Results 1  10
of
10
Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs
"... δHyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4point 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 2δ. They play an important role in geometric group theory, g ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
(Show Context)
δHyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4point 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 2δ. 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 dG is δhyperbolic if the metric space (V, dG) is δhyperbolic. In this paper, using our Layering Partition technique, we provide a simpler construction of distance approximating trees of δhyperbolic graphs on n vertices with an additive error O(δ log n) and show that every nvertex δhyperbolic graph has an additive O(δ log n)spanner with at most O(δn) edges. As a consequence, we show that the family of δhyperbolic graphs with n vertices enjoys an O(δ log n)additive routing labeling scheme with O(δ log 2 n) bit labels and O(log δ) time routing protocol, and an easier constructable O(δ log n)additive distance labeling scheme with O(log 2 n) bit labels and constant time distance decoder.
On the Hyperbolicity of SmallWorld and TreeLike Random Graphs
"... Abstract. Hyperbolicity is a property of a graph that may be viewed as being a “soft ” version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. Here, we consider Gromov’s notion of δhype ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
(Show Context)
Abstract. Hyperbolicity is a property of a graph that may be viewed as being a “soft ” version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. Here, we consider Gromov’s notion of δhyperbolicity, and we establish several positive and negative results for smallworld and treelike random graph models. In particular, we show that smallworld random graphs built from underlying grid structures do not have strong improvement in hyperbolicity, even when the rewiring greatly improves decentralized navigation. On the other hand, for a class of treelike graphs called ringed trees that have constant hyperbolicity, adding random links among the leaves in a manner similar to the smallworld graph constructions may easily destroy the hyperbolicity of the graphs, except for a class of random edges added using an exponentially decaying probability function based on the ring distance among the leaves. Our study provides the first significant analytical results on the hyperbolicity of a rich class of random graphs, which shed light on the relationship between hyperbolicity and navigability of random graphs, as well as on the sensitivity of hyperbolic δ to noises in random graphs.
Hyperbolicity and chordality of a graph
 2p + ⌊ε/2⌋, and d = 2p + ⌊ε/2⌋ + 1, which gives S1 = 2, S2 = 4p + 2 ⌊ε/2⌋, and so h(a, b, c, d) = ε − 2 − 2 ⌊ε/2⌋  ≤ 2. Let us now assume that S2 = max {S1, S2, S3}. Since S1 + S3 = 4p + ε, the
"... Let G be a connected graph with the usual shortestpath metric d. The graph G is δhyperbolic provided for any vertices x,y,u,v in it, the two larger of the three sums d(u,v) + d(x,y),d(u,x) + d(v,y) and d(u,y) + d(v,x) differ by at most 2δ. The graph G is kchordal provided it has no induced cycle ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
Let G be a connected graph with the usual shortestpath metric d. The graph G is δhyperbolic provided for any vertices x,y,u,v in it, the two larger of the three sums d(u,v) + d(x,y),d(u,x) + d(v,y) and d(u,y) + d(v,x) differ by at most 2δ. The graph G is kchordal provided it has no induced cycle of length greater than k. Brinkmann, Koolen and Moulton find that every 3chordal graph is 1hyperbolic and that graph is not 1hyperbolic if and only if it contains one of two special graphs 2 as an isometric subgraph. For every k ≥ 4, we show that a kchordal graph must be ⌊ k 2 ⌋ k−2
Cop and robber game and hyperbolicity
, 2013
"... Abstract. In this note, we prove that all copwin graphs G in the game in which the robber and the cop move at different speeds s and s ′ with s ′ < s, are δhyperbolic with δ = O(s2). We also show that the dependency between δ and s is linear if s − s ′ = Ω(s) and G obeys a slightly stronger co ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract. In this note, we prove that all copwin graphs G in the game in which the robber and the cop move at different speeds s and s ′ with s ′ < s, are δhyperbolic with δ = O(s2). We also show that the dependency between δ and s is linear if s − s ′ = Ω(s) and G obeys a slightly stronger condition. This solves an open question from the paper J. Chalopin et al., Cop and robber games when the robber can hide and ride, SIAM J. Discr. Math. 25 (2011) 333–359. Since any δhyperbolic graph is copwin for s = 2r and s ′ = r + 2δ for any r> 0, this establishes a new –gametheoretical – characterization of Gromov hyperbolicity. We also show that for weakly modular graphs the dependency between δ and s is linear for any s ′ < s. Using these results, we describe a simple constantfactor approximation of the hyperbolicity δ of a graph on n vertices in O(n2) time when the graph is given by its distancematrix. 1.
Horoball Hulls and Extents in Positive Definite Space
"... The space of positive definite matrices P(n) is a Riemannian manifold with variable nonpositive curvature. It includes Euclidean space and hyperbolic space as submanifolds, and poses significant challenges for the design of algorithms for data analysis. In this paper, we develop foundational geometr ..."
Abstract
 Add to MetaCart
(Show Context)
The space of positive definite matrices P(n) is a Riemannian manifold with variable nonpositive curvature. It includes Euclidean space and hyperbolic space as submanifolds, and poses significant challenges for the design of algorithms for data analysis. In this paper, we develop foundational geometric structures and algorithms for analyzing collections of such matrices. A key technical contribution of this work is the use of horoballs, a natural generalization of halfspaces for nonpositively curved Riemannian manifolds. We propose generalizations of the notion of a convex hull and a centerpoint and approximations of these structures using horoballs and based on novel decompositions of P(n). This leads to an algorithm for approximate hulls using a generalization of extents.
Diameters, Centers, and Approximating Trees of δHyperbolic Geodesic Spaces and Graphs (Extended Abstract)
, 2008
"... δHyperbolic metric spaces have been defined by M. Gromov via a simple 4point condition: for any four points u, v, w, x, the two larger of the sums d(u, v) +d(w, x),d(u, w) + d(v, x),d(u, x) +d(v, w) differbyatmost2δ. Given a finite set S of points of a δhyperbolic space, we present simple and fas ..."
Abstract
 Add to MetaCart
δHyperbolic metric spaces have been defined by M. Gromov via a simple 4point condition: for any four points u, v, w, x, the two larger of the sums d(u, v) +d(w, x),d(u, w) + d(v, x),d(u, x) +d(v, w) differbyatmost2δ. Given a finite set S of points of a δhyperbolic space, we present simple and fast methods for approximating the diameter of S with an additive error 2δ and computing an approximate radius and center of a smallest enclosing ball for S with an additive error 3δ. These algorithms run in linear time for classical hyperbolic spaces and for δhyperbolic graphs and networks. Furthermore, we show that for δhyperbolic graphs G =(V,E) with uniformly bounded degrees of vertices, the exact center of S can be computed in linear time O(E). We also provide a simple construction of distance approximating trees of δhyperbolic graphs G on n vertices with an additive error O(δ log 2 n). This construction has an additive error comparable with that given by Gromov for npoint δhyperbolic spaces, but can be implemented in O(E) time (instead of O(n 2)). Finally, we establish that several geometrical classes of graphs have bounded hyperbolicity.
IN
, 2014
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
Abstract
 Add to MetaCart
(Show Context)
HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Wenjie Fang
"... Hyperbolicity is a property of a graph that may be viewed as being a “soft ” version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. Here, we consider Gromov’s notion of δhyperbolicity, ..."
Abstract
 Add to MetaCart
(Show Context)
Hyperbolicity is a property of a graph that may be viewed as being a “soft ” version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. Here, we consider Gromov’s notion of δhyperbolicity, and we establish several positive and negative results for smallworld and treelike random graph models. First, we study the hyperbolicity of the class of Kleinberg smallworld random graphs KSW (n, d, γ), where n is the number of vertices in the graph, d is the dimension of the underlying base grid B, and γ is the smallworld parameter such that each node u in the graph connects to another node v in the graph with probability proportional to 1/dB(u, v)γ with dB(u, v) being the grid distance from u to v in the base grid B. We show that when γ = d, the parameter value allowing efficient decentralized routing in Kleinberg’s smallworld network, with probability 1 − o(1) the hyperbolic δ is Ω((log n) 11.5(d+1)+ε) for any ε> 0 independent of n. Comparing to the diameter of Θ(log n) in this case, it indicates that hyperbolicity is not significantly improved comparing to graph diameter even when the longrange connections greatly improves decentralized navigation. We also show that for other values of γ the hyperbolic δ is either at the same level or very close to the graph diameter, indicating poor hyperbolicity in these graphs as well.