### Abstract

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}.

Original language | English |
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Title of host publication | Computing and Combinatorics - 20th International Conference, COCOON 2014, Proceedings |

Publisher | Springer Verlag |

Pages | 25-36 |

Number of pages | 12 |

ISBN (Print) | 9783319087825 |

DOIs | |

Publication status | Published - Jan 1 2014 |

Event | 20th International Computing and Combinatorics Conference, COCOON 2014 - Atlanta, GA, United States Duration: Aug 4 2014 → Aug 6 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8591 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 20th International Computing and Combinatorics Conference, COCOON 2014 |
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Country | United States |

City | Atlanta, GA |

Period | 8/4/14 → 8/6/14 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

_{∞}-Discrepancy analysis of polynomial-time deterministic samplers emulating rapidly mixing chains. In

*Computing and Combinatorics - 20th International Conference, COCOON 2014, Proceedings*(pp. 25-36). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8591 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-08783-2_3

**L _{∞}-Discrepancy analysis of polynomial-time deterministic samplers emulating rapidly mixing chains.** / Shiraga, Takeharu; Yamauchi, Yukiko; Kijima, Shuji; Yamashita, Masafumi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

_{∞}-Discrepancy analysis of polynomial-time deterministic samplers emulating rapidly mixing chains. in

*Computing and Combinatorics - 20th International Conference, COCOON 2014, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8591 LNCS, Springer Verlag, pp. 25-36, 20th International Computing and Combinatorics Conference, COCOON 2014, Atlanta, GA, United States, 8/4/14. https://doi.org/10.1007/978-3-319-08783-2_3

_{∞}-Discrepancy analysis of polynomial-time deterministic samplers emulating rapidly mixing chains. In Computing and Combinatorics - 20th International Conference, COCOON 2014, Proceedings. Springer Verlag. 2014. p. 25-36. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-08783-2_3

}

TY - GEN

T1 - L∞-Discrepancy analysis of polynomial-time deterministic samplers emulating rapidly mixing chains

AU - Shiraga, Takeharu

AU - Yamauchi, Yukiko

AU - Kijima, Shuji

AU - Yamashita, Masafumi

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84904756566&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904756566&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-08783-2_3

DO - 10.1007/978-3-319-08783-2_3

M3 - Conference contribution

AN - SCOPUS:84904756566

SN - 9783319087825

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 25

EP - 36

BT - Computing and Combinatorics - 20th International Conference, COCOON 2014, Proceedings

PB - Springer Verlag

ER -