Probabilistic Bounds on the Length of a Longest Edge in Delaunay Graphs of Random Points in d-Dimensions

TL;DR

This paper establishes probabilistic bounds on the length of the longest edge in Delaunay graphs of randomly distributed points in any dimension, extending known planar results to higher dimensions with tight asymptotic bounds.

Contribution

It provides the first comprehensive bounds on long edge lengths in Delaunay graphs across multiple dimensions, including boundary effects, with asymptotic tightness.

Findings

Bounds hold with high probability in any dimension

Boundary effects are similar across dimensions

Results are asymptotically tight for relevant probabilities

Abstract

Motivated by low energy consumption in geographic routing in wireless networks, there has been recent interest 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 random in domains with and without boundary. The results obtained are asymptotically tight for all relevant values of such probability and constant number of dimensions, and show that the overhead produced by boundary nodes in the plane holds also for higher dimensions. To our knowledge, this is the first comprehensive study on the lengths of long edges in Delaunay graphs