Relay Nodes in Wireless Sensor Networks

Abstract

Motivated by application to wireless sensor networks, we study the following problem. We are given a set S of wireless sensor nodes, given as a set of points in the two-dimensional plane, and real numbers 0 < r ≤ R. We must place a minimum-size set Q of wireless relay nodes in the two dimensional plane to connect S, where connectivity is explained formally next. The nodes of S can communicate to nodes within distance r, and the relay nodes of Q can communicate within distance R. Once the nodes of Q are placed, they together with S induce an undirected graph G = (V,E) defined as follows: V = S ∪ Q, and $E = \{ uv | u,v \in Q \mbox{ and } ||u,v|| \leq R \} \cup \{ xu | x \in S \mbox{ and } u \in (Q \cup S) \mbox{ and } ||u,x|| \leq r\}$, where ||u,v|| denotes the Euclidean distance from u to v. G must be connected.

It was shown in [1] that an algorithm based on Minimum Spanning Tree achieves approximation ratio 7. We improve the analysis of this algorithm to 6, and propose a post-processing heuristic called Post-Order Greedy to practically improve the performance of the approximation algorithms. Experiments on random instances give Post-Order Greedy applied after Minimum Spanning Tree an average improvement of up to 23%. Applying Post-Order Greedy after an optimum Steiner Tree algorithm results in another circa 6% improvement considering the same instances.