On the cover time of random geometric graphs

Chen Avin, Gunes Ercal

Research output: Contribution to journalConference articlepeer-review

26 Scopus citations

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.

Original languageEnglish
Pages (from-to)677-689
Number of pages13
JournalLecture Notes in Computer Science
Volume3580
DOIs
StatePublished - 1 Jan 2005
Externally publishedYes
Event32nd International Colloquium on Automata, Languages and Programming, ICALP 2005 - Lisbon, Portugal
Duration: 11 Jul 200515 Jul 2005

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science (all)

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