Nearly Tight Low Stretch Spanning Trees

Abraham, Ittai; Bartal, Yair; Neiman, Ofer

doi:10.1109/focs.2008.62

Cited by 81 publications

(192 citation statements)

References 16 publications

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“…With a slight adaptation of the metric embeddings terminology to our particular setting, the basic idea in this approach is to compute a random spanning tree T ⊆ G, sampled from a distribution T over a set of spanning trees in a way that pairwise distances do not get "stretched" by much in expectation. This line of work [2,4] has evolved into a near-optimal bound due to Abraham, Bartal, and Neiman [1], who showed how to sample a random spanning tree such that the expected stretch is O(log n) uniformly over all vertex pairs, that is,…”

Section: Approximate Shortest Pathsmentioning

confidence: 99%

Approximation Algorithms and Hardness Results for Shortest Path Based Graph Orientations

Blokh

Segev

Sharan

2012

Combinatorial Pattern Matching

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Abstract. The graph orientation problem calls for orienting the edges of an undirected graph so as to maximize the number of pre-specified source-target vertex pairs that admit a directed path from the source to the target. Most algorithmic approaches to this problem share a common preprocessing step, in which the input graph is reduced to a tree by repeatedly contracting its cycles. While this reduction is valid from an algorithmic perspective, the assignment of directions to the edges of the contracted cycles becomes arbitrary, and the connecting source-target paths may be arbitrarily long. In the context of biological networks, the connection of vertex pairs via shortest paths is highly motivated, leading to the following variant: Given an undirected graph and a collection of source-target vertex pairs, assign directions to the edges so as to maximize the number of pairs that are connected by a shortest (in the original graph) directed path. Here we study this variant, provide strong inapproximability results for it and propose an approximation algorithm for the problem, as well as for relaxations of it where the connecting paths need only be approximately shortest.

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Section: Approximate Shortest Pathsmentioning

confidence: 99%

Approximation Algorithms and Hardness Results for Shortest Path Based Graph Orientations

Blokh

Segev

Sharan

2012

Combinatorial Pattern Matching

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“…With a slight adaptation of the metric embeddings terminology to our particular setting, the basic idea in this approach is to compute a random spanning tree T 4 G, sampled from a distribution T over a set of spanning trees in a way that pairwise distances do not get ''stretched'' by much in expectation. This line of work (Alon et al, 1995;Elkin et al, 2008) has evolved into a near-optimal bound due to Abraham et al (2008), who showed how to sample a random spanning tree such that the expected stretch is Õ (log n) uniformly over all vertex pairs, that is,…”

Section: Approximate Shortest Pathsmentioning

confidence: 99%

“…We begin by computing a random spanning tree T using the embedding method of Abraham et al (2008). With respect to this tree, let P small 4 P be the collection of pairs whose shortest path distances have not been significantly stretched beyond a factor of w(n), which will be formally defined as 954 BLOKH ET AL.…”

Section: Approximate Shortest Pathsmentioning

confidence: 99%

The Approximability of Shortest Path-Based Graph Orientations of Protein–Protein Interaction Networks

Blokh

Segev

Sharan

2013

Journal of Computational Biology

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The graph orientation problem calls for orienting the edges of an undirected graph so as to maximize the number of prespecified source-target vertex pairs that admit a directed path from the source to the target. Most algorithmic approaches to this problem share a common preprocessing step, in which the input graph is reduced to a tree by repeatedly contracting its cycles. Although this reduction is valid from an algorithmic perspective, the assignment of directions to the edges of the contracted cycles becomes arbitrary and, consequently, the connecting source-target paths may be arbitrarily long. In the context of biological networks, the connection of vertex pairs via shortest paths is highly motivated, leading to the following variant: Given an undirected graph and a collection of source-target vertex pairs, assign directions to the edges so as to maximize the number of pairs that are connected by a shortest (in the original graph) directed path. Here we study this variant, provide strong inapproximability results for it, and propose approximation algorithms for the problem, as well as for relaxations where the connecting paths need only be approximately shortest.

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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%

“…T ). 2 In [4], it was shown that for every weighted graph, it is possible to find a spanning tree which has constant average distortion.…”

Section: Introductionmentioning

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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Nearly Tight Low Stretch Spanning Trees

Cited by 81 publications

References 16 publications

Approximation Algorithms and Hardness Results for Shortest Path Based Graph Orientations

Approximation Algorithms and Hardness Results for Shortest Path Based Graph Orientations

The Approximability of Shortest Path-Based Graph Orientations of Protein–Protein Interaction Networks

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

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