Verifying Groups in Linear Time

Shai Evra, Shay Gadot, Ohad Klein, Ilan Komargodski

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Consider the following problem: Given an n × n multiplication table, decide whether it is a Cayley multiplication table of a group. Among deterministic algorithms for this problem, the best known algorithm is implied by F. W. Light's associativity test (1949) and has running time of O(n2log n). Allowing randomization. the best known algorithm has running time of O(n2log(1/Δ)), where Δ > 0 is the error probability of the algorithm (Rajagopalan and Schulman, FOCS 1996, SICOMP 2000). In this work, we improve upon both of the above known algorithms. Specifically, we present a deterministic algorithm for the above problem whose running time is O(n2). This performance is optimal up to constants. A central tool we develop is an efficient algorithm for finding a subset A of a group G satisfying A2=G while |A|=O(√{|G|}).

Original languageEnglish
Title of host publicationProceedings - 2024 IEEE 65th Annual Symposium on Foundations of Computer Science, FOCS 2024
PublisherIEEE Computer Society
Pages2131-2147
Number of pages17
ISBN (Electronic)9798331516741
StatePublished - 2024
Event65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024 - Chicago, United States
Duration: 27 Oct 202430 Oct 2024

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Conference

Conference65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024
Country/TerritoryUnited States
CityChicago
Period27/10/2430/10/24

Bibliographical note

Publisher Copyright:
© 2024 IEEE.

Keywords

  • Computational group theory

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