How to Compute Times of Random Walks based Distributed Algorithms

TL;DR

This paper introduces new methods and algorithms for precisely computing hitting and cover times in weighted graphs, enhancing the analysis of random walk-based distributed algorithms with improved efficiency and applicability to dynamic networks.

Contribution

It generalizes hitting and cover times for weighted graphs, and provides the first algorithms with optimal or near-optimal complexity for their computation, including robustness to topological changes.

Findings

New generalizations of hitting and cover times for weighted graphs.

An algorithm to compute n^2 hitting times with O(n^3) complexity.

An algorithm to compute cover times with O(n^3 2^n) complexity.

Abstract

Random walk based distributed algorithms make use of a token that circulates in the system according to a random walk scheme to achieve their goal. To study their efficiency and compare it to one of the deterministic solutions, one is led to compute certain quantities, namely the hitting times and the cover time. Until now, only bounds on these quantities were defined. First, this paper presents two generalizations of the notions of hitting and cover times to weighted graphs. Indeed, the properties of random walks on symmetrically weighted graphs provide interesting results on random walk based distributed algorithms, such as local load balancing. Both of these generalization are proposed to precisely represent the behaviour of these algorithms, and to take into account what the weights represent. Then, we propose an algorithm to compute the n^2 hitting times on a weighted graph of n vertices, which we improve to obtain a O(n^3) complexity. This complexity is the lowest up to now. This algorithm computes both of the generalizations that we propose for the hitting times on a weighted graph. Finally, we provide the first algorithm to compute the cover time (in both senses) of a graph. We improve it to achieve a complexity of O(n^3 2^n). The algorithms that we present are all robust to a topological change in a limited number of edges. This property allows us to use them on dynamic graphs.