Spanners, Weak Spanners, and Power Spanners for Wireless Networks

Abstract

For c ∈ $\mathbb R$, a c-spanner is a subgraph of a complete Euclidean graph satisfying that between any two vertices there exists a path of weighted length at most c times their geometric distance. Based on this property to approximate a complete weighted graph, sparse spanners have found many applications, e.g., in FPTAS, geometric searching, and radio networks. In a weakc-spanner, this path may be arbitrary long but must remain within a disk 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 geometric distance of the direct 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 fails: There exists a family of c ₁-power spanners that are no weak C-spanners and also a family of weak c ₂-spanners that are no C-spanners for any fixed C (and thus no uniform spanners, either). However the deepest result of the present work reveals that any weak spanner is also a uniform power spanner. 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 (·,2)-power spanners coincide with the usual power spanners and (·,1)-power spanners are classical spanners. Interestingly, these (·,δ)-power spanners form a strict hierarchy where the above results still hold for any δ ≥ 2; some even hold for δ > 1 while counterexamples exist for δ < 2. We show that every self-similar curve of fractal dimension d > δ is no (C,δ)-power spanner for any fixed C, in general.