A parallel randomized approximation scheme for shortest paths

Klein, Philip N.; Sairam, Sri

doi:10.1145/129712.129785

Cited by 27 publications

(12 citation statements)

References 14 publications

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“…Klein [20] extended the work of Ullman and Yannakakis [27] to the weighted case, obtaining a randomized PRAM algorithm that can (1 + )-approximate shortest paths between k pairs of vertices in O( √ n −2 log n · log * n) time using (kE log n) 2 processors (where is constant). When this algorithm is run sequentially, the time is not as good as known sequential algorithms for the exact problem.…”

Section: 1mentioning

confidence: 99%

Near-Linear Time Construction of Sparse Neighborhood Covers

Awerbuch

Berger

Cowen³

et al. 1998

SIAM J. Comput.

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Abstract. This paper introduces a near-linear time sequential algorithm for constructing a sparse neighborhood cover. This implies analogous improvements (from quadratic to near-linear time) for any problem whose solution relies on network decompositions, including small edge cuts in planar graphs, approximate shortest paths, and weight-and distance-preserving graph spanners. In particular, an O(log n) approximation to the k-shortest paths problem on an n-vertex, E-edge graph is obtained that runs inÕ (n + E + k) time.

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Section: 1mentioning

confidence: 99%

Near-Linear Time Construction of Sparse Neighborhood Covers

Awerbuch

Berger

Cowen³

et al. 1998

SIAM J. Comput.

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“…The work performed depends on the size of the cover. To achieve an efficient parallel construction of pairwise covers, we refine the definition to be with respect to an additional "path size" parameter C. For a parameter 1 5 C 5 n , the C-limited distance between a pair of vertices is the weight of the minimum-weight path connecting them among paths consisting of at most C edges (this terminology is from [7]). In pairwise covers with parameter e, we have a relaxed requirement that pairs of vertices where the !-limited distance between them is a t most W must be both contained in at least one cluster.…”

Section: Overviewmentioning

confidence: 99%

“…The weights U, are obtained by scaling and rounding up the weights w . The reduction is based on a technique and ideas from Klein and Sairam [7]. ...…”

Section: Computing An (E P W ;)-Covermentioning

confidence: 99%

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

Cohen

Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science

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The distance between two vertices in a weighted graph is the weight of a minimum-weight path between them. A path has stretch t if its weight is at most t times the distance between its end points. We consider a weighted undirected graph G = (V, E) and present algorithms that compute paths with stretch 2 5 t 5 log n.We present a q((m + k)n('+')lt) time randomized algorithm that finds paths between k specified pairs of vertices and a O ( ( m + ns)n2(1t10&-mtc)/t) deterministic algorithm that finds paths from s specified sources to all other vertices (for any fixed c > 0), where n = IV( and m = IEl. ?_'his improves significantly over the slower O(min{k, n } m ) exact shortest paths algorithms and a previous O(mnG41t + kn3'/') time algorithm by Awerbuch et al. A t-spanner of a graph G is a set of weighted edges on the vertices of C 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 &spanners of size O(nlt('tc)/t) in 6(mn('sc)/t) expected time (for any fixed 6 > 0), what constitutes a faster construction (by a factor of n3t2/t/m) of sparser spanners than was previously attainable. We also provide efficient parallel constructions. Our algorithms are based on new structures called pairwise-covers and a novel approach to construct them efficiently.

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“…The basic sparsification technique works best with paths that are all roughly of same length. To accommodate this, we use a grouping technique (see, e.g., [14]). …”

Section: 2mentioning

confidence: 99%

A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs

Klein

Subramanian

1998

Algorithmica

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Abstract. In this paper we give a fully dynamic approximation scheme for maintaining all-pairs shortest paths in planar networks. Given an error parameter ε such that 0 < ε, our algorithm maintains approximate all-pairs shortest paths in an undirected planar graph G with nonnegative edge lengths. The approximate paths are guaranteed to be accurate to within a 1 + ε factor. The time bounds for both query and update for our algorithm is O(ε −1 n 2/3 log 2 n log D), where n is the number of nodes in G and D is the sum of its edge lengths. The time bound for the queries is worst case, while that for the additions is amortized. Our approximation algorithm is based upon a novel technique for approximately representing all-pairs shortest paths among a selected subset of the nodes by a sparse substitute graph.

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A parallel randomized approximation scheme for shortest paths

Cited by 27 publications

References 14 publications

Near-Linear Time Construction of Sparse Neighborhood Covers

Near-Linear Time Construction of Sparse Neighborhood Covers

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

A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs

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