On the Exponent of the All Pairs Shortest Path Problem

Alon, Noga; Galil, Zvi; Margalit, Oded

doi:10.1006/jcss.1997.1388

Cited by 142 publications

(155 citation statements)

References 9 publications

(14 reference statements)

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“…Alon, Galil and Margalit [1] presented an algorithm for APSP in directed graphs with weights in {−1, 0, 1} of runtimeÕ(n ω+3 2 ). This was improved and generalized to integer weights in [−M, M ] by Zwick [26], whose algorithm has runtimeÕ(M…”

Section: B Related Workmentioning

confidence: 99%

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Improved Distance Sensitivity Oracles via Fast Single-Source Replacement Paths

Grandoni

Williams

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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Abstract-A distance sensitivity oracle is a data structure which, given two nodes s and t in a directed edge-weighted graph G and an edge e, returns the shortest length of an s-t path not containing e, a so called replacement path for the triple (s, t, e). Such oracles are used to quickly recover from edge failures.In this paper we consider the case of integer weights in the interval [−M, M ], and present the first distance sensitivity oracle that achieves simultaneously subcubic preprocessing time and sublinear query time. More precisely, for a given parameter α ∈ [0, 1], our oracle has preprocessing timeÕ(M n

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Section: B Related Workmentioning

confidence: 99%

“…DSOs allow quick recovery from (single) edge failures. 1 In some variants of the problem the data structure also returns the replacement path itself. Our results can be modified to return paths as well, in time linear in the number of path edges.…”

Section: Introductionmentioning

confidence: 99%

Improved Distance Sensitivity Oracles via Fast Single-Source Replacement Paths

Grandoni

Williams

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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“…The distance δ(a i , a j ) calculated using (3) is the shortest distance between the arcs a i and a j .…”

Section: Lemmamentioning

confidence: 99%

“…Alon et al [3] have reported a sub-cubic algorithm for computing all pairs shortest distances on directed graphs with integer edge-lengths.…”

Section: Introductionmentioning

confidence: 99%

An optimal parallel algorithm for solving all-pairs shortest paths problem on circular-arc graphs

Saha

Pal

2005

JAMC

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Abstract. The shortest-paths problem is a fundamental problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one vertex to another often gives the best way to route a message between the vertices. This paper presents an O(n 2 ) time sequential algorithm and an O(n 2 /p + log n) time parallel algorithm on EREW PRAM model for solving all pairs shortest paths problem on circular-arc graphs, where p and n represent respectively the number of processors and the number of vertices of the circular-arc graph.

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“…More speci cally, if w ij = k in the path or witness matrix W = fw ij g, it means that the path from i to j goes through k. Therefore a recursive function path i; j is de ned by path i; k , k, path k;j if path i; j = k 0 and nil if path i; j = 0 , where a path is de ned by a list of vertices excluding endpoints. In the following sections, we record k in w ij whenever we can nd k such that a path from i to j is modi ed or newly set up by paths from i to k and from k to j. fIf S = ;, the i-th row need not be computed in step sg 6`1 := d3`=2e; 7 for i := 0 to n , 1 do for j := 0 to n , 1 2 re m , , n 0 by repeated squaring in the cruising phase, where n 0 is the smallest integer in this series of`such that`n. The key observation in the cruising phase is that we only need to check S i at line 8 whose size is not larger than 2n=`, since the correct distances between`+ 1 and d3`=2e can beobtained as the sum d ì We design a parallel algorithm on an EREW-PRAM for a directed graph with unit edge costs.…”

Section: Basic De Nitionsmentioning

confidence: 99%

Subcubic Cost Algorithms for the All Pairs Shortest Path Problem

Takaoka¹

1998

Algorithmica

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In this paper we give three sub-cubic cost algorithms for the all pairs shortest distance APSD and path APSP problems. The rst is a parallel algorithm that solves the APSD problem for a directed graph with unit edge costs in O log 2 n time with O n p log n processors where = 2:688 on an EREW-PRAM. The second parallel algorithm solves the APSP, and consequently APSD, problem for a directed graph with non-negative general costs real numbers in O log 2 n time with o n 3 subcubic cost. Previously this cost was greater than O n 3 . Finally we improve with respect to M the complexity O M n of a sequential algorithm for a graph with edge costs up to M into O M 1=3 n 6+! =3 log n 2=3 log log n 1=3 in the APSD problem, where ! = 2 :376.

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On the Exponent of the All Pairs Shortest Path Problem

Cited by 142 publications

References 9 publications

Improved Distance Sensitivity Oracles via Fast Single-Source Replacement Paths

Improved Distance Sensitivity Oracles via Fast Single-Source Replacement Paths

An optimal parallel algorithm for solving all-pairs shortest paths problem on circular-arc graphs

Subcubic Cost Algorithms for the All Pairs Shortest Path Problem

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