A Generalized Cover Time for Random Walks on Graphs

  • Cyril Banderier
  • Robert P. Dobrow
Conference paper

Abstract

Given a random walk on a graph, the cover time is the first time (number of steps) that every vertex has been hit (covered) by the walk. Define the marking time for the walk as follows. When the walk reaches vertex vi, a coin is flipped and with probability pi the vertex is marked (or colored). We study the time that every vertex is marked. (When all the pi’s are equal to 1, this gives the usual cover time problem.) General formulas are given for the marking time of a graph. Connections are made with the generalized coupon collector’s problem. Asymptotics for small p i ’s are given. Techniques used include combinatorics of random walks, theory of determinants, analysis and probabilistic considerations.

Keywords

Kato 

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References

  1. 1.
    Aldous (David J.) and Fill (James). — Reversible Markov Chains and Random Walks on Graphs. - Book in preparation. Available at http://www.stat.berkeley.edu/users/aldous/book.html.
  2. 2.
    Boneh (Amon) and Hofri (Micha). — The coupon-collector problem revisited. Comm. Statist. Stochastic Models,vol. 13, n° 1, 1997, pp. 39–66.Google Scholar
  3. 3.
    Csörgö (Sândor). — A rate of convergence for coupon collectors. Acta Sci. Math. (Szeged),vol. 57, n° 1–4, 1993, pp. 337–351.Google Scholar
  4. 4.
    Feige (Uriel). — Collecting coupons on trees, and the cover time of random walks. Comput. Complexity,vol. 6, n° 4, 1996/97, pp. 341–356.Google Scholar
  5. 5.
    Flajolet (Philippe), Gardy (Danièle), and Thimonier (Lois).–Birthday paradox, coupon collectors, caching algorithms and self-organizing search. Discrete Applied Mathematics, vol. 39, n° 3, 1992, pp. 207–229.Google Scholar
  6. 6.
    Kato (Tosio). — Perturbation theory for linear operators. - Springer-Verlag, 1995.Google Scholar
  7. 7.
    Nath (Harmindar B.).–Waiting time in the coupon-collector’s problem. Austral. J. Statist., vol. 15, 1973, pp. 132–135.Google Scholar
  8. 8.
    Sen (Pranab Kumar). — Invariance principles for the coupon collector’s problem: a martingale approach. Ann. Statist,vol. 7, n° 2, 1979, pp. 372–380.Google Scholar
  9. 9.
    von Schelling (Hermann).–Coupon collecting for unequal probabilities. Amer. Math. Monthly, vol. 61, 1954, pp. 306–311.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Cyril Banderier
    • 1
  • Robert P. Dobrow
    • 2
  1. 1.Algorithms ProjectINRIA (Rocquencourt)France
  2. 2.Clarkson UniversityPotsdamUSA

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