The power of two choices for random walks

Georgakopoulos, A., Haslegrave, J., Sauerwald, T. and Sylvester, J. (2022) The power of two choices for random walks. Combinatorics, Probability and Computing, 31(1), pp. 73-100. (doi: 10.1017/s0963548321000183)

Abstract

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 O(n log log n) on bounded degree expanders, and of order O(n( log log n) 2 ) on the Erdos–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.

• Item Type: Articles
• Additional Information: A.G. and J.H. were supported by ERC Starting Grant no. 639046 (RGGC). J.H. was also partially supported by UK Research and Innovation Future Leaders Fellowship MR/S016325/1. T.S. and J.S. were supported by ERC Starting Grant no. 679660 (DYNAMIC MARCH).
• 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
• Journal Name: Combinatorics, Probability and Computing
• Publisher: Cambridge University Press
• ISSN: 0963-5483
• ISSN (Online): 1469-2163
• Published Online: 28 May 2021
• Copyright Holders: Copyright © 2021 The Authors
• First Published: First published in Combinatorics, Probability and Computing 31(1): 73-100
• Publisher Policy: Reproduced under a Creative Commons License