Maintaning maximal matching with lookahead

TL;DR

This paper introduces a deterministic algorithm for maintaining a maximal matching in fully dynamic graphs with lookahead, achieving an amortized update time of O(log m) by leveraging future update information.

Contribution

It presents the first deterministic algorithm for dynamic maximal matching with lookahead, significantly improving update times under certain conditions.

Findings

Achieves O(log m) amortized update time with lookahead

Provides a deterministic approach for dynamic maximal matching

Enhances efficiency over trivial algorithms in dynamic graph settings

Abstract

In this paper we study the problem of fully dynamic maximal matching with lookahead. In a fully dynamic $n$-vertex graph setting, we have to handle updates (insertions and removals of edges), and answer queries regarding the current graph, preferably with a better time bound than that when running the trivial deterministic algorithm with worst-case time of $O(m)$ (where $m$ is the all-time maximum number of the edges) and recompute the matching from scratch each time a query arrives. We show that a maximal matching can be maintained in an (undirected) general graph with a deterministic amortized update cost of $O(\log m)$, provided that a lookahead of length $m$ is available, i.e. we can ``take a peek'' at the next $m$ update operations in advance.