Dynamic Euclidean bottleneck matching

Abstract

A fundamental question in computational geometry is for a set of input points in the Euclidean space, that is subject to discrete changes (insertion/deletion of points at each time step), whether it is possible to maintain an exact/approximate minimum weight perfect matching and/or bottleneck matching (a perfect matching that minimizes the length of the longest matched edge), in sublinear update time. In this work, we answer this question in the affirmative for points on a real line and for points in the plane with a bounded geometric spread.

For a set P of n points on a line, we show that there exists a dynamic algorithm that maintains an exact bottleneck matching of P and supports insertion and deletion in $O(\log n)$ time. Moreover, we show that a modified version of this algorithm maintains an exact minimum-weight perfect matching with $O(\log n)$ update (insertion and deletion) time. Next, for a set P of n points in the plane, we show that a ($6\sqrt{2}$)-factor approximate bottleneck matching of $P_k$, at each time step k, can be maintained in $O(\log \Delta )$ amortized time per insertion and $O(\log \Delta +|P_k\left|\right)$ amortized time per deletion, where Δ is the geometric spread of P (the ratio between the diameter of P and the distance between the closest pair of points in P).