Fast and efficient computation of additively weighted voronoi cells for applications in molecular biology

Abstract

This paper is concerned with the efficient computation of additively weighted Voronoi cells for applications in molecular biology. We propose a projection map for the representation of these cells leading to a surprising insight into their geometry. We present a randomized algorithm computing one such cell amidst n other spheres in expected time O(n2 log n). Since the best known upper bound on the complexity such a cell is O(n2), this is optimal up to a logarithmic factor. However, the experimentally observed behavior of the complexity of these cells is linear in n. In this case our algorithm performs the task in expected time O(n log2 n). A variant of this algorithm was implemented and performs well on problem instances from molecular biology.