Speeding up random walks with neighborhood exploration

We consider the following marking process (rw-rand) made by a random walk on an undirected graph G. Upon arrival at a vertex v, it marks v if unmarked and otherwise it marks a randomly chosen unmarked neighbor of v. We also consider a variant of this process called rw-r-rank. Here each vertex is assigned a global random rank rst and then in each step, the walk marks the lowest ranked unmarked neighbor of the currently visited vertex. Depending on the degree and the expansion of the graph, we prove several upper bounds on the time required by these processes to mark all vertices. For instance, if G is a hypercube or random graph, our processes mark all vertices in time O(n), signicantly speeding up the Θ(n log n)-cover time of standard random walks. ∗ partially supported by NSERC. †

• Type: Journal
• Year: 2010
• Where: SODA
• Authors: Petra Berenbrink, Colin Cooper, Robert Elsaesser, Tomasz Radzik, Thomas Sauerwald
• Keywords: Algorithms, Random Walks, SODA 2010, Undirected Graph G, Unmarked Neighbor