## On the Cover Time of Random Geometric Graphs (2005)

Venue: | In: ICALP. (2005 |

Citations: | 17 - 4 self |

### BibTeX

@INPROCEEDINGS{Avin05onthe,

author = {Chen Avin and Gunes Ercal},

title = {On the Cover Time of Random Geometric Graphs},

booktitle = {In: ICALP. (2005},

year = {2005},

pages = {677--689}

}

### OpenURL

### Abstract

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 ≥ ropt 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 G(n, r) via the power of certain constructed flows. 1

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Citation Context ...d width associated with cover time for random geometric graphs. Our results improve upon bounds on the cover time obtained through bounding the mixing-time and spectral gap of random geometric graphs =-=[27, 20, 19]-=-, as cover time can be bounded by the spectral gap [9]. In particular, the spectral gap method only guarantees optimal cover time of G(n,r) for r = Θ(1). Aside from that, our results also have importa... |

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Citation Context ...he partial cover time through (2). A powerful method used to bound resistance is by bounding the power of a current flow in the network. The following definitions and propositions from the literature =-=[8, 32, 35]-=- help to formalize this method. Definition 1 (Power of a flow). Given an electrical network (V,E,ρ), with resistance ρ(e) for each edge e, a flow c from a source u to a sink v is a function from V × V... |

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Citation Context ...d width associated with cover time for random geometric graphs. Our results improve upon bounds on the cover time obtained through bounding the mixing-time and spectral gap of random geometric graphs =-=[27, 20, 19]-=-, as cover time can be bounded by the spectral gap [9]. In particular, the spectral gap method only guarantees optimal cover time of G(n,r) for r = Θ(1). Aside from that, our results also have importa... |

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