### Abstract

Motivated by a derandomization of Markov chain Monte Carlo (MCMC), this paper investigates deterministic random walks, which is a deterministic process analogous to a random walk. While there are some progress on the analysis of the vertex-wise discrepancy (i.e., L_{∞} discrepancy), little is known about the total variation discrepancy (i.e., Li discrepancy), which plays a significant role in the analysis of an FPRAS based on MCMC. This paper investigates upper bounds of the L_{1} discrepancy between the expected number of tokens in a Markov chain and the number of tokens in its corresponding deterministic random walk. First, we give a simple but nontrivial upper bound O(mt∗) of the L_{1} discrepancy for any ergodic Markov chains, where m is the number of edges of the transition diagram and t∗ is the mixing time of the Markov chain. Then, we give a better upper bound O(m√t∗ log t∗) for non-oblivious deterministic random walks, if the corresponding Markov chain is ergodic and lazy. We also present some lower bounds.

Original language | English |
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Title of host publication | 13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016 |

Editors | James Allen Fill, Mark Daniel Ward |

Publisher | Society for Industrial and Applied Mathematics Publications |

Pages | 138-148 |

Number of pages | 11 |

ISBN (Electronic) | 9781510819696 |

DOIs | |

Publication status | Published - 2016 |

Event | 13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016 - Arlington, United States Duration: Jan 11 2016 → … |

### Publication series

Name | 13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016 |
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### Other

Other | 13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016 |
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Country | United States |

City | Arlington |

Period | 1/11/16 → … |

### All Science Journal Classification (ASJC) codes

- Applied Mathematics
- Materials Chemistry
- Discrete Mathematics and Combinatorics

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## Cite this

*13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016*(pp. 138-148). (13th Workshop on Analytic Algorithmics and Combinatorics 2016, ANALCO 2016). Society for Industrial and Applied Mathematics Publications. https://doi.org/10.1137/1.9781611974324.13