L ∞ -Discrepancy Analysis of Polynomial-Time Deterministic Samplers Emulating Rapidly Mixing Chains

  • Takeharu Shiraga
  • Yukiko Yamauchi
  • Shuji Kijima
  • Masafumi Yamashita
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8591)


Markov chain Monte Carlo (MCMC) is a standard technique to sample from a target distribution by simulating Markov chains. In an analogous fashion to MCMC, this paper proposes a deterministic sampling algorithm based on deterministic random walk, such as the rotor-router model (a.k.a. Propp machine). For the algorithm, we give an upper bound of the point-wise distance (i.e., infinity norm) between the “distributions” of a deterministic random walk and its corresponding Markov chain in terms of the mixing time of the Markov chain. As a result, for uniform sampling of #P-complete problems, such as 0-1 knapsack solutions, linear extensions, matchings, etc., for which rapidly mixing chains are known, our deterministic algorithm provides samples with a “distribution” with a point-wise distance at most ε from the target distribution, in time polynomial in the input size and ε − 1.


rotor-router model #P-complete Markov chain Monte Carlo mixing time 


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  1. 1.
    Akbari, H., Berenbrink, P.: Parallel rotor walks on finite graphs and applications in discrete load balancing. In: Proc. SPAA 2013, pp. 186–195 (2013)Google Scholar
  2. 2.
    Angel, O., Holroyd, A.E., Martin, J., Propp, J.: Discrete low discrepancy sequences, arXiv:0910.1077Google Scholar
  3. 3.
    Bubley, R., Dyer, M.: Faster random generation of linear extensions. Discrete Mathematics 201, 81–88 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Cooper, J., Doerr, B., Friedrich, T., Spencer, J.: Deterministic random walks on regular trees. Random Structures & Algorithms 37, 353–366 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Cooper, J., Doerr, B., Spencer, J., Tardos, G.: Deterministic random walks on the integers. European Journal of Combinatorics 28, 2072–2090 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cooper, J., Spencer, J.: Simulating a random walk with constant error. Combinatorics, Probability and Computing 15, 815–822 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Doerr, B., Friedrich, T.: Deterministic random walks on the two-dimensional grid. Combinatorics, Probability and Computing 18, 123–144 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Friedrich, T., Gairing, M., Sauerwald, T.: Quasirandom load balancing. SIAM Journal on Computing 41, 747–771 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Gopalan, P., Klivans, A., Meka, R., Stefankovic, D., Vempala, S., Vigoda, E.: An FPTAS for #knapsack and related counting problems. In: Proc. FOCS 2011, pp. 817–826 (2011)Google Scholar
  10. 10.
    Holroyd, A.E., Propp, J.: Rotor walks and Markov chains. In: Lladser, M., Maier, R.S., Mishna, M., Rechnitzer, A. (eds.) Algorithmic Probability and Combinatorics, pp. 105–126. The American Mathematical Society (2010)Google Scholar
  11. 11.
    Hosoya, H.: Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. Bulletin of the Chemical Society of Japan 44, 2332–2339 (1971)CrossRefGoogle Scholar
  12. 12.
    Jerrum, M., Sinclair, A.: Approximation algorithms for NP-hard problems. In: Hochbaum, D.S. (ed.) The Markov Chain Monte Carlo Method: An Approach to Approximate Counting and Integration. PWS Publishing (1996)Google Scholar
  13. 13.
    Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. Journal of the ACM 51, 671–697 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Karzanov, A., Khachiyan, L.: On the conductance of order Markov chains. Order 8, 7–15 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Kijima, S., Koga, K., Makino, K.: Deterministic random walks on finite graphs. In: Proc. ANALCO 2012, pp. 16–25 (2012)Google Scholar
  16. 16.
    Levine, D.A., Peres, Y., Wilmer, E.L.: Markov Chain and Mixing Times. American Mathematical Society (2008)Google Scholar
  17. 17.
    Morris, B., Sinclair, A.: Random walks on truncated cubes and sampling 0-1 knapsack solutions. SIAM Journal on Computing 34, 195–226 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Rabani, Y., Sinclair, A., Wanka, R.: Local divergence of Markov chains and analysis of iterative load balancing schemes. In: Proc. FOCS 1998, pp. 694–705 (1998)Google Scholar
  19. 19.
    Shiraga, T., Yamauchi, Y., Kijima, S., Yamashita, M.: Deterministic random walks for rapidly mixing chains, arXiv:1311.3749Google Scholar
  20. 20.
    Sinclair, A.: Algorithms for Random Generation & Counting, A Markov chain approach. Birkhäuser (1993)Google Scholar
  21. 21.
    Tijdeman, R.: The chairman assignment problem. Discrete Math. 32, 323–330 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM Journal on Computing 8, 410–421 (1979)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Takeharu Shiraga
    • 1
  • Yukiko Yamauchi
    • 1
  • Shuji Kijima
    • 1
  • Masafumi Yamashita
    • 1
  1. 1.Graduate School of Information Science and Electrical EngineeringKyushu UniversityJapan

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