On the cover time of random geometric graphs

  • Authors:

  • Chen Avin;Gunes Ercal

  • Affiliations:

  • Computer Science Department, University of California, Los Angeles, Los Angeles, CA;Computer Science Department, University of California, Los Angeles, Los Angeles, CA

  • Venue:
  • ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
  • Year:
  • 2005

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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 ${\mathcal G}(n,r)$ 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 $r \geq r_{\rm opt} {\mathcal G}(n,r)$ has optimal cover time of Θ(n log n) 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(rcon). We are able to draw our results by giving a tight bound on the electrical resistance of ${\mathcal G}(n,r)$ via the power of certain constructed flows.