Kinetic Pie Delaunay Graph and Its Applications

Abstract

We construct a new proximity graph, called the Pie Delaunay graph, on a set of n points which is a super graph of Yao graph and Euclidean minimum spanning tree (EMST). We efficiently maintain the Pie Delaunay graph where the points are moving in the plane. We use the kinetic Pie Delaunay graph to create a kinetic data structure (KDS) for maintenance of the Yao graph and the EMST on a set of n moving points in 2-dimensional space. Assuming x and y coordinates of the points are defined by algebraic functions of at most degree s, the structure uses O(n) space, O(nlogn) preprocessing time, and processes O(n ² λ _2s + 2(n)β _s + 2(n)) events for the Yao graph and O(n ² λ _2s + 2(n)) events for the EMST, each in O(log² n) time. Here, λ _s (n) = nβ _s (n) is the maximum length of Davenport-Schinzel sequences of order s on n symbols. Our KDS processes nearly cubic events for the EMST which improves the previous bound O(n ⁴) by Rahmati et al. [1].