### Abstract

Motivated by a derandomization of Markov chain Monte Carlo (MCMC), this paper investigates a deterministic random walk, which is a deterministic process analogous to a random walk. There is some recent progress in the analysis of the vertex-wise discrepancy (i.e., L_{∞}-discrepancy), while little is known about the total variation discrepancy (i.e., L_{1}-discrepancy), which plays an important role in the analysis of an FPRAS based on MCMC. This paper investigates 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(mt^{⁎}) 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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Pages (from-to) | 63-74 |

Number of pages | 12 |

Journal | Theoretical Computer Science |

Volume | 699 |

DOIs | |

Publication status | Published - Nov 7 2017 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Theoretical Computer Science*,

*699*, 63-74. https://doi.org/10.1016/j.tcs.2016.11.017