Spanners, Weak Spanners, and Power Spanners for Wireless Networks

For $c in REAL$, 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. For geometric searching, it turned out to suffice whether the radius rather than the length of some path between any two vertices is bounded relatively to their geometric distance; this is the defining property of weak spanners. Finally regarding radio network applications, a power spanner accounts for the total energy afforded for a wireless transmission with the requirement that the sum of the squares of the lengths of some path between any two planar vertices must be bounded relatively to the square of their geometric distance (or higher powers up to 6 or even 8).<br><br>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, surprisingly, any weak spanner is also a uniform power spanner. 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 $(cdot,2)$-power spanners coincide with the usual power spanners and $(cdot,1)$-power spanners are classical spanners. Interestingly, these $(cdot,delta)$-power spanners form a strict hierarchy where the above results still hold for any $deltageq2$; some even hold for $delta>1$ while counterexamples reveal others to fail for $delta<2$. In fact we show that in general every self-similar curve of fractal dimension $d>delta$ is no $(C,delta)$-power spanner for any fixed $C$.