Dynamic Rectangular Point Location, with an Application to the Closest Pair Problem

Abstract

In the k -dimensional rectangular point location problem, we have to store a set of n non-overlapping axes-parallel hyperrectangles in a data structure, such that the following operations can be performed efficienctly: point location queries, insertions and deletions of hyperrectangles, and splitting and merging of hyperrectangles. A linear size data structure is given for this problem, allowing queries to be solved in O ((log n ) k − 1 log log n ) time, and allowing the four update operations to be performed in O ((log n ) 2 log log n ) amortized time. If only queries, insertions, and split operations have to be supported, the log log n factors disappear. The data structure is based on the skewer tree of Edelsbrunner et al . (1986, Comput. J. 29 , 76-82) and uses dynamic fractional cascading. This result is used to obtain a linear size data structure that maintains the closest pair in a set of n points in k -dimensional space, when points are inserted. This structure has an O ((log n ) k − 1 ) amortized insertion time. This leads to an on-line algorithm for computing the closest pair in a point set in O ( n (log n ) k − 1 ) time. In the planar case, these two latter results are optimal.