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 is an element of 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 C-1-spanner and a C-2-power spanner for appropriate C-1, C-2 depending only on c but not on the graph under consideration, we show that the converse is not true: there exists a family of c(1)-power spanners that are not weak C-spanners and also a family of weak c(2)-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, delta)-power spanners where the sum of the delta th powers of the lengths has to be bounded, so (c, 2)-power spanners coincide with the usual power spanners and (c, I)-power spanners are classical spanners. Interestingly, these (c, delta)-power spanners form a strict hierarchy where the above results still hold for any delta >= D some even hold for delta > 1 while counter-examples exist for delta < D. We show that every self-similar curve of fractal dimension D-f >= delta is not a (C, delta)-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. (c) 2006 Elsevier B.V. All rights reserved.