On approximating arbitrary metrices by tree metrics

Bartal, Yair

doi:10.1145/276698.276725

Cited by 388 publications

(364 citation statements)

References 39 publications

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“…Their aim is to efficiently minimize energies of the form D + V . They consider the class of r-hierarchically well-separated tree (r-HST) metrics (Bartal, 1998) which are a special case of metrics defined above. We discuss r-HST metrics in Section 8, but for now it is enough to understand that they are a special case of metrics and that each has an associated constant r > 1.…”

Section: Observation 1 Feige's Hardness Results Is Evidence That No Pomentioning

confidence: 99%

Minimizing Energies with Hierarchical Costs

et al. 2012

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Computer vision is full of problems elegantly expressed in terms of energy minimization. We characterize a class of energies with hierarchical costs and propose a novel hierarchical fusion algorithm. Hierarchical costs are natural for modeling an array of difficult problems. For example, in semantic segmentation one could rule out unlikely object combinations via hierarchical context. In geometric model estimation, one could penalize the number of unique model families in a solution, not just the number of models-a kind of hierarchical MDL criterion. Hierarchical fusion uses the well-known α-expansion algorithm as a subroutine, and offers a much better approximation bound in important cases.

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Section: Observation 1 Feige's Hardness Results Is Evidence That No Pomentioning

confidence: 99%

Minimizing Energies with Hierarchical Costs

et al. 2012

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“…Probabilistic embedding into trees [9,10,11,19] and spanning trees [7,15,2,6] has been intensively studied, and found numerous applications to approximation and online algorithms, and to fast linear system solvers. While our distortion guarantee does not match the best known worst-case bounds, which are O(log n) for arbitrary trees andÕ(log n) for spanning trees, we give the first probabilisitc embeddings into spanning trees with polylogarithmic scaling distortion in which all the spanning trees in the support of the distribution are light.…”

Section: Related Workmentioning

confidence: 99%

On Notions of Distortion and an Almost Minimum Spanning Tree with Constant Average Distortion

Bartal

Filtser

Neiman

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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Minimum Spanning Trees of weighted graphs are fundamental objects in numerous applications. In particular in distributed networks, the minimum spanning tree of the network is often used to route messages between network nodes. Unfortunately, while being most efficient in the total cost of connecting all nodes, minimum spanning trees fail miserably in the desired property of approximately preserving distances between pairs. While known lower bounds exclude the possibility of the worst case distortion of a tree being small, it was shown in [4] that there exists a spanning tree with constant average distortion. Yet, the weight of such a tree may be significantly larger than that of the MST. In this paper, we show that any weighted undirected graph admits a spanning tree whose weight is at most (1 + ρ) times that of the MST, providing constant average distortion O(1/ρ 2 ). 1 The constant average distortion bound is implied by a stronger property of scaling distortion, i.e., improved distortion for smaller fractions of the pairs. The result is achieved by first showing the existence of a low weight spanner with small prioritized distortion, a property allowing to prioritize the nodes whose associated distortions will be improved. We show that prioritized distortion is essentially equivalent to coarse scaling distortion via a general transformation, which has further implications and may be of independent interest. In particular, we obtain an embedding for arbitrary metrics into Eu-

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“…Other related results refer to probabilistic approximations of metric spaces using tree metrics [6,12]. The idea is to build a set of trees such that their union makes up a t-spanner with high probability.…”

Section: T-spanner Construction Algorithmsmentioning

confidence: 99%

t-Spanners for metric space searching

Navarro¹,

Paredes²,

Chávez³

2007

Data & Knowledge Engineering

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The problem of Proximity Searching in Metric Spaces consists in finding the elements of a set which are close to a given query under some similarity criterion. In this paper we present a new methodology to solve this problem, which uses a t-spanner G ′ (V, E) as the representation of the metric database. A t-spanner is a subgraph G ′ (V, E) of a graph G(V, A), such that E ⊆ A and G ′ approximates the shortest path costs over G within a precision factor t.Our key idea is to regard the t-spanner as an approximation to the complete graph of distances among the objects, and to use it as a compact device to simulate the large matrix of distances required by successful search algorithms such as AESA. The t-spanner properties imply that we can use shortest paths over G ′ to estimate any distance with bounded error factor t.For this sake, several t-spanner construction, updating, and search algorithms are proposed and experimentally evaluated. We show that our technique is competitive against current approaches. For example, in a metric space of documents our search time is only 9% over AESA, yet we need just 4% of its space requirement. Similar results are obtained in other metric spaces.Finally, we conjecture that the essential metric space property to obtain good tspanner performance is the existence of clusters of elements, and enough empirical evidence is given to support this claim. This property holds in most real-world metric spaces, so we expect that t-spanners will display good behavior in most practical applications. Furthermore, we show that t-spanners have a great potential for improvements.

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On approximating arbitrary metrices by tree metrics

Cited by 388 publications

References 39 publications

Minimizing Energies with Hierarchical Costs

Minimizing Energies with Hierarchical Costs

On Notions of Distortion and an Almost Minimum Spanning Tree with Constant Average Distortion

t-Spanners for metric space searching

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