A linear-time algorithm to compute geodesics in solvable Baumslag-Solitar groups

TL;DR

This paper introduces a linear-time algorithm for converting words into geodesic form in solvable Baumslag-Solitar groups, optimizing computational efficiency for group theoretic problems.

Contribution

It provides the first linear-time algorithm for geodesic computation in BS(1,p), improving previous methods in efficiency and complexity.

Findings

Algorithm runs in linear time for geodesic conversion

Uses O(n log n) space complexity

Applicable to standard generators of BS(1,p)

Abstract

We present an algorithm to convert a word of length $n$ in the standard generators of the solvable Baumslag-Solitar group $BS(1,p)$ into a geodesic word, which runs in linear time and $O(n\log n)$ space on a random access machine.