Relay Nodes in Wireless Sensor Networks

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 (2 call 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 U Q, and E = {uv vertical bar u, v epsilon Q and parallel to u, v parallel to <= R} boolean OR {xu vertical bar x epsilon S and ut epsilon (Q boolean OR S) and parallel to u, x parallel to <= r}, where parallel to u, v parallel to 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 oil random instances give Post-Order Greedy applied after Minimum Spanning Tree all 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.