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

AbstractIn 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