Illinois Journal of Mathematics

Sublinear time algorithms in the theory of groups and semigroups

Vladimir Shpilrain

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Sublinear time algorithms represent a new paradigm in computing, where an algorithm must give some sort of an answer after inspecting only a small portion of the input. The most typical situation where sublinear time algorithms are considered is property testing. There are several interesting contexts where one can test properties in sublinear time. A canonical example is graph colorability. To tell that a given graph is not k-colorable, it is often sufficient to inspect just one vertex with incident edges: if the degree of a vertex is greater than k, then the graph is not k-colorable.

It is a challenging and interesting task to find algebraic properties that could be tested in sublinear time. In this paper, we address several algorithmic problems in the theory of groups and semigroups that may admit sublinear time solution, at least for “most” inputs.

Article information

Illinois J. Math. Volume 54, Number 1 (2010), 187-197.

First available in Project Euclid: 9 March 2011

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Mathematical Reviews number (MathSciNet)

Primary: 20F10: Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70] 68Q17: Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) [See also 68Q15]
Secondary: 20M05: Free semigroups, generators and relations, word problems [See also 03D40, 08A50, 20F10] 68W30: Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 17-08, 33F10]


Shpilrain, Vladimir. Sublinear time algorithms in the theory of groups and semigroups. Illinois J. Math. 54 (2010), no. 1, 187--197.

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  • N. Alon, E. Fischer, M. Krivelevich and M. Szegedy, Efficient testing of large graphs, Combinatorica 20 (2000), 451–476.
  • M. Autord and P. Dehornoy, On the distance between the expressions of a permutation, preprint.
  • J. S. Birman, Braids, links and mapping class groups, Ann. of Math. Studies, vol. 82, Princeton University Press, 1974.
  • J. W. Cannon, W. J. Floyd and W. R. Parry, Introductory notes on Richard Thompson's groups, L'Enseignement Mathematique (2) 42 (1996), 215–256.
  • K. T. Chen, R. H. Fox and R. C. Lyndon, Free differential calculus. IV. The quotient groups of the lower central series, Ann. of Math. (2) 68 (1958), 81–95.
  • M. Garey and J. Johnson, Computers and intractability, a guide to NP-completelness, W. H. Freeman, 1979.
  • O. Goldreich, S. Goldwasser and D. Ron, Property testing and its connection to learning and approximation, J. ACM 45 (1998), 653–750.
  • Yu. Gurevich and S. Shelah, Expected computation time for Hamiltonian Path Problem, SIAM J. Comput. 16 (1987), 486–502.
  • I. Kapovich, A. Myasnikov, P. Schupp and V. Shpilrain, Generic-case complexity, decision problems in group theory and random walks, J. Algebra 264 (2003), 665–694.
  • I. Kapovich and P. Schupp, Genericity, the Arzhantseva–Ol'shanskii method and the isomorphism problem for one-relator groups, Math. Ann. 331 (2005), 1–19.
  • I. Kapovich, P. Schupp and V. Shpilrain, Generic properties of Whitehead's algorithm and isomorphism rigidity of random one-relator groups, Pacific J. Math. 223 (2006), 113–140.
  • O. Kharlampovich and M. Sapir, Algorithmic problems in varieties, Internat. J. Algebra Comput. 5 (1995), 379–602.
  • R. Lyndon and P. Schupp, Combinatorial group theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition.
  • W. Magnus, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, Math. Ann. 111 (1935), 259–280.
  • A. I. Mal'cev, Nilpotent semigroups, Uchen. Zapiski Ivanovsk. Ped. Inst. 4 (1953), 107–111 (Russian).
  • B. H. Neumann and T. Taylor, Subsemigroups of nilpotent groups, Proc. Roy. Soc. Ser. A 274 (1963), 1–4.
  • R. Rubinfeld, Sublinear time algorithms, ICM 2006, invited talk; available at http://
  • L. Shneerson, Relatively free semigroups of intermediate growth, J. Algebra 235 (2001), 484–546.
  • V. Shpilrain, Magnus embeddings for semigroups, Internat. J. Algebra Comput. 6 (1996), 155–163.
  • J. R. Stallings, Whitehead graphs on handlebodies, Geometric group theory down under (Canberra, 1996), de Gruyter, Berlin, 1999, pp. 317–330.