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The power of two choices for random walks

Part of: Markov processes Discrete mathematics in relation to computer science Theory of computing Graph theory

Published online by Cambridge University Press:  28 May 2021

Agelos Georgakopoulos ,
John Haslegrave ,
Thomas Sauerwald  and
John Sylvester
Agelos Georgakopoulos
Affiliation:
Mathematics Institute, University of Warwick, Coventry, UK
John Haslegrave
Affiliation:
Mathematics Institute, University of Warwick, Coventry, UK
Thomas Sauerwald*
Affiliation:
Department of Computer Science & Technology, University of Cambridge, Cambridge, UK
John Sylvester
Affiliation:
Department of Computer Science & Technology, University of Cambridge, Cambridge, UK
*
*Corresponding author. Email: [email protected]
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Abstract

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We apply the power-of-two-choices paradigm to a random walk on a graph: rather than moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours. We prove that this allows the controller to significantly accelerate the hitting and cover times in several natural graph classes. In particular, we show that the cover time becomes linear in the number n of vertices on discrete tori and bounded degree trees, of order $${\mathcal O}(n\log \log n)$$ on bounded degree expanders, and of order $${\mathcal O}(n{(\log \log n)^2})$$ on the Erdős–Rényi random graph in a certain sparsely connected regime. We also consider the algorithmic question of computing an optimal strategy and prove a dichotomy in efficiency between computing strategies for hitting and cover times.

MSC classification

Primary: 05C81: Random walks on graphs 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 68R10: Graph theory (including graph drawing) 68Q17: Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 1 , January 2022 , pp. 73 - 100
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

A preliminary version of this paper appeared at The 11th Innovations in Theoretical Computer Science Conference (ITCS 2020), volume 151 of LIPIcs, pages 76:1–76:19 [22]

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