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Geometric Spanners with Applications in Wireless Networks
, 2005
"... 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 cspanner is a subgraph of the complete Euclidean graph satisfying the condition that between any two ..."
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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 cspanner is a subgraph of the complete Euclidean graph satisfying the condition that between any two vertices there exists a path of length at most ctimes 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 cspanner, this path may be arbitrarily long, but must remain within a disk or sphere of radius ctimes the Euclidean distance between the vertices. Finally in a cpower 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 ctimes the square of the Euclidean distance of the direct edge or communication link. While it is known that any cspanner is also both a weak C1spanner and a C2power 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 c1power spanners that are not weak Cspanners and also a family of weak c2spanners that are not Cspanners for any fixed C. However a main result of this paper
Spanners, weak spanners, and power spanners for wireless networks
 PROC. OF 15TH ANNUAL INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION (ISAAC’04
, 2004
"... For, aspanner is a subgraph of a complete Euclidean graph satisfying that between any two vertices there exists a path of weighted length at most times their geometric distance. Based on this property to approximate a complete weighted graph, sparse spanners have found many applications, e.g., in ..."
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For, aspanner is a subgraph of a complete Euclidean graph satisfying that between any two vertices there exists a path of weighted length at most 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 weakspanner, this path may be arbitrary long but must remain within a disk of radiustimes the Euclidean distance between the vertices. Finally in apower 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 mosttimes the square of the geometric distance of the direct link. While it is known that anyspanner is also both a weakspanner and apower spanner (for appropriate depending only on but not on the graph under consideration), we show that the converse fails: There exists a family ofpower spanners that are no weakspanners and also a family of weakspanners that are nospanners for any fixed (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 consideringpower spanners where the sum of theth powers of the lengths has to be bounded; sopower spanners coincide with the usual power spanners andpower spanners are classical spanners. Interestingly, thesepower spanners form a strict hierarchy where the above results still hold for any; some even hold! for while counterexamples exist #" $ for. We show that every selfsimilar curve of fractal % & dimension is ' nopower spanner for any fixed, in general.