Fast algorithms for constructing t-spanners and paths with stretch t

Cohen, Edith

doi:10.1109/sfcs.1993.366822

Cited by 69 publications

(94 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2 The underlying network and capacities are arbitrary. As we will later see, uniform multicommodity flow problems arise in many important applications and their structure is sufficiently robust that the methods that we develop in this paper have already proven useful in the study of more general multicommodity flow problems [Klein et al 1989;1993;Garg et al 1996;Plotkin and Tardos 1993].…”

Section: Our Max-flow Min-cut Resultsmentioning

confidence: 89%

Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms

Leighton

Rao

1999

J. ACM

694

666

View full text Add to dashboard Cite

Abstract. In this paper, we establish max-flow min-cut theorems for several important classes of multicommodity flow problems. In particular, we show that for any n-node multicommodity flow problem with uniform demands, the max-flow for the problem is within an O(log n) factor of the upper bound implied by the min-cut. The result (which is existentially optimal) establishes an important analogue of the famous 1-commodity max-flow min-cut theorem for problems with multiple commodities. The result also has substantial applications to the field of approximation algorithms. For example, we use the flow result to design the first polynomial-time (polylog n-times-optimal) approximation algorithms for well-known NP-hard optimization problems such as graph partitioning, min-cut linear arrangement, crossing number, VLSI layout, and minimum feedback arc set. Applications of the flow results to path routing problems, network reconfiguration, communication in distributed networks, scientific computing and rapidly mixing Markov chains are also described in the paper.

show abstract

Section: Our Max-flow Min-cut Resultsmentioning

confidence: 89%

Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms

Leighton

Rao

1999

J. ACM

694

666

View full text Add to dashboard Cite

show abstract

“…The latter requirement guarantees that any two points of the metric space will be connected in T by a path that consists of only a small number of edges or hops. This guarantee turns out to be particularly important for routing [37,1], computing almost shortest paths [22,23], and in other applications. Another parameter that plays an important role in many applications is the maximum (vertex) degree of the constructed tree [6,14,8,37].…”

Section: Background and Main Resultsmentioning

confidence: 99%

Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners

Dinitz

Elkin

Solomon

2009

Discrete Comput Geom

View full text Add to dashboard Cite

We show that for every n-point metric space M and positive integer k, there exists a spanning tree T with unweighted diameter O(k) and weight, and a spanning tree T ′ with weight w(T ′ ) = O(k) · w(M ST (M )) and unweighted diameter O(k · n 1/k ). These trees also achieve an optimal maximum degree. Furthermore, we demonstrate that these trees can be constructed efficiently.We prove that these tradeoffs between unweighted diameter and weight are tight up to constant factors in the entire range of parameters. Moreover, our lower bounds apply to a basic 1-dimensional Euclidean space.Our lower bounds for the particular case of unweighted diameter O(log n) are of independent interest, settling a long-standing open problem in Computational Geometry. In STOC'95 Arya et al. devised a construction of Euclidean Spanners with unweighted diameter O(log n) and weight O(log n) · w(M ST (M )). In SODA'05 Agarwal et al. showed that this result is tight up to a factor of O(log log n). We close this gap and show that the result of Arya et al. is tight up to constant factors.Finally, our upper bounds imply improved approximation algorithms for the minimum h-hop spanning tree and bounded diameter minimum spanning tree problems for metric spaces.

show abstract

“…There are also typically close upper and lower bounds on the radius of each cluster. Just a few of the clustering algorithms with this sort of behavior can be found in [AP92,Coh93,PR10].…”

Section: Graph Clusteringmentioning

confidence: 99%

Better Distance Preservers and Additive Spanners

Bodwin¹,

Williams²

2015

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

View full text Add to dashboard Cite

We make improvements to the upper bounds on several popular types of distance preserving graph sketches. These sketches are all various restrictions of the additive pairwise spanner problem, in which one is given an undirected unweighted graph G, a set of node pairs P , and an error allowance +β, and one must construct a sparse subgraphThe first part of our paper concerns pairwise distance preservers, which make the restriction β = 0 (i.e. distances must be preserved exactly). Our main result here is an upper bound of |H| = O(n 2/3 |P | 2/3 + n|P | 1/3 ) when G is undirected and unweighted. This improves on existing bounds whenever |P | = ω(n 3/4 ), and it is the first such improvement in the last ten years.We then devise a new application of distance preservers to graph clustering algorithms, and we apply this algorithm to subset spanners, which require P = S × S for some node subset S, and (standard) spanners, which require P = V × V . For both of these objects, our construction generalizes the best known bounds when the error allowance is constant, and we obtain the strongest polynomial error/sparsity tradeoff that has yet been reported (in fact, for subset spanners, ours is the first nontrivial construction that enjoys improved sparsity from a polynomial error allowance).We leave open a conjecture that O(n 2/3 |P | 2/3 + n) pairwise distance preservers are possible for undirected unweighted graphs. Resolving this conjecture in the affirmative would improve and simplify our upper bounds for all the graph sketches mentioned above. * gbodwin@stanford.edu †

show abstract

Fast algorithms for constructing t-spanners and paths with stretch t

Cited by 69 publications

References 9 publications

Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms

Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms

Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners

Better Distance Preservers and Additive Spanners

Contact Info

Product

Resources

About