The Parallel Dynamic Complexity of the Abelian Cayley Group Membership Problem

Abstract

Let G be a finite group given as input by its multiplication table. For a subset S ⊆ G and an element g ∈ G the Cayley Group Membership Problem (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. Barrington et al [Barrington et al., 2001] have shown that for abelian groups the CGM problem can be solved in O(log log |G|) parallel time. 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 ⟨S⟩. Though the static version of the CGM problem can be easily reduced to digraph reachability, the reduction does not carry over to the dynamic setting. We obtain the following results:  
1) First, we consider the more general problem of Monoid Membership, where G is a monoid input by its multiplication table. When G is a commutative monoid we show there is a deterministic dynamic AC⁰ 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 AC⁰ algorithm for abelian CGM that supports polylog(|G|) 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 AC⁰ algorithm for abelian CGM. 
4) Applying these algorithms we obtain analogous results for the dynamic abelian Group Isomorphism.  We can also handle sub-linearly many changes to the multiplication table for G, utilizing the hamming distance between multiplication tables of any two distinct groups.