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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 ..."
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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 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.
Sparse spanners vs. compact routing.
 In Proc. 23th ACM Symp. on Parallel Algorithms and Architectures (SPAA),
, 2011
"... ABSTRACT Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables ofΘ( ..."
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ABSTRACT Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables ofΘ(
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 ..."
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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
Low distortion Delaunay embedding of trees in hyperbolic plane
 IN PROC. 19TH INTERNATIONAL SYMPOSIUM ON GRAPH DRAWING
, 2012
"... This paper considers the problem of embedding trees into the hyperbolic plane. We show that any tree can be realized as the Delaunay graph of its embedded vertices. Particularly, a weighted tree can be embedded such that the weight on each edge is realized as the hyperbolic distance between its e ..."
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This paper considers the problem of embedding trees into the hyperbolic plane. We show that any tree can be realized as the Delaunay graph of its embedded vertices. Particularly, a weighted tree can be embedded such that the weight on each edge is realized as the hyperbolic distance between its embedded vertices. Thus the embedding preserves the metric information of the tree along with its topology. Further, the distance distortion between non adjacent vertices can be made arbitrarily small – less than a (1 + ε) factor for any given ε. Existing results on low distortion of embedding discrete metrics into trees carry over to hyperbolic metric through this result. The Delaunay character implies useful properties such as guaranteed greedy routing and realization as minimum spanning trees.
Computing with Advice: when Knowledge Helps
, 2013
"... In several areas of computer science the possibility and efficiency of the solution is determined by information that is not accessible to the algorithm. Traditionally, a qualitative approach to the study of this information has been pursued, in which the impact of enhancing the algorithm with vario ..."
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In several areas of computer science the possibility and efficiency of the solution is determined by information that is not accessible to the algorithm. Traditionally, a qualitative approach to the study of this information has been pursued, in which the impact of enhancing the algorithm with various specific types of information has been studied. Recently, a number of authors have proposed a quantitative approach, where the amount of the added information is studied in relation with the improvement of the quality or efficiency of the solution. We survey several recent examples of this approach from the area of distributed and online computing.
THEME Networks and TelecommunicationsTable of contents
"... 4.1.2. Connected graph searching 2 ..."
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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 ..."
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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.