A two-dimensional kinetic triangulation with near-quadratic topological changes

Abstract A triangulation of a set S of points in the plane is a subdivision of the convex hull of S intotriangles whose vertices are points of S. Given a set S of n points in R2, each moving independently,we wish to maintain a triangulation of S. The triangulation needs to be updated periodically asthe points in S move, so the goal is to maintain a triangulation with a small number of topologicalevents, each being the insertion or deletion of an edge. We propose a kinetic data structure (KDS) that processes n22O(plog n*log log n) topological events with high probability if the trajectories of inputpoints are algebraic curves of fixed degree. Each topological event can be processed in O(log n)time. This is the first known KDS for maintaining a triangulation that processes a near-quadratic number of topological events, and almost matches the \Omega (n2) lower bound [1]. The number of topological events can be reduced to nk * 2O(plog k*log log n) if only k of the points are moving