On the cover time of random geometric graphs

Abstract

The cover time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of ad-hoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have been the subject of much study. A random geometric graph G(n, r) is obtained by placing n points uniformly at random on the unit square and connecting two points iff their Euclidean distance is at most r. The phase transition behavior with respect to the radius r of such graphs has been of special interest. We show that there exists a critical radius ropt such that for any r ≥ r opt G(n, r) has optimal cover time of Θ(rcon) with high probability, and, importantly, ropt = Θ(rCOn) where rCOn denotes the critical radius guaranteeing asymptotic connectivity. Moreover, since a disconnected graph has infinite cover time, there is a phase transition and the corresponding threshold width is O(r con). We are able to draw our results by giving a tight bound on the electrical resistance of G(n, r) via the power of certain constructed flows. © Springer-Verlag Berlin Heidelberg 2005.