A dual approach to detect polyhedral intersections in arbitrary dimensions

Abstract

This paper presents a dual approach to detect intersections of hyperplanes and convex polyhedra in arbitrary dimensions. Ind dimensions, the time complexities of the dual algorithms areO(2^d logn) for the hyperplane-polyhedron intersection problem, andO((2d)^d−1 log^d−1 n) for the polyhedron- polyhedron intersection problem. These results are the first of their kind ford > 3. In two dimensions, these time bounds are achieved with linear space and preprocessing. In three dimensions, the hyperplane-polyhedron intersection problem is also solved with linear space and preprocessing; quadratic space and preprocessing, however, is required for the polyhedron-polyhedron intersection problem. For generald, the dual algorithms require$O(n^{2^d } )$ space and preprocessing. All of these results readily extend to unbounded polyhedra.