Diameter and Broadcast Time of Random Geometric Graphs in Arbitrary Dimensions
 Tobias Friedrich,
 Thomas Sauerwald,
 Alexandre Stauffer
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A random geometric graph (RGG) is defined by placing n points uniformly at random in [0,n ^{1/d }]^{ d }, and joining two points by an edge whenever their Euclidean distance is at most some fixed r. We assume that r is larger than the critical value for the emergence of a connected component with Ω(n) nodes. We show that, with high probability (w.h.p.), for any two connected nodes with a Euclidean distance of \(\omega (\frac{\log n}{r^{d1}} )\) , their graph distance is only a constant factor larger than their Euclidean distance. This implies that the diameter of the largest connected component is Θ(n ^{1/d }/r) w.h.p. We also prove that the condition on the Euclidean distance above is essentially tight.
We also analyze the following randomized broadcast algorithm on RGGs. At the beginning, only one node from the largest connected component of the RGG is informed. Then, in each round, each informed node chooses a neighbor independently and uniformly at random and informs it. We prove that w.h.p. this algorithm informs every node in the largest connected component of an RGG within Θ(n ^{1/d }/r+logn) rounds.
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 Title
 Diameter and Broadcast Time of Random Geometric Graphs in Arbitrary Dimensions
 Journal

Algorithmica
Volume 67, Issue 1 , pp 6588
 Cover Date
 20130901
 DOI
 10.1007/s004530129710y
 Print ISSN
 01784617
 Online ISSN
 14320541
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Random geometric graphs
 Diameter
 Randomized rumor spreading
 Industry Sectors
 Authors

 Tobias Friedrich ^{(1)}
 Thomas Sauerwald ^{(2)}
 Alexandre Stauffer ^{(3)}
 Author Affiliations

 1. FriedrichSchillerUniversität Jena, Jena, Germany
 2. MaxPlanckInstitut für Informatik, Saarbrücken, Germany
 3. Microsoft Research, Redmond, WA, USA