Parallel Dynamic Maximal Matching

TL;DR

This paper introduces the first parallel dynamic algorithm for maintaining a maximal matching in graphs, capable of processing multiple updates simultaneously with polylogarithmic depth and amortized work.

Contribution

It presents a novel randomized parallel algorithm for dynamic maximal matching that handles arbitrary batch updates efficiently, extending to hypergraphs.

Findings

Processes batch updates in poly(log n) depth

Achieves amortized poly(log n) work per update

Generalizes to hypergraphs with rank r

Abstract

We present the first (randomized) parallel dynamic algorithm for maximal matching, which can process an arbitrary number of updates simultaneously. Given a batch of edge deletion or insertion updates to the graph, our parallel algorithm adjusts the maximal matching to these updates in $poly(\log n)$ depth and using $poly(\log n)$ amortized work per update. That is, the amortized work for processing a batch of $k$ updates is $kpoly(\log n)$, while all this work is done in $poly(\log n)$ depth, with high probability. This can be seen as a parallel counterpart of the sequential dynamic algorithms for constant-approximate and maximal matching [Onak and Rubinfeld STOC'10; Baswana, Gupta, and Sen FOCS'11; and Solomon FOCS'16]. Our algorithm readily generalizes to maximal matching in hypergraphs of rank $r$ -- where each hyperedge has at most $r$ endpoints -- with a $poly(r)$ increase in work, while retaining the $poly(\log n)$ depth.