The Power of Two Choices for Random Walks

TL;DR

This paper demonstrates that applying the power-of-two-choices paradigm to random walks on graphs significantly accelerates hitting and cover times across various graph classes, with implications for algorithmic strategy optimization.

Contribution

It introduces the application of the power-of-two-choices paradigm to random walks, showing substantial improvements in cover and hitting times on different graph types.

Findings

Cover time is linear in n on tori and trees.

Cover time is O(n log log n) on expanders.

Cover time is O(n (log log n)^2) on sparse Erdős-Rényi graphs.

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 $\mathcal{O}(n \log \log n)$ on bounded degree expanders, and of order $\mathcal{O}(n (\log \log n)^2)$ on the Erd\H{o}s-R\'{e}nyi 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.