Maintaining the minimal distance of a point set in less than linear time

Abstract

Consider a set of n points in d-dimensional space. It is shown that the ordered sequence of O(n^{2/3}) smallest distances defined by these points can be computed in optimal O(nlog n) time and O(n) space. Here, distances are measured in an arbitrary L_{t}-metric, where 1\leq t\leq\infty. This result is used to give a dynamic data structure of linear size, that maintains the minimal distance of the n points in O(n^{2/3}log n) time per update.