Sublinear time algorithms in the theory of groups and semigroups

TL;DR

This paper explores the potential for sublinear time algorithms to test algebraic properties in groups and semigroups, extending property testing concepts beyond graph problems.

Contribution

It introduces the idea of applying sublinear algorithms to algebraic structures, identifying problems that can be efficiently tested for typical inputs.

Findings

Certain algebraic properties can be tested in sublinear time for most inputs.

Sublinear algorithms can be applied to group and semigroup problems.

The approach extends property testing to algebraic contexts.

Abstract

Sublinear time algorithms represent a new paradigm in computing, where an algorithm must give some sort of an answer after inspecting only a small portion of the input. The most typical situation where sublinear time algorithms are considered is property testing. There are several interesting contexts where one can test properties in sublinear time. A canonical example is graph colorability. To tell that a given graph is not k-colorable, it is often sufficient to inspect just one vertex with incident edges: if the degree of a vertex is greater than k, then the graph is not k-colorable. It is a challenging and interesting task to find algebraic properties that could be tested in sublinear time. In this paper, we address several algorithmic problems in the theory of groups and semigroups that may admit sublinear time solution, at least for "most" inputs.