Output-sensitive well-separated pair decompositions for dynamic point sets

Abstract

The well-separated pair decomposition (WSPD) is a fundamental structure in computational geometry. Given a set P of n points in d-dimensional space and a positive separation parameter s, an s-WSPD is a concise representation of all the O(n2) pairs of P requiring only O(sdn) storage. The WSPD has numerous applications in spatial data processing, such as computing spanner graphs, minimum spanning trees, shortest-path oracles, and statistics on interpoint distances. We consider the problem of maintaining a WSPD when points are inserted to or deleted from P. Worst-case arguments suggest that the addition or deletion of a single point could result in the generation (or removal) up to Ω(sd) pairs, which can be unacceptably high in many applications. Fortunately, the actual number of well separated pairs can be significantly smaller in practice, particularly when the points are well clustered. This suggests the importance of being able to respond to insertions and deletions in a manner that is output sensitive, that is, whose running time depends on the actual number of pairs that have been added or removed. We present the first output-sensitive algorithms for maintaining a WSPD of a point set under insertion and deletion. We show that our algorithms are nearly optimal, in the sense that these operations can be performed in time that is roughly equal to the number of changes to the WSPD.