Summary: Computational Aspects of Helly's Theorem and its Relatives
David Avis* and Michael E. Houle**
*School of Computer Science, McGill University
Montr'eal, Qu'ebec, Canada H3A 2A7
**Department of Computer Science, University of Newcastle,
Newcastle, New South Wales 2308, Australia
Abstract
This paper investigates computational aspects of the well­known convexity theorem due to
Helly, which states that the existence of a point in the common intersection of n convex sets is
guaranteed by the existence of points in the common intersection of each combination of d + 1
of these sets. Given an oracle which accepts d + 1 convex sets and either returns a point in
their common intersection, or reports its non­existence, we give two algorithms which compute a
point in the common intersection of n such sets. The first algorithm runs in O(n d+1 T ) time and
O(n d ) space, where T is the time required for a single call to the oracle. The second algorithm
is a multi­stage variant of the first by which the space complexity may be reduced to O(n) at
the expense of an increase in the time complexity by a factor independent of n.
We also show how these algorithms may be adapted to construct linear and spherical sepa­
rators of a collection of sets, and to construct a translate of a given object which either contains,
is contained by, or intersects a collection of convex sets.
1 Introduction

Source: Avis, David - School of Computer Science, McGill University

Collections: Computer Technologies and Information Sciences