Approximation algorithms for geometric networks

Abstract

We study various combinatorial and geometric problems that have applications in ad hoc wireless networks. From a combinatorial point of view, we are particularly interested in dominating set and coloring problems. We study the impact of solving these problems in a distributed manner on unit disk graphs. We also discuss the importance for the nodes in the network to know their location in the plane. More specifically, we provide algorithms that take advantage of the geometric properties of the model without knowing the actual geometry of a given instance. From a geometric point of view, we are interested in computing graphs that approximate shortest paths in geometric networks. Such a graph is called a spanner. The geometric networks under consideration include unit disk graphs, complete k-partite graphs and complete graphs on additively weighted point sets. For each of these three types of graphs, we provide algorithms that compute spanners that have a linear number of edges and a constant spanning ratio. The spanner we propose for unit disk graphs has bounded out-degree and, when applied to complete graphs, admits a local routing strategy. The spanner we propose for complete k-partite graphs has a spanning ratio of at most 5 + e, while 3 — e is a lower bound for any solution to that problem. For additively weighted point sets, we study two spanners. One has a spanning ratio of (1 + e) and the other has constant spanning ratio and admits a plane embedding. Finally, we are interested in the problem of computing geometric spanners under the combinatorial constraint that the output graph must have bounded chromatic number k. When k = 2, 3, 4, we provide algorithms that are optimal in the sense that the upper bound on the spanning ratio of the output graph is also a lower bound for any given algorithm computing a spanner with the given chromatic numbers. We also give upper and lower bounds for greater values of k. Additionally, we consider an on-line variant of the problem where the points are given one after another, and the color of a point must be assigned at the moment when the point is given; thus, later on, the color of a point cannot be changed. This makes the problem more difficult. Consequently, the bounds are higher, but still tight for k = 2, 3, 4.