hypothesis. 00:18:42.846 --> 00:18:45.470 align:middle line:90% 00:18:45.470 --> 00:18:49.160 align:middle line:84% And by applying the theorem, we can then 00:18:49.160 --> 00:18:52.500 align:middle line:84% deduce the concentration bound here. 00:18:52.500 --> 00:18:54.960 align:middle line:84% So that finishes the proof of this theorem 00:18:54.960 --> 00:18:57.720 align:middle line:84% on the concentration of the chromatic number 00:18:57.720 --> 00:19:00.480 align:middle line:90% of a random graph. 00:19:00.480 --> 00:19:01.000 align:middle line:90% All right. 00:19:01.000 --> 00:19:04.020 align:middle line:84% So in this video, we saw the statement 00:19:04.020 --> 00:19:06.420 align:middle line:84% of the bounded differences inequality, which 00:19:06.420 --> 00:19:09.570 align:middle line:84% is a important and versatile tool that 00:19:09.570 --> 00:19:13.740 align:middle line:84% is used all over probabilistic combinatorics. 00:19:13.740 --> 00:19:18.090 align:middle line:84% It says that the concentration of a function 00:19:18.090 --> 00:19:20.310 align:middle line:90% of independent inputs-- 00:19:20.310 --> 00:19:24.300 align:middle line:84% or this, as long as the function has the property that it doesn't 00:19:24.300 --> 00:19:27.490 align:middle line:84% change very much if you only change a single coordinate, 00:19:27.490 --> 00:19:30.030 align:middle line:84% then the output as a random variable 00:19:30.030 --> 00:19:33.450 align:middle line:84% is highly concentrated around its mean. 00:19:33.450 --> 00:19:35.940 align:middle line:84% And we saw three different examples 00:19:35.940 --> 00:19:40.040 align:middle line:84% applying this bounded differences inequality. 00:19:40.040 --> 00:19:46.000 align:middle line:90%