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

• Authors: Petra Berenbrink;Colin Cooper;Tom Friedetzky
• Venue: PODC '12 Proceedings of the 2012 ACM symposium on Principles of distributed computing
• Year: 2012

In this paper, 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 (or E-process). We assume there is a rule A, which tells the walk which unvisited edge to use whenever there are several unvisited edges. 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 the class of connected, even degree graphs G of constant maximum degree, we characterize the vertex cover time of the E-process in terms of the edge expansion rate of G, as measured by eigenvalue gap 1-λmax of the transition matrix of a simple random walk on G. Denote by l-good, the property that every vertex is in at least one vertex induced cycle of length l. In particular, for even degree expander graphs, of bounded maximum degree, we have the following result. Let G be an n vertex l-good expander graph. Any E-process on G has cover time CG(E -process)= O (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. With high probability random r-regular graphs, r ≥ 4 even, are expanders for which l = Ω(log n). Thus, for almost all such graphs, the vertex cover time of the E-process is Θ(n). This improves the vertex cover time of such graphs by a factor of log n, compared to the Ω(n log n) cover time of any weighted random walk.