A simple, faster method for kinetic proximity problems

Abstract

For a set of n points in the plane, this paper presents simple kinetic data structures (KDSs) for solutions to some fundamental proximity problems, namely, the all nearest neighbors problem, the closest pair problem, and the Euclidean minimum spanning tree (EMST) problem. Also, the paper introduces KDSs for maintenance of two well-studied sparse proximity graphs, the Yao graph and the Semi-Yao graph.We use sparse graph representations, the Pie Delaunay graph and the Equilateral Delaunay graph, to provide new solutions for the proximity problems. Then we design KDSs that efficiently maintain these sparse graphs on a set of n moving points, where the trajectory of each point is assumed to be a polynomial function of constant maximum degree s. We use the kinetic Pie Delaunay graph and the kinetic Equilateral Delaunay graph to create KDSs for maintenance of the Yao graph, the Semi-Yao graph, all the nearest neighbors, the closest pair, and the EMST. Our KDSs use O(n) space and O(nlog⁡n) preprocessing time.We provide the first KDSs for maintenance of the Semi-Yao graph and the Yao graph. Our KDS processes O(n2β2s+2(n)) (resp. O(n3β2s+22(n)log⁡n)) events to maintain the Semi-Yao graph (resp. the Yao graph); each event can be processed in amortized time O(log⁡n). Here, βs(n)=λs(n)/n is an extremely slow-growing function and λs(n) is the maximum length of Davenport–Schinzel sequences of order s on n symbols.Our KDS for maintenance of all the nearest neighbors and the closest pair processes O(n2β2s+22(n)log⁡n) events. For maintenance of the EMST, our KDS processes O(n3β2s+22(n)log⁡n) events. For all three of these problems, each event can be handled in amortized time O(log⁡n).Our deterministic kinetic approach for maintenance of all the nearest neighbors improves by an O(log2⁡n) factor the previous randomized kinetic algorithm by Agarwal, Kaplan, and Sharir. Furthermore, our KDS is simpler than their KDS, as we reduce the problem to one-dimensional range searching, as opposed to using two-dimensional range searching as in their KDS.For maintenance of the EMST, our KDS improves the previous KDS by Rahmati and Zarei by a near-linear factor in the number of events.