Dynamic Euclidean Bottleneck Matching

TL;DR

This paper presents efficient dynamic algorithms for maintaining approximate bottleneck matchings in Euclidean spaces, achieving polylogarithmic update times for points on a line and in the plane with bounded spread.

Contribution

It introduces novel dynamic algorithms for approximate bottleneck matchings on a line and in the plane with bounded spread, with improved update times.

Findings

Line case: O(log n) update time for bottleneck matching.

Plane case: O(log Δ) amortized update time with approximation factor.

Modified algorithm maintains minimum-weight matching with similar efficiency.

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 approximate bottleneck matching 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 a 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 a minimum-weight 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 $O(\log{\Delta} + |P_k|)$ amortized time per deletion, where $\Delta$ is the geometric spread of $P$.