Choice and Bias in Random Walks

Georgakopoulos, A., Haslegrave, J., Sauerwald, T. and Sylvester, J. (2020) Choice and Bias in Random Walks. In: 11th Innovations in Theoretical Computer Science Conference (ITCS 2020), Seattle, WA, USA, 12-14 Jan 2020, p. 76. ISBN 9783959771344 (doi: 10.4230/LIPICS.ITCS.2020.76)

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Abstract

We analyse the following random walk process inspired by the power-of-two-choice paradigm: starting from a given vertex, at each step, unlike the simple random walk (SRW) that always moves to a randomly chosen neighbour, we have the choice between two uniformly and independently chosen neighbours. We call this process the choice random walk (CRW). We first prove that for any graph, there is a strategy for the CRW that visits any given vertex in expected time O(|E|). Then we introduce a general tool that quantifies by how much the probability of a rare event in the simple random walk can be boosted under a suitable CRW strategy. We believe this result to be of independent interest, and apply it here to derive an almost optimal O(n log log n) bound for the cover time of bounded-degree expanders. This tool also applies to so-called biased walks, and allows us to make progress towards a conjecture of Azar et al. [STOC 1992]. Finally, we prove the following dichotomy: computing an optimal strategy to minimise the hitting time of a vertex takes polynomial time, whereas computing one to minimise the cover time is NP-hard.

Item Type:Conference Proceedings
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Sylvester, Dr John
Authors: Georgakopoulos, A., Haslegrave, J., Sauerwald, T., and Sylvester, J.
College/School:College of Science and Engineering > School of Computing Science
ISSN:1868-8969
ISBN:9783959771344
Copyright Holders:Copyright © 2020 Agelos Georgakopoulos, John Haslegrave, Thomas Sauerwald, and John Sylvester
First Published:First published in Proceedings of the 11th Innovations in Theoretical Computer Science Conference (ITCS 2020): 76
Publisher Policy:Reproduced under a Creative Commons License

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