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ALGORITHMICA

2007

2007

The unit ball random geometric graph G = Gd p(λ, n) has as its vertices n points distributed independently and uniformly in the unit ball in Rd, with two vertices adjacent if and only if their ℓp-distance is at most λ. Like its cousin the Erd˝os-R´enyi random graph, G has a connectivity threshold: an asymptotic value for λ in terms of n, above which G is connected and below which G is disconnected. In the connected zone, we determine upper and lower bounds for the graph diameter of G. Speciﬁcally, almost always, diamp(B)(1− o(1))/λ ≤ diam(G) ≤ diamp(B)(1 + O((ln ln n/ ln n)1/d))/λ, where diamp(B) is the ℓp-diameter of the unit ball B. We employ a combination of methods from probabilistic combinatorics and stochastic geometry.

Related Content

Added |
08 Dec 2010 |

Updated |
08 Dec 2010 |

Type |
Journal |

Year |
2007 |

Where |
ALGORITHMICA |

Authors |
Robert B. Ellis, Jeremy L. Martin, Catherine H. Yan |

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