The Parallel Dynamic Complexity of the Abelian Cayley Group Membership Problem

TL;DR

This paper investigates the parallel complexity of the Abelian Cayley Group Membership problem, especially in dynamic settings, providing efficient algorithms for membership testing under various update scenarios.

Contribution

It introduces new deterministic and randomized parallel algorithms for dynamic Abelian CGM, extending to monoid membership and group isomorphism.

Findings

Deterministic constant-time parallel algorithm for monoid membership.

Randomized parallel algorithm for abelian CGM with polylogarithmic updates.

Deterministic algorithm for limited updates (O(log n / log log n)).

Abstract

Let $G$ be a finite group given as input by its multiplication table. For a subset $S$ of $G$ and an element $g\in G$ the Cayley Group Membership Problem (denoted CGM) is to check if $g$ belongs to the subgroup generated by $S$. While this problem is easily seen to be in polynomial time, pinpointing its parallel complexity has been of research interest over the years. In this paper we further explore the parallel complexity of the abelian CGM problem, with focus on the dynamic setting: the generating set $S$ changes with insertions and deletions and the goal is to maintain a data structure that supports efficient membership queries to the subgroup $\angle{S}$. We obtain the following results: 1. We first consider the more general problem of Monoid Membership. When $G$ is a commutative monoid we give a deterministic dynamic algorithm constant time parallel algorithm for membership testing that supports $O(1)$ insertions and deletions in each step. 2. Building on the previous result we show that there is a dynamic randomized constant-time parallel algorithm for abelian CGM that supports polylogarithmically many insertions/deletions to $S$ in each step. 3. If the number of insertions/deletions is at most $O(\log n/\log\log n)$ then we obtain a deterministic dynamic constant-time parallel algorithm for the problem. 4. We obtain analogous results for the dynamic abelian Group Isomorphism.