Notes on diameters, centers, and approximating trees of δ-hyperbolic geodesic spaces and graphs

Chepoi, Victor; Dragan, Feodor F.; Estellon, Bertrand; Habib, Michel; Vaxès, Yann

doi:10.1016/j.endm.2008.06.046

Cited by 92 publications

(176 citation statements)

References 2 publications

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“…Krauthgamer and Lee [18] studied the nearest neighbor problem for points in δ -hyperbolic space. Chepoi et al [9,10] advanced this line of research, providing algorithms for computing the diameter and minimum enclosing ball of collections of points in δ -hyperbolic space. Work by Billera et al [6] showed how to model the space of phylogenetic trees as a specific CAT(0) space [8, II.1]; work by Owen and Provan investigated how to efficiently compute geodesics in such a space [24].…”

Section: Related Workmentioning

confidence: 99%

Horoball Hulls and Extents in Positive Definite Space

Fletcher

Moeller

Phillips

et al. 2011

Lecture Notes in Computer Science

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Abstract. 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 non-positively 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.

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Section: Related Workmentioning

confidence: 99%

Horoball Hulls and Extents in Positive Definite Space

Fletcher

Moeller

Phillips

et al. 2011

Lecture Notes in Computer Science

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“…Their results also apply to δ-hyperbolic spaces. In [3], Chepoi et al present schemes for computing an additive approximation of the diameter, center, and radius of δ-hyperbolic spaces and graphs. They also show that several graph classes are δ-hyperbolic and present a linear-time algorithm for approximating trees of n-node δ-hyperbolic graphs with O(δ log n) additive distortion.…”

Section: Related Workmentioning

confidence: 99%

Metric Embedding, Hyperbolic Space, and Social Networks

Verbeek

Suri

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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We consider the problem of embedding an undirected graph into hyperbolic space with minimum distortion. A fundamental problem in its own right, it has also drawn a great deal of interest from applied communities interested in empirical analysis of large-scale graphs. In this paper, we establish a connection between distortion and quasi-cyclicity of graphs, and use it to derive lower and upper bounds on metric distortion. Two particularly simple and natural graphs with large quasi-cyclicity are n-node cycles and n × n square lattices, and our lower bound shows that any hyperbolic-space embedding of these graphs incurs a multiplicative distortion of at least Ω(n/ log n). This is in sharp contrast to Euclidean space, where both of these graphs can be embedded with only constant multiplicative distortion. We also establish a relation between quasi-cyclicity and δ-hyperbolicity of a graph as a way to prove upper bounds on the distortion. Using this relation, we show that graphs with small quasi-cyclicity can be embedded into hyperbolic space with only constant additive distortion. Finally, we also present an efficient (linear-time) randomized algorithm for embedding a graph with small quasi-cyclicity into hyperbolic space, so that with high probability at least a (1 − ε) fraction of the node-pairs has only constant additive distortion. Our results also give a plausible theoretical explanation for why social networks have been observed to embed well into hyperbolic space: they tend to have small quasi-cyclicity.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

On the Hyperbolicity of Small-World and Treelike Random Graphs

Chen¹,

Fang²,

Hu³

et al. 2013

Internet Mathematics

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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 δ-hyperbolicity, and we establish several positive and negative results for small-world and tree-like random graph models. In particular, we show that small-world 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 tree-like graphs called ringed trees that have constant hyperbolicity, adding random links among the leaves in a manner similar to the small-world 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.

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Notes on diameters, centers, and approximating trees of δ-hyperbolic geodesic spaces and graphs

Cited by 92 publications

References 2 publications

Horoball Hulls and Extents in Positive Definite Space

Horoball Hulls and Extents in Positive Definite Space

Metric Embedding, Hyperbolic Space, and Social Networks

On the Hyperbolicity of Small-World and Treelike Random Graphs

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