Kinetic Pie Delaunay Graph and Its Applications
We construct a new proximity graph, called the Pie Delaunay graph, on a set of n points which is a super graph of Yaograph and Euclidean minimum spanning tree (EMST). We efficiently maintain the PieDelaunaygraph where the points are moving in the plane. We use the kinetic PieDelaunaygraph to create a kinetic data structure (KDS) for maintenance of the Yaograph 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(n2λ2s + 2(n)βs + 2(n)) events for the Yaograph and O(n2λ2s + 2(n)) events for the EMST, each in O(log2n) time. Here, λs(n) = nβs(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(n4) by Rahmati etal. .
KeywordsEuclidean minimum spanning tree Yao graph Pie Delaunay triangulation kinetic data structures
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