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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an unpublished manuscript.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