Voronoi diagrams of moving points in the plane

Abstract

In this paper, we consider the dynamic Voronoi diagram problem. In this problem, a given set of planar points are moving and our objective is to find the Voronoi diagram of these moving points at any time t. A preprocessing algorithm and a query processing algorithm are presented in this paper. Assume that the points are in k-motion, and it takes O(k) time to find the roots of a polynomial with degree O(k). The preprocessing algorithm uses $O(k^2 n^3 logn\cdot2^{O(\alpha (n)^{5k + 1} )} )$ time to process moving functions of given points, and uses $O(k^2 n^3 \cdot2^{O(\alpha (n)^{5k + 1} )} )$ space to store the preprocessing result where α(n) is the functional inverse of Ackermann's function. The query processing algorithm is designed to report the Voronoi diagram of these points for a query time t. It takes O(n) time which is optimal.