An optimal algorithm for the on-line closest-pair problem

We give an algorithm that computes the closest pair in a set ofn points ink-dimensional space on-line, inO(n logn) time. The algorithm only uses algebraic functions and, therefore, is optimal. The algorithm maintains a hierarchical subdivision ofk-space into hyperrectangles, which is stored in a binary tree. Centroids are used to maintain a balanced decomposition of this tree.These authors were supported by the ESPRIT II Basic Research Actions Program, under Contract No. 3075 (project ALCOM).This author was supported in part by the National Science and Engineering Research Council of Canada.