Optimal cover time for a graph-based coupon collector process

Abstract

In this paper we study the following covering process defined over an arbitrary directed graph. Each node is initially uncovered and is assigned a random integer rank drawn from a suitable range. The process then proceeds in rounds. In each round, a uniformly random node is selected and its lowest-ranked uncovered outgoing neighbor, if any, is covered. We prove that if each node has in-degree Θ(d) and out-degree O(d), then with high probability, every node is covered within O(n + n log n/d) rounds, matching a lower bound due to Alon. Alon has also shown that, for a certain class of d-regular expander graphs, the upper bound holds no matter what method is used to choose the uncovered neighbor. In contrast, we show that for arbitrary d-regular graphs, the method used to choose the uncovered neighbor can affect the cover time by more than a constant factor. © Springer-Verlag Berlin Heidelberg 2005.