Average-case complexity of the Whitehead problem for free groups

The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this paper, we address the average-case complexity (i.e., the expected runtime) of algorithms that sol...

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Bibliographic Details
Published in:Communications in algebra Vol. 51; no. 2; pp. 799 - 806
Main Author: Shpilrain, Vladimir
Format: Journal Article
Language:English
Published: Abingdon Taylor & Francis 01.02.2023
Taylor & Francis Ltd
Subjects:
Algorithms
Automorphisms
average-case complexity
Complexity
Free group
Whitehead problem
ISSN:0092-7872, 1532-4125
Online Access:Get full text
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Summary:The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this paper, we address the average-case complexity (i.e., the expected runtime) of algorithms that solve a well-known problem, the Whitehead problem in a free group, which is: given two elements of a free group, find out whether there is an automorphism that takes one element to the other. First we address a special case of the Whitehead problem, namely deciding if a given element of a free group is part of a free basis. We show that there is an algorithm that, on a cyclically reduced input word, solves this problem and has constant (with respect to the length of the input) average-case complexity. For the general Whitehead problem, we show that the classical Whitehead algorithm has linear average-case complexity if the rank of the free group is 2. We argue that the same should be true in a free group of any rank but point out obstacles to establishing this general result.
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ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2022.2113791