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FAST ALGORITHMS FOR CONSTRUCTING t-SPANNERS AND PATHS WITH STRETCH t.

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Title: FAST ALGORITHMS FOR CONSTRUCTING t-SPANNERS AND PATHS WITH STRETCH t.
Authors: Cohen, Edith
Source: SIAM Journal on Computing; 1998, Vol. 28 Issue 1, p210, 27p
Subject Terms: Algorithms, Graphic methods
Abstract: The distance between two vertices in a weighted graph is the weight of a minimum- weight path between them (where the weight of a path is the sum of the weights of the edges in the path). A path has stretch t if its weight is at most t times the distance between its end points. We present algorithms that compute paths of stretch 2 ≤ t ≤ log n on undirected graphs G = (V, E) with nonnegative weights. The stretch t is of the form t = β(2 + ∊), where β is integral and &espi;' > 0 is at least as large as some fixed ∊ > 0. We present an Õ((m + k) n[SUP2 + ∊]/t]) time randomized algorithm that finds paths between k specified pairs of vertices and an n[SUP2(1 + log[SUBn] m + ∊)/t]) deterministic algorithm that finds paths from s specified sources to all other vertices (for any fixed ∊ > 0), where n = |V| and m = |E|. This improves significantly over the slower Õ(min{k,n}m) exact shortest paths algorithms and a previous O(mn&frac64t;] + , kn[SUP&fr4ac32t;]) time algorithm by Awerbuch et al. [Proc. 34th IEEE Annual Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 1993, pp. 638-6471. A t-spanner of a graph G is a set of weighted edges on the vertices of G such that distances in the spanner are not smaller and within a factor of t from the corresponding distances in G. Previous work was concerned with bounding the size and efficiently constructing t-spanners. We construct t-spanners of size Õ(n[SUP1+(2 + ∊)]/t) in Õ(mn(2 + ∊)/t) expected time (for any fixed ∊ > 0), which constitutes a faster construction (by a factor of n[SUP3+2/t]/m) of sparser spanners than was previously attainable. We also provide efficient parallel constructions. Our algorithms are based on pairwise covers and a novel approach to construct them efficiently. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: The distance between two vertices in a weighted graph is the weight of a minimum- weight path between them (where the weight of a path is the sum of the weights of the edges in the path). A path has stretch t if its weight is at most t times the distance between its end points. We present algorithms that compute paths of stretch 2 ≤ t ≤ log n on undirected graphs G = (V, E) with nonnegative weights. The stretch t is of the form t = β(2 + ∊), where β is integral and &espi;' > 0 is at least as large as some fixed ∊ > 0. We present an Õ((m + k) n[SUP2 + ∊]/t]) time randomized algorithm that finds paths between k specified pairs of vertices and an n[SUP2(1 + log[SUBn] m + ∊)/t]) deterministic algorithm that finds paths from s specified sources to all other vertices (for any fixed ∊ > 0), where n = |V| and m = |E|. This improves significantly over the slower Õ(min{k,n}m) exact shortest paths algorithms and a previous O(mn&frac64t;] + , kn[SUP&fr4ac32t;]) time algorithm by Awerbuch et al. [Proc. 34th IEEE Annual Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 1993, pp. 638-6471. A t-spanner of a graph G is a set of weighted edges on the vertices of G such that distances in the spanner are not smaller and within a factor of t from the corresponding distances in G. Previous work was concerned with bounding the size and efficiently constructing t-spanners. We construct t-spanners of size Õ(n[SUP1+(2 + ∊)]/t) in Õ(mn(2 + ∊)/t) expected time (for any fixed ∊ > 0), which constitutes a faster construction (by a factor of n[SUP3+2/t]/m) of sparser spanners than was previously attainable. We also provide efficient parallel constructions. Our algorithms are based on pairwise covers and a novel approach to construct them efficiently. [ABSTRACT FROM AUTHOR]
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  Group: Ab
  Data: <i>Copyright of SIAM Journal on Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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