E. Cohen and U. Zwick. All-pairs small-stretch paths. Journal of Algorithms, 38:335-353, 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....distance and the shortest LCA ancestral distance. We calculate the allpairs shortest ancestral distance in O(n 2:575 ) and all pairs shortest LCA ancestral distance in O(n 2:688 ) There has been considerable interest in fast (subcubic) algorithms for the all pairs shortest path problem [Fre76, Sei92, AGM91, Zwi98, CZ97, CZ96, Zwi99]. We give subcubic bounds for the all pairs longest path problem in DAGs and a matrix multiplication lower bound. Experimental Results. We implement and measure the performance of various LCA algorithms for rooted trees and DAGs. We compare three di erent tree LCA algorithms, 1) a naive ....

E. Cohen and U. Zwick. All-pairs small-stretch paths. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pages 93-102, New Orleans, Louisiana, 5-7 January 1997.

....distance and the shortest LCA ancestral distance. We calculate the allpairs shortest ancestral distance in O(n 2:575 ) and all pairs shortest LCA ancestral distance in O(n 2:688 ) There has been considerable interest in fast (subcubic) algorithms for the all pairs shortest path problem [Fre76, Sei92, AGM91, Zwi98, CZ97, CZ96, Zwi99]. We give subcubic bounds for the all pairs longest path problem in DAGs and a matrix multiplication lower bound. Experimental Results. We implement and measure the performance of various LCA algorithms for rooted trees and DAGs. We compare three di erent tree LCA algorithms, 1) a naive ....

E. Cohen and U. Zwick. All-pairs small-stretch paths. In Proceedings of the 36th Annual Symposium on Foundations of Computer Science, pages 452{ 461, Burlington, Vermont, 14-16 October 1996.

...., where jd(u; v)j is the length of the shortest u Gamma v path. The approximate all pairs shortest path problem involves a tradeoff of stretch against time short paths with stretch bounded by a constant are computed in time less than it would take to compute exact all pairs shortest paths (see [1, 2, 6, 8, 9, 10]) The compact routing problem considers instead a tradeoff of stretch for space, in the setting where each node locally stores its own routing tables. The stretch of a compact routing algorithm is defined as the maximum stretch over the routes for all pairs of nodes in the network. Clearly if ....

....node i is simply a function of the routing information stored at i and the address of its final destination. The principal ingredients of our algorithm include the following: ffl The O(log n) greedy approximation to dominating set, coupled with truncated and full Dijkstra s algorithms as used in [1, 2, 9, 10] and in the same fashion as to how it is used in [4] ffl A new density dependent algorithm for landmark selection. It is possible to give a careful distributed implementation of our algorithms, along the same lines as in [5] The improvements we present are at the algorithmic, not at the protocol ....

E. Cohen and U. Zwick. All-pairs small-stretch paths. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 93--102, New Orleans, Louisiana, 5--7 Jan. 1997.

....may be longer that the shortest path between the nodes. The compact routing problem instead considers a tradeoff of route lengths for space, in the setting where each node locally stores its own routing tables. Much recent work has been done on fast constructions of approximate shortest paths (see [1, 2, 5, 7, 8, 10]) and on compact routing schemes with sublinear maximum or average space (see [3, 4, 6, 9, 11, 13, 14] for undirected weighted and unweighted graphs. However, there have been no previous schemes that achieved any savings of time or space over the exact schemes for either weighted or unweighted ....

....of time or space over the exact schemes for either weighted or unweighted directed graphs. Why are digraphs so hard It is shown in [10] that distinguishing between distances 2 and 1 in even unweighted directed graphs is at least as hard as Boolean matrix multiplication. Cohen and Zwick write in [8]: Thus finding all pairs estimated distances of any finite stretch in directed graphs is at least as hard as Boolean matrix multiplication. It is not surprising, therefore, that the approximation methods used here, and elsewhere, work only for undirected graphs. Supported in part by ONR grant ....

E. Cohen and U. Zwick. All-pairs small-stretch paths. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 93--102, New Orleans, Louisiana, 5--7 Jan. 1997.

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E. Cohen and U. Zwick. All-pairs small-stretch paths. Journal of Algorithms, 38:335--353, 2001.

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E. Cohen and U. Zwick. All-pairs small-stretch paths. Journal of Algorithms, 38:335--353, 2001.

No context found.

E. Cohen and U. Zwick. All-pairs small-stretch paths. J. Algorithms, 38:335--353, 2001.

No context found.

E. Cohen and U. Zwick. All-pairs small-stretch paths. Journal of Algorithms, 38:335-353, 2001.

No context found.

E. Cohen and U. Zwick. All-pairs small-stretch paths. In Proceedings of the 8th Annual ACMSIAM Symposium on Discrete Algorithms, New Orleans, Louisiana, pages 93--102, 1997.

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E. Cohen and U. Zwick. All pairs small stretch paths. Submitted for publication, 1996.

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E. Cohen and U. Zwick. All-pairs small-stretch paths. Journal of Algorithms, 38:335-353, 2001.

....graph on n vertices is at least as hard as computing the Boolean product of two n=3 Theta n=3 matrices. Our result is therefore very close to being optimal. Algorithms for approximating the distances between all pairs of vertices in a weighted undirected graph were obtained by Cohen and Zwick [6]. They present an O(n 2 ) algorithm for finding paths with stretch at most 3, an O(n 7=3 ) algorithm for finding paths of stretch 7=3, and an O(n 3=2 m 1=2 ) algorithm for finding paths of stretch 2. The algorithms of Cohen and Zwick [6] use ideas obtained by Aingworth, Chekuri, ....

....graph were obtained by Cohen and Zwick [6] They present an O(n 2 ) algorithm for finding paths with stretch at most 3, an O(n 7=3 ) algorithm for finding paths of stretch 7=3, and an O(n 3=2 m 1=2 ) algorithm for finding paths of stretch 2. The algorithms of Cohen and Zwick [6] use ideas obtained by Aingworth, Chekuri, Indyk and Motwani [2] and by Dor, Halperin and Zwick [10] that design algorithms that approximate distances in unweighted undirected graphs with a small additive error. As can be seen from their running times, these algorithms are all purely ....

E. Cohen and U. Zwick. All-pairs small-stretch paths. In Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, Louisiana, pages 93-- 102, 1997.

....graph on n vertices is at least as hard as computing the Boolean product of two n=3 n=3 matrices. Our result is therefore very close to being optimal. Algorithms for approximating the distances between all pairs of vertices in a weighted undirected graph were obtained by Cohen and Zwick [CZ97]. They present an O(n 2 ) algorithm for nding paths with stretch at most 3, an O(n 7=3 ) algorithm for nding paths of stretch 7=3, and an O(n 3=2 m 1=2 ) algorithm for nding paths of stretch 2. The algorithms of Cohen and Zwick [CZ97] use ideas obtained by Aingworth, ....

....graph were obtained by Cohen and Zwick [CZ97] They present an O(n 2 ) algorithm for nding paths with stretch at most 3, an O(n 7=3 ) algorithm for nding paths of stretch 7=3, and an O(n 3=2 m 1=2 ) algorithm for nding paths of stretch 2. The algorithms of Cohen and Zwick [CZ97] use ideas obtained by Aingworth, Chekuri, Indyk and Motwani [ACIM99] and by Dor, Halperin and Zwick [DHZ00] who designed algorithms that approximate distances in unweighted undirected graphs with a small additive error. As can be seen from their running times, these algorithms are all purely ....

E. Cohen and U. Zwick. All-pairs small-stretch paths. In Proceedings of the 8th Annual ACMSIAM Symposium on Discrete Algorithms, New Orleans, Louisiana, pages 93-102, 1997.

.... algorithm for finding approximate distances between all pairs of vertices with an additive error of at most 2M , where m is the number of edges in the graph, and an O(n 2 ) time algorithm for finding all the distances with an additive error of at most O(M log n) are given by Cohen and Zwick [2]. These two algorithms do not use fast matrix multiplication and work again only on undirected graphs. The rest of this paper is organized as follows. In the next section we present a simple O(n ) time algorithm, called USP, for finding all the distances in unweighted undirected graphs. ....

E. Cohen and U. Zwick. All-pairs small-stretch paths. In Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, Louisiana, pages 93-- 102, 1997.

....stretch 3 distances in a weighted undirected graph on n vertices. Extended and improved results for weighted graphs, including an O(n 2 ) time algorithm for finding stretch 3 distances and an O(n 3=2 m 1=2 ) time algorithm for finding stretch 2 distances appear in Cohen and Zwick [CZ97] 2 Preliminaries The work of Aingworth et al. ACIM96] is based on the following simple observation: there is a small set of vertices that dominates all the high degree vertices of a graph. A set of vertices D is said to dominate a set U if every vertex in U has a neighbor in D. This ....

....distances are stretched by a factor of at most 7=3. For any k, the distances are stretched by a factor of at most 3. By taking k = Theta(log n) we get an O(n 2 ) time algorithm for finding stretch 3 approximate distances. An extension of this algorithm for weighted graph is presented in [CZ97] 5 Boolean Matrix Multiplication Let A and B be two Boolean n Theta n matrices. Construct a graph G A;B = V; E) with V = fu 1 ; u n g [ fv 1 ; v n g [ fw 1 ; w n g ; E = f(u i ; v k ) j a ik = 1g [ f(v k ; w j ) j b kj = 1g : The graph corresponding to two 3 Theta 3 ....

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E. Cohen and U. Zwick. All-pairs small-stretch paths. In Proceedings of the 8th Annual ACMSIAM Symposium on Discrete Algorithms, New Orleans, Louisiana, pages 93--102, 1997.

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E. Cohen and U. Zwick. All-pairs small-stretch paths. Journal of Algorithms, 38:335-353, 2001.

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E. Cohen and U. Zwick. All-pairs small-stretch paths. In In Proceedings of the Eighth Annual ACMSIAM Symposium on Discrete Algorithms, pages 93-102, New Orleans, Louisiana, Jan. 1997.

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