Kinetic data structures for all nearest neighbors and closest pair in the plane

Abstract

This paper presents a kinetic data structure (KDS) for solutions to the all nearest neighbors problem and the closest pair problem in the plane. For a set P of n moving points where the trajectory of each point is an algebraic function of constant maximum degree s, our kinetic algorithm uses O(n) space and O(n log n) preprocessing time, and processes O(n2β22s+2(n)log n) events with total processing time O(n2β22s+2(n)log2 n), where βs(n) is an extremely slow-growing function. In terms of the KDS performance criteria, our KDS is efficient, responsive (in an amortized sense), and compact.Our deterministic kinetic algorithm for the all nearest neighbors problem improves by an O(log2 n) factor the previous randomized kinetic algorithm by Agarwal, Kaplan, and Sharir. The improvement is obtained by using a new sparse graph representation, the Pie Delaunay graph, to reduce the problem to one-dimensional range searching, as opposed to using two-dimensional range searching as in the previous work.