Spanners, Weak Spanners, and Power Spanners for Wireless Networks Christian Schindelhauer  , Klaus Volbert ∗ , and Martin Ziegler ∗ Heinz Nixdorf Institute, Paderborn University Institute of Computer Science, {schindel, kvolbert, ziegler}@uni-paderborn.de Abstract. For c ∈ 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 weak c-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 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 fails: There exists a fam- ily of c1-power spanners that are no weak C-spanners and also a family of weak c2-spanners that are no C-spanners for any fixed C (and thus no uniform span- ners, 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. 1 Motivation Spanners have appeared in Computer Science with the advent of Computational Geom- etry [4, 18], raised further in interest as a tool for approximating NP-hard problems [13] and, quite recently, for routing and topology control in ad-hoc networks [1, 12, 8, 7, 11]. Roughly speaking, they approximate the complete Euclidean graph on a set of geomet- ric vertices while having only linearly many edges. The formal condition for a c-spanner  Partially supported by the DFG-Sonderforschungsbereich 376 and by the EU within 6th Framework Programme under contract 001907 “Dynamically Evolving, Large Scale Infor- mation Systems” (DELIS). R. Fleischer and G. Trippen (Eds.): ISAAC 2004, LNCS 3341, pp. 805–821, 2004. c  Springer-Verlag Berlin Heidelberg 2004