On the Longest Edge of Gabriel Graphs in Wireless Ad Hoc Networks

Abstract

Abstract-In wireless ad hoc networks, without fixed infrastructures, virtual backbones are constructed and maintained to efficiently operate such networks. The Gabriel graph (GG) is one of widely used geometric structures for topology control in wireless ad hoc networks. If all nodes have the same maximal transmission radii, the length of the longest edge of the GG is the critical transmission radius such that the GG can be constructed by localized and distributed algorithms using only 1-hop neighbor information. In this paper, we assume a wireless ad hoc network is represented by a Poisson point process with mean n on a unit-area disk, and nodes have the same maximal transmission radii. We give three asymptotic results on the length of the longest edge of the GG. First, weshow that the ratio of the length of the longest edge to ffiffiffiffiffi ffi ln n n q is asymptotically almost surely equal to 2. Next, we show that for any , the expected number of GG edges whose lengths are at least 2 ffiffiffiffiffiffiffiffiffi ffi ln nþ n q is asymptotically equal to 2e À . This implies that ! 1 is an asymptotically almost sure sufficient condition for constructing the GG by 1-hop information. Last, we prove that the number of long edges is asymptotically Poisson with mean 2e À . Therefore, the probability of the event that the length of the longest edge is less than 2 ffiffiffiffiffiffiffiffiffi ffi ln nþ n q is asymptotically equal to exp À2eIndex Terms-Wireless ad hoc network, Gabriel graph, asymptotic probability distribution, the longest edge, poisson point process, topology control.

Joint network lifetime and delay optimization for topology control in heterogeneous wireless multi-hop networks

“…Most existing topology control schemes apply geometric proximity graphs (such as Relative Neighbor Graph (RNG) [1], Gabriel Graph (GG) [2], Delaunay triangulation graph [3], and Minimum Spanning Tree (MST) [4]) to build sparse, but connected links, such as [5][6][7][8][9][10][11][12][13][14].…”

“…The area of energy-saving topology control [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] has attracted a great deal of attention. Nodes in a wireless multi-hop network collaboratively determine their transmission power and define the network topology by forming the proper neighbor relation under the constraint of network connectivity and the criteria with respect to energy efficiency.…”

Asymptotic distribution of critical transmission radius for greedy forward routing

“…Two key techniques used in our remaining proof are the Palm theory for Poisson processes (see, e.g., Theorem 1.6 in [8]) and the Brun's sieve (see, e.g., Theorem 10 in [12]), which are stated below.…”

“…We will frequently change the integral variables using a technique introduced in [12]. Consider a tree topology on planar points 1 , 2 , ⋅ ⋅ ⋅ , , and assume without loss of generality that −1 is an edge in this tree.…”

On the Longest RNG Edge of Wireless Ad Hoc Networks

“…The asymptotic distributions of λ(M ST (P n )) and λ(GG(P n )) were derived in [11] (based on an earlier result [2]) and in [16] respectfully. Specifically, for any constant c,…”