Probabilistic bounds on the length of a longest edge in delaunay graphs of random points in d-dimensions

Motivated by low energy consumption in geographic routing in wireless networks, there has been recent in- terest in determining bounds on the length of edges in the Delaunay graph of randomly distributed points. Asymptotic results are known for random networks in planar domains. In this paper, we obtain upper and lower bounds that hold with parametric probability in any dimension, for points distributed uniformly at ran- dom in domains with and without boundary. The re- sults obtained are asymptotically tight for all relevant values of such probability and constant number of di- mensions, and show that the overhead produced by boundary nodes in the plane holds also for higher di- mensions. To our knowledge, this is the first compre- hensive study on the lengths of long edges in Delaunay graphs.