Geometric spanners with applications in wireless networks

In this paper we investigate the relations between spanners, weak spanners, and power spanners in R D for any dimension D and apply our results to topology control in wireless networks. For c ∈ R, a c-spanner is a subgraph of the complete Euclidean graph satisfying the condition that between any two vertices there exists a path of length at most c-times their Euclidean distance. Based on this ability to approximate the complete Euclidean graph, sparse spanners have found many applications, e.g., in FPTAS, geometric searching, and radio networks. In a weak c-spanner, this path may be arbitrarily long, but must remain within a disk or sphere of radius c-times the Euclidean distance between the vertices. Finally in a c-power spanner, the total energy consumed on such a path, where the energy is given by the sum of the squares of the edge lengths on this path, must be at most c-times the square of the Euclidean distance of the direct edge or communication link. While it is known that any c-spanner is also both a weak C1-spanner and a C2-power spanner for appropriate C1, C2 depending only on c but not on the graph under consideration, we show that the converse is not true: there exists a family of c1-power spanners that are not weak C-spanners and also a family of weak c2-spanners that are not C-spanners for any fixed C. However a main result of this paper reveals that any weak c-spanner is also a C-power spanner for an appropriate constant C. We further generalize the latter notion by considering (c, δ)-power spanners where the sum of the δ-th powers of the lengths has to be bounded; so (c, 2)-power spanners coincide with the usual power spanners and (c, 1)-power spanners are classical spanners. Interestingly, these (c, δ)-power spanners form a strict hierarchy where the above results still hold for any δ ≥ D; some even hold for δ > 1 while counter-examples exist for δ < D. We show that every self-similar curve of fractal dimension Df > δ is not a (C, δ)-power spanner for any fixed C, in general. Finally, we consider the sparsified Yao-graph (SparsY-graph or YY) that is a well-known sparse topology for wireless networks. We prove that all SparsY-graphs are weak c-spanners for a constant c and hence they allow us to approximate energy-optimal wireless networks by a constant factor.