Random walks which prefer unvisited edges. Exploring high girth even degree expanders in linear time

TL;DR

This paper introduces a modified random walk, called an edge-process, that prefers unvisited edges, and proves it achieves faster cover times on certain high-girth, even-degree expander graphs, independent of the edge selection rule.

Contribution

It establishes that edge-processes on even degree expanders have a cover time of n + (n log n)/L, improving previous bounds and showing the process's efficiency regardless of the rule used.

Findings

Edge-processes have cover time O(n) on high-girth expanders.

Random regular graphs with degree at least 4 are expanders with L = Omega(log n).

The process improves cover time from Omega(n log n) to Theta(n) for most such graphs.

Abstract

We consider a modified random walk which uses unvisited edges whenever possible, and makes a simple random walk otherwise. We call such a walk an edge-process. We assume there is a rule A, which tells the walk which unvisited edge to use whenever there is a choice. In the simplest case, A is a uniform random choice over unvisited edges incident with the current walk position. However we do not exclude arbitrary choices of rule A. For example, the rule could be determined on-line by an adversary, or could vary from vertex to vertex. For even degree expander graphs, of bounded maximum degree, we have the following result. Let G be an n vertex even degree expander graph, for which every vertex is in at least one vertex induced cycle of length L. Any edge-process on G has cover time (n+ (n log n)/L). This result is independent of the rule A used to select the order of the unvisited edges, which can be chosen on-line by an adversary. As an example, With high probability, random r-regular graphs, (r at least 4, even), are expanders for which L = Omega(log n). Thus, for almost all such graphs, the vertex cover time of the edge-process is Theta(n). This improves the vertex cover time of such graphs by a factor of log n, compared to the Omega(n log n) cover time of any weighted random walk.