Efficient algorithms for highly compressed data: The Word Problem in Generalized Higman Groups is in P

TL;DR

This paper extends power circuit data structures to arbitrary bases, enabling efficient solutions to the word problem in generalized Higman groups, maintaining polynomial time complexity.

Contribution

It generalizes power circuits for any base q≥2 and applies this to solve the word problem in broader Higman groups efficiently.

Findings

Power circuits are generalized to work with any base q≥2.

The word problem in generalized Higman groups H_f(1,q) is solvable in polynomial time.

The time complexity remains O(n^6) for these generalized groups.

Abstract

This paper continues the 2012 STACS contribution by Diekert, Ushakov, and the author. We extend the results published in the proceedings in two ways. First, we show that the data structure of power circuits can be generalized to work with arbitrary bases q>=2. This results in a data structure that can hold huge integers, arising by iteratively forming powers of q. We show that the properties of power circuits known for q=2 translate to the general case. This generalization is non-trivial and additional techniques are required to preserve the time bounds of arithmetic operations that were shown for the case q=2. The extended power circuit model permits us to conduct operations in the Baumslag-Solitar group BS(1,q) as efficiently as in BS(1,2). This allows us to solve the word problem in the generalization H_4(1,q) of Higman's group, which is an amalgamated product of four copies of the Baumslag-Solitar group BS(1,q) rather than BS(1,2) in the original form. As a second result, we allow arbitrary numbers f>=4 of copies of BS(1,q), leading to an even more generalized notion of Higman groups H_f(1,q). We prove that the word problem of the latter can still be solved within the O(n^6) time bound that was shown for H_4(1,2).