Distance Labeling in Hyperbolic Graphs

Abstract

Abstract.A graph G is δ-hyperbolic if for any four vertices u, v, x, y of G the two larger of the three distance sums dG(u, v) + dG(x, y), dG(u, x) + dG(v, y), dG(u, y) + dG(v, x) differ by at most δ, and the smallest δ 0 for which G is δ-hyperbolic is called the hyperbolicity of G. In this paper, we construct a distance labeling scheme for bounded hyperbolicity graphs, that is a vertex labeling such that the distance between any two vertices of G can be estimated from their labels, without any other source of information. More precisely, our scheme assigns labels of O(log 2 n) bits for bounded hyperbolicity graphs with n vertices such that distances can be approximated within an additive error of O(log n). The label length is optimal for every additive error up to n ε . We also show a lower bound of Ω(log log n) on the approximation factor, namely every s-multiplicative approximate distance labeling scheme on bounded hyperbolicity graphs with polylogarithmic labels requires s = Ω(log log n).

On the Hyperbolicity of Small-World and Treelike Random Graphs

“…Since trees are an important class of graphs and since tree-like graphs arise in numerous applications, the idea of hyperbolicity has received attention in a range of applications. For example, it has found usefulness in the visualization of the Internet, the Web, and other large graphs [22,26,31]; it has been applied to questions of compact routing, navigation, and decentralized search in Internet graphs and small-world social networks [11,19,1,20,8]; and it has been applied to a range of other problems such as distance estimation, sensor networks, and traffic flow and congestion minimization [2,13,27,10].…”

“…Thus, our work is intended to fill this gap. Second, a number of algorithmic studies show that good graph hyperbolicity leads to efficient distance labeling and routing schemes [6,11,9,7,21,8], and the routing infrastructure of the Internet is also empirically shown to be hyperbolic [2]. Thus, it is interesting to further investigate if efficient routing capability implies good graph hyperbolicity.…”

“…There are a number of works that connect graph hyperbolicity with efficient distance labeling and routing schemes [6,11,9,7,21,8]. Understanding the relationship between graph hyperbolicity and the ability of efficient routing is one motivation of our research.…”

Gromov hyperbolicity of periodic planar graphs

“…For example, it has been shown empirically in [20] that the internet topology embeds with better accuracy into hyperbolic space than into Euclidean space of comparable dimension. A few algorithmic problems in hyperbolic spaces and hyperbolic graphs have been considered in recent papers (see [21,22,23,24]). Another interesting application of these spaces is secure transmission of information on the internet (see [9,10,11]).…”

Gromov hyperbolicity of planar graphs

“…This theory was mainly applied to the study of automatic groups [31], which appear in computational science. The concept of hyperbolicity appears also in discrete mathematics, in particular, a number of algorithmic problems in hyperbolic spaces and hyperbolic graphs has been considered in recent papers [12,13,15,28]. Another application of these spaces is secure transmission of information on the internet [21][22][23], playing a significant role in the spread of viruses through the network [21,23].…”