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. 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 H4(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 H4(1, 2).

Improved Parallel Algorithms for Generalized Baumslag Groups

“…As a first step, we show how to compute compact base-q signed-digit representations in AC 0 (see Theorem 7). Moreover, we not only unify [17] and [23] but also prove improved complexity bounds:…”

“…Based on these striking results, Diekert, Laun and Ushakov [8] improved the running time for the word problem of the Baumslag group from O(n 7 ) down to O(n 3 ) and described a polynomial-time algorithm for the word problem of the Higman group H4 [13]. Subsequently, more applications of power circuits to similar groups emerged: In [17] Laun gave a polynomial-time solution for the word problem of generalized Baumslag groups G1,q = a, b | bab −1 a = a q bab −1 for q ≥ 1 and also for generalized Higman groups. In order to do so, he generalized power circuits to arbitrary bases q ≥ 2 and adapted the corresponding algorithms from [27,8].…”

“…Contribution. In this work we combine the results of [17] and [23] by considering the parallel complexity of the word problem of generalized Baumslag groups G1,q = a, b | bab −1 a = a q bab −1 . As a first step, we show how to compute compact base-q signed-digit representations in AC 0 (see Theorem 7).…”

“…For this we allow operations in a SIMD (single instruction multiple data) fashion: many operations of the same type are performed on the same power circuit in parallel in TC 0 . Furthermore, we improve the algorithm to get power circuits of quasi-linear size -thus, close to the optimal size as in the sequential algorithms [8,17].…”

Parallel algorithms for power circuits and the word problem of the Baumslag group

“…In [31] these algorithms have been implemented in C++. Subsequently, more applications of power circuits to these groups emerged: in [20] a polynomial time solution to the word problem in generalized Baumslag and Higman groups is given, in [12,11] the conjugacy problem of the Baumslag group is shown to be strongly generically in P and in [2] the same is done for the conjugacy problem of the Higman group. Here "generically" roughly means that the algorithm works for most inputs (for details on the concept of generic complexity, see [17]).…”