Speeding up the FMMR perfect sampling algorithm: A case study revisited

  1. Robert P. Dobrow1,
  2. James Allen Fill2,†,*

Article first published online: 14 AUG 2003

DOI: 10.1002/rsa.10096

Issue

Random Structures & Algorithms

Random Structures & Algorithms

Volume 23, Issue 4, pages 434–452, December 2003

Additional Information

How to Cite

Dobrow, R. P. and Fill, J. A. (2003), Speeding up the FMMR perfect sampling algorithm: A case study revisited. Random Struct. Alg., 23: 434–452. doi: 10.1002/rsa.10096

Author Information

  1. 1

    Mathematics and Computer Science Department, Carleton College, Northfield, Minnesota 55057

  2. 2

    Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, Maryland 21218-2682

  1. †

    Research for this author supported by NSF Grants DMS-9803780 and DMS-0104167, and for both authors by The Johns Hopkins University's Acheson J. Duncan Fund for the Advancement of Research in Statistics.

*Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, Maryland 21218-2682

  • †

    Research for this author supported by NSF Grants DMS-9803780 and DMS-0104167, and for both authors by The Johns Hopkins University's Acheson J. Duncan Fund for the Advancement of Research in Statistics.

Publication History

  1. Issue published online: 11 NOV 2003
  2. Article first published online: 14 AUG 2003
  3. Manuscript Accepted: 15 NOV 2002
  4. Manuscript Received: 14 MAY 2002

Funded by

  • NSF. Grant Numbers: DMS-9803780, DMS-0104167
  • The Johns Hopkins University's Acheson J. Duncan Fund

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Keywords:

  • perfect simulation;
  • exact sampling;
  • rejection sampling;
  • Markov chain Monte Carlo;
  • FMMR algorithm;
  • Fill's algorithm;
  • move-to-front rule;
  • coupling from the past;
  • Propp–Wilson algorithm;
  • running time;
  • monotonicity;
  • separation;
  • strong stationary time;
  • partially ordered set

Abstract

In a previous paper by the second author, two Markov chain Monte Carlo perfect sampling algorithms—one called coupling from the past (CFTP) and the other (FMMR) based on rejection sampling—are compared using as a case study the move-to-front (MTF) self-organizing list chain. Here we revisit that case study and, in particular, exploit the dependence of FMMR on the user-chosen initial state. We give a stochastic monotonicity result for the running time of FMMR applied to MTF and thus identify the initial state that gives the stochastically smallest running time; by contrast, the initial state used in the previous study gives the stochastically largest running time. By changing from worst choice to best choice of initial state we achieve remarkable speedup of FMMR for MTF; for example, we reduce the running time (as measured in Markov chain steps) from exponential in the length n of the list nearly down to n when the items in the list are requested according to a geometric distribution. For this same example, the running time for CFTP grows exponentially in n. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2003

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