Electronic Communications in Probability

Note: Random-to-front shuffles on trees

Anders Bjorner

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A Markov chain is considered whose states are orderings of an underlying fixed tree and whose transitions are local ``random-to-front'' reorderings, driven by a probability distribution on subsets of the leaves. The eigenvalues of the transition matrix are determined using Brown's theory of random walk on semigroups.

Article information

Electron. Commun. Probab., Volume 14 (2009), paper no. 4, 36-41.

Accepted: 4 February 2009
First available in Project Euclid: 6 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60C05: Combinatorial probability 05E99: None of the above, but in this section

Markov chain shuffle random-to-front random walk tree semigroup eigenvalue

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Bjorner, Anders. Note: Random-to-front shuffles on trees. Electron. Commun. Probab. 14 (2009), paper no. 4, 36--41. doi:10.1214/ECP.v14-1445. https://projecteuclid.org/euclid.ecp/1465234713

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  • Allen, Brian; Munro, Ian. Self-organizing binary search trees. J. Assoc. Comput. Mach. 25 (1978), no. 4, 526–535.
  • Bidigare, Pat; Hanlon, Phil; Rockmore, Dan. A combinatorial description of the spectrum for the Tsetlin library and its generalization to hyperplane arrangements. Duke Math. J. 99 (1999), no. 1, 135–174.
  • Bjorner,A. Random walks, arrangements, cell complexes, greedoids, and self-organizing libraries, in “Building Bridges” (eds. M. Grotschel and G. O. H. Katona), Bolyai Soc. Math. Studies 19 (2008), Springer (Berlin) and Janos Bolyai Math. Soc. (Budapest), pp.165–203.
  • Brown, Kenneth S. Semigroups, rings, and Markov chains. J. Theoret. Probab. 13 (2000), no. 3, 871–938.
  • Brown, Kenneth S.; Diaconis, Persi. Random walks and hyperplane arrangements. Ann. Probab. 26 (1998), no. 4, 1813–1854.
  • Dobrow, Robert P.; Fill, James Allen. On the Markov chain for the move-to-root rule for binary search trees. Ann. Appl. Probab. 5 (1995), no. 1, 1–19.
  • Fill, James Allen; Holst, Lars. On the distribution of search cost for the move-to-front rule. Random Structures Algorithms 8 (1996), no. 3, 179–186.
  • Stanley, Richard P. Enumerative combinatorics. Vol. 1.With a foreword by Gian-Carlo Rota.Corrected reprint of the 1986 original.Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, Cambridge, 1997. xii+325 pp. ISBN: 0-521-55309-1; 0-521-66351-2