Reconstructing approximate tree metrics

Abraham, Ittai; Balakrishnan, Mahesh; Kühn, Fabian; Malkhi, Dahlia; Ramasubramanian, Venugopalan; Talwar, Kunal

doi:10.1145/1281100.1281110

Cited by 38 publications

(88 citation statements)

References 31 publications

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“…This definition is in contrast to the -four point condition defined in [1], which is a condition on the lengths of matchings on four points rather than the lengths of edges on three points. In particular, a metric (V, d) satisfies the -four point…”

Section: Definitions and Resultsmentioning

confidence: 99%

“…Our work is heavily influenced by the work of Abraham et al [1], who gave an embedding of metrics satisfying the related -four point condition into tree metrics. While our techniques are not the same (since we focus on ultrametrics), the motivation (network latency prediction) is, and part of the original motivation for this work was an attempt to make a distributed version of the algorithm of [1].…”

Section: Related Workmentioning

confidence: 99%

“…We take our inspiration from the work of Abraham et al [1] who tried to use embeddings into tree metrics instead of into Euclidean space. Through experimental measurements they discovered that many networks have latencies that "locally" look like trees, and in particular satisfy an -four point condition that they invented.…”

Section: Introductionmentioning

confidence: 99%

“…We define an -three point condition (as opposed to the four point condition of [1]) and show that there are good embeddings into ultrametrics when is small. Since ultrametrics are a special case of tree metrics this is also a tree embedding, but they have the extra property that they can be represented by spanning trees, so no Steiner nodes are required.…”

Section: Introductionmentioning

confidence: 99%

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Online, Dynamic, and Distributed Embeddings of Approximate Ultrametrics

Dinitz

2008

Lecture Notes in Computer Science

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Abstract. The theoretical computer science community has traditionally used embeddings of finite metrics as a tool in designing approximation algorithms. Recently, however, there has been considerable interest in using metric embeddings in the context of networks to allow network nodes to have more knowledge of the pairwise distances between other nodes in the network. There has also been evidence that natural network metrics like latency and bandwidth have some nice structure, and in particular come close to satisfying an -three point condition or anfour point condition. This empirical observation has motivated the study of these special metrics, including strong results about embeddings into trees and ultrametrics. Unfortunately all of the current embeddings require complete knowledge about the network up front, and so are less useful in real networks which change frequently. We give the first metric embeddings which have both low distortion and require only small changes in the structure of the embedding when the network changes. In particular, we give an embedding of semimetrics satisfying an -three point condition into ultrametrics with distortion (1 + ) log n+4 and the property that any new node requires only O(n 1/3 ) amortized edge swaps, where we use the number of edge swaps as a measure of "structural change". This notion of structural change naturally leads to small update messages in a distributed implementation in which every node has a copy of the embedding. The natural offline embedding has only (1+ ) log n distortion but can require Ω(n) amortized edge swaps per node addition. This online embedding also leads to a natural dynamic algorithm that can handle node removals as well as insertions.

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Section: Definitions and Resultsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Online, Dynamic, and Distributed Embeddings of Approximate Ultrametrics

Dinitz

2008

Lecture Notes in Computer Science

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“…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].…”

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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Metric tree‐like structures in real‐world networks: an empirical study

2015

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Based on solid theoretical foundations, we present strong evidence that a number of real-world networks, taken from different domains (such as Internet measurements, biological data, web graphs, and social and collaboration networks) exhibit tree-like structures from a metric point of view. We investigate a few graph parameters, namely, the tree-distortion and the tree-stretch, the tree-length and the tree-breadth, Gromov's hyperbolicity, the cluster-diameter and the cluster-radius in a layering partition of a graph; such parameters capture and quantify this phenomenon of being metrically close to a tree. By bringing all those parameters together, we provide efficient means for detecting such metric tree-like structures in large-scale networks. We also show how such structures can be used. For example, they are helpful in efficient and compact encoding of approximate distance and almost shortest path information and in quick and accurate estimation of diameters and radii of those networks. Estimating the diameter and estimating the radius of a graph (or distances between arbitrary vertices) are fundamental primitives in many network and graph mining algorithms.

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Reconstructing approximate tree metrics

Cited by 38 publications

References 31 publications

Online, Dynamic, and Distributed Embeddings of Approximate Ultrametrics

Online, Dynamic, and Distributed Embeddings of Approximate Ultrametrics

On the Hyperbolicity of Small-World and Treelike Random Graphs

Metric tree‐like structures in real‐world networks: an empirical study

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