Dynamic Euclidean Bottleneck Matching

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 $P$.