Efficient Construction of (d+1,3d)-Ruling Set in Wireless Ad Hoc Networks

Abstract

Let G = (V,E) be a graph and V′ ⊆ V . We call V′ a (d,b)-ruling set if for all v _i ,v _j  ∈ V′ the distance dist(v _i ,v _j ) ≥ d and V′ is b-dominating set (i.e. each vertex of G can be reached form some vertex from V′ by a path of length at most b).

Here we consider the problem of finding (d,b)-ruling set in a Unit Disc Graph. Unit Disc Graphs (UDG) are the most natural class of graphs to model wireless ad hoc networks (for example cell phones networks, wireless sensor networks etc.). The set of vertices of UDG is a set of points on the plane and two vertices, say v and w, are connected by an edge if and only if they are at the distance at most one in Euclidean norm ($\left\|v,w\right\| \leq 1$). We also assume that every vertex knows its position on the plane and, for such setting, present a deterministic algorithm which finds (d + 1,3d)-ruling set in UDG, and works in O(poly(d)) time (where d is a parameter).

As a result we are able improve time complexity (from O(log|V(G)|) to O(poly(d))) of several known algorithms for UDG, such as algorithm determining k-dominating set, maximum matching , minimum connected dominating set, minimum spanning tree and regular clustering.

This work was supported by grant N206 017 32/2452 for years 2007-2010.