2024/01/17 by Vladimir Shpilrain, Shpilrain, Vladimir
Computer Science · Engineering · Mathematics · #Computational Complexity (cs.CC) #FOS: Computer and information sciences #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #graph theory and CDMA systems #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2401.09218
openalex publication_date 2024/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this survey, we address the worst-case, average-case, and generic-case time complexity of the word problem and some other algorithmic problems in several classes of groups and show that it is often the case that the average-case complexity of the word problem is linear with respect to the length of an input word, which is as good as it gets if one considers groups given by generators and defining relations. At the same time, there are other natural algorithmic problems, for instance, the geodesic (decision) problem or Whitehead's automorphism problem, where the average-case time complexity can be sublinear, even constant.