Core Concepts
Finitely generated linear groups, metabelian groups, and free solvable groups admit randomized streaming algorithms with logarithmic space complexity for their word problems. The class of finitely generated groups with a logspace randomized streaming algorithm for the word problem is closed under several group-theoretic constructions. In contrast, Thompson's group F has only a linear space randomized streaming algorithm for its word problem.
Abstract
The paper studies deterministic and randomized streaming algorithms for word problems of finitely generated groups. The main results are:
For finitely generated linear groups, metabelian groups, and free solvable groups, the authors show the existence of randomized streaming algorithms with logarithmic space complexity for their word problems.
The class of finitely generated groups with a logspace randomized streaming algorithm for the word problem is closed under several group-theoretical constructions: finite extensions, graph products, and wreath products by finitely generated abelian groups.
In contrast, the authors provide an example of a finitely presented group, Thompson's group F, where the word problem has only a linear space randomized streaming algorithm.
The deterministic streaming space complexity of a group's word problem is directly linked to the growth function of the group. The randomized streaming space complexity is lower bounded by the logarithm of the logarithm of the group's growth function.
Randomized streaming algorithms for subgroup membership problems in free groups and direct products of free groups are also studied.
The paper provides a comprehensive analysis of the streaming complexity of group word problems, identifying classes of groups with efficient randomized streaming algorithms as well as examples of groups with inherent limitations.