From: Jeff Erickson
Subject: Re: Random sampling on multidimensional polytopes
Date: Wed, 03 Nov 1999 16:11:06 -0600
Newsgroups: comp.theory,sci.math
Nick Maclaren wrote:
> There are two known, simple algorithms that give "true" uniformity
> and independence, based only on a volume calculation breakdown.
> But that is, I agree, non-trivial to do efficiently in general -
> I can't remember its complexity, but O(N^5) sounds plausible.
Computing the exact volume of a polytope represented as the convex hull
of its vertices is a #P-complete problem, even if every coordinate is 0
or 1. No polynomial-time exact algorithm is known or likely to ever be
known.
> The two methods are, of course, decomposition into simplices and
> interior/exterior testing/rejection.
Triangulation (= decomposing into simplicies) is fine when the dimension
is small, but the number of simplices can grows exponentially with the
dimension. Specifically, decomposing a d-dimensional polytope with
either n facets or n vertices requires Theta(n^{\floor{d/2}}) simplices
in the worst case.
Interior/exterior testing/rejection suffers from similar problems. As
you've already pointed out, the problem is finding an appropriate
simpler bounding polytope. Unfortunately, as the dimension rises, it
becomes exponentially harder to approximate polytopes effectively.
--
Jeff Erickson mailto:jeffe@cs.uiuc.edu
Computer Science Department http://www.uiuc.edu/~jeffe
University of Illinois, Urbana-Champaign
==============================================================================
From: leydold@statrix2.wu-wien.ac.at (Josef Leydold)
Subject: Re: Random sampling on multidimensional polytopes
Date: 4 Nov 1999 06:50:05 GMT
Newsgroups: comp.theory,sci.math
Nick Maclaren (nmm1@cus.cam.ac.uk) wrote:
:
: In article <7vpp4b$lnm@hermes.acs.unt.edu>, srt@nospam.unt.edu writes:
: |> In comp.theory Vu Ha wrote:
: |>
: |> > I am interested in the problem of random sampling of points of a
: |> > multidimensional polytope, according to the uniform distribution.
: |>
:
: The two methods are, of course, decomposition into simplices and
: interior/exterior testing/rejection.
:
For simple polytopes a sweep plane algorithm might be a possible solution. See
J. Leydold and W. Hoermann, A Sweep-Plane Algorithm for Generating random
tuples in simple polytopes, Math. Comp. 67(224), pp. 1617--1635, 1998
The details of the algorithm are rather complicated. hope it helps.
Josef
--
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