Efficient Algorithms for Highly Compressed Data: The Word Problem in Generalized Higman Groups Is in P

Abstract

This paper continues the 2012 STACS contribution by Diekert, Ushakov, and the author as well as the 2012 IJAC publication by the same authors. We extend the results published there 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 in polynomial time. The group H ₄(1,q) is an amalgamated product of four copies of the Baumslag-Solitar group BS(1,q) rather than BS(1,2) in the original group H ₄=H ₄(1,2).

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 ${\mathcal {O}}(n^{6})$ time bound that was shown for H ₄(1,2).