Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.3 4.3bsd-beta 6/6/85; site ucbvax.BERKELEY.EDU Path: utzoo!watmath!clyde!burl!ulysses!ucbvax!ernie!raghavan From: raghavan@ernie.BERKELEY.EDU (Prabhakar Raghavan) Newsgroups: net.math Subject: Re: Probability Problem (reward!) Message-ID: <10863@ucbvax.BERKELEY.EDU> Date: Fri, 1-Nov-85 00:59:53 EST Article-I.D.: ucbvax.10863 Posted: Fri Nov 1 00:59:53 1985 Date-Received: Sat, 2-Nov-85 07:17:49 EST References: <511@aero.ARPA> Sender: usenet@ucbvax.BERKELEY.EDU Reply-To: raghavan@ernie.UUCP (Prabhakar Raghavan) Organization: University of California, Berkeley Lines: 33 In article <511@aero.ARPA> sinclair@aero.UUCP (William S. Sinclair) writes: >We select n points from a 2-dimensional uniform distribution, say {0 to 1} >in both x and y. Now we form the convex hull of these, and it will contain >k of the points. What is the probability that k of the n points will be on >the convex hull? > >Hint: if n=3, p(k,n)=1.0 >Also, as n goes to infinity, the expected value for k goes up like sqrt(n). The latter hint is incorrect. Bentley, Kung, Schkolnick and Thompson (JACM 1978) showed that the expected number of points on the hull is O(log n) for a large class of distributions on the unnit square including the uniform distribution. In d-dimensions, it's O(log**d-1 n). The expected value O(sqrt(n)) holds when the points are uniformly distributed in a circle (I vaguely recall that the result is due to Renyi and somebody else). Intuitively, the corners in the square allow a few points to `dominate' a large number of the other points, resulting in fewer points on the hull; in a circle, no such corners exist. More recently, Luc Devroye of McGill Univ Computer Science has shown that the variance of the number of points on the hull (uniform on a square case) is O(log**2 n). Thus, with very high probability, the number of points on the hull is O(log n). I don't have a published reference for this, although I've read an unpublished manuscript. (For those that are interested, Devroye also showed that for points uniformly distributed in a d-dimensional hypercube, the variance is O(log**(2d-2) n) ). The significance of the result of Bentley &al is that there are convex hull algorithms that can be proved to run in expected linear time. .... Prabhakar Raghavan