Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs
Résumé
$\delta$-Hyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4-point condition: for any four points $u,v,w,x$, the two larger of the distance sums $d(u,v)+d(w,x), d(u,w)+d(v,x), d(u,x)+d(v,w)$ differ by at most $2\delta$. They play an important role in geometric group theory, geometry of negatively curved spaces, and have recently become of interest in several domains of computer science, including algorithms and networking. For example, (a) it has been shown empirically that the internet topology embeds with better accuracy into a hyperbolic space than into an Euclidean space of comparable dimension, (b) every connected finite graph has an embedding in the hyperbolic plane so that the greedy routing based on the virtual coordinates obtained from this embedding is guaranteed to work. A connected graph $G=(V,E)$ equipped with standard graph metric $d_G$ is $\delta$-{\it hyperbolic} if the metric space $(V,d_G)$ is $\delta$-hyperbolic. In this paper, using our Layering Partition technique, we provide a simpler construction of distance approximating trees of $\delta$-hyperbolic graphs on $n$ vertices with an additive error $O(\delta\log n)$ and show that every $n$-vertex $\delta$-hyperbolic graph has an additive $O(\delta \log n)$-spanner with at most $O(\delta n)$ edges. As a consequence, we show that the family of $\delta$-hyperbolic graphs with $n$ vertices enjoys an $O(\delta\log n)$-additive routing labeling scheme with $O(\delta\log^2n)$ bit labels and $O(\log\delta)$ time routing protocol, and an easier constructable $O(\delta\log n)$-additive distance labeling scheme with $O(\log^2n)$ bit labels and constant time distance decoder.