The Cover Time of Deterministic Random Walks

TL;DR

This paper investigates the cover time of deterministic walks on graphs, providing bounds and analyzing how their speed compares to random walks across different graph structures.

Contribution

It introduces techniques for bounding cover times of deterministic walks and compares their efficiency to random walks on various graph classes.

Findings

Deterministic walks can be faster, slower, or equal in speed compared to random walks depending on the graph topology.

The paper provides upper and lower bounds for cover times on key graph classes.

Short-term visit times of deterministic walks are also analyzed.

Abstract

The rotor router model is a popular deterministic analogue of a random walk on a graph. Instead of moving to a random neighbor, the neighbors are served in a fixed order. We examine how fast this "deterministic random walk" covers all vertices (or all edges). We present general techniques to derive upper bounds for the vertex and edge cover time and derive matching lower bounds for several important graph classes. Depending on the topology, the deterministic random walk can be asymptotically faster, slower or equally fast as the classic random walk. We also examine the short term behavior of deterministic random walks, that is, the time to visit a fixed small number of vertices or edges.