On the All-Pairs-Shortest-Path Problem in Unweighted Undirected Graphs

Seidel, Raimund

doi:10.1006/jcss.1995.1078

Cited by 229 publications

(176 citation statements)

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“…A simple randomized algorithm for computing (not necessarily maximum) witnesses for Boolean matrix multiplication, in essentially the same time required to perform the product, is given by Seidel [17]. Alon and Naor [2] gave a deterministic algorithm for the problem.…”

Section: Maximum Witnesses For Boolean Matrix Multiplicationmentioning

confidence: 99%

All-Pairs Bottleneck Paths in Vertex Weighted Graphs

2009

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Let G = (V, E, w) be a directed graph, where w : V → R is an arbitrary weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u, v the bottleneck weight, or the capacity, from u to v, denoted c (u, v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem we have to find the bottleneck weights for all ordered pairs of vertices. Our main result is an O(n 2.575 ) time algorithm for the APBP problem. The exponent is derived from the exponent of fast matrix multiplication. Our algorithm is the first sub-cubic algorithm for this problem. Unlike the sub-cubic algorithm for the all-pairs shortest paths (APSP) problem, that only applies to bounded (or relatively small) integer edge or vertex weights, the algorithm presented for APBP problem works for arbitrary large vertex weights.The APBP problem has numerous applications, and several interesting problems that have recently attracted attention can be reduced to it, with no asymptotic loss in the running times of the known algorithms for these problems. Some examples are a result of Vassilevska and Williams [STOC 2006] on finding a triangle of maximum weight, a result of Bender et al. [SODA 2001] on computing least common ancestors in DAGs and a result of Kowaluk and Lingas [ICALP 2005] on finding maximum witnesses for boolean matrix multiplication. Thus, the APBP problem provides a uniform framework for these applications. For some of these problems, we can in fact show that their complexity is equivalent to that of the APBP problem.A slight modification of our algorithm enables us to compute shortest paths of maximum bottleneck weight. Let d(u, v) denote the (unweighted) distance from u to v, and let sc (u, v) denote the maximum bottleneck weight of a path from u to v having length d (u, v). The AllPairs Bottleneck Shortest Paths (APBSP) problem is to compute sc (u, v) for all ordered pairs of vertices. We present an algorithm for the APBSP problem whose running time is O(n 2.86 ).

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Section: Maximum Witnesses For Boolean Matrix Multiplicationmentioning

confidence: 99%

All-Pairs Bottleneck Paths in Vertex Weighted Graphs

2009

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“…P's runtime in the above diameter protocol matches the best known unverifiable diameter algorithm up to a low-order additive term [89,107], and the communication is just polylog (n).…”

Section: Comparison To Prior Workmentioning

confidence: 83%

“…For example, using the protocol of Theorem 8.7.2 as a primitive, we give a natural protocol for computing the diameter of an unweighted directed graph G. V's runtime in this protocol is O(m log n), where m is the number of edges in G, P's runtime matches the best known unverifiable diameter algorithm up to a low-order additive term [89,107], and the total communication is just polylog (n). We know of no other protocol achieving this.…”

Section: Introductionmentioning

confidence: 99%

Practical verified computation with streaming interactive proofs

Cormode

Mitzenmacher

Thaler

2012

Proceedings of the 3rd Innovations in Theoretical Computer Science Conference

153

269

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When delegating computation to a service provider, as in the cloud computing paradigm, we seek some reassurance that the output is correct and complete. Yet recomputing the output as a check is inefficient and expensive, and it may not even be feasible to store all the data locally. We are therefore interested in what can be validated by a streaming (sublinear space) user, who cannot store the full input, or perform the full computation herself. Our aim in this work is to advance a recent line of work on "proof systems" in which the service provider proves the correctness of its output to a user. The goal is to minimize the time and space costs of both parties in generating and checking the proof. Only very recently have there been attempts to implement such proof systems, and thus far these have been quite limited in functionality.Here, our approach is two-fold. First, we describe a carefully chosen instantiation of one of the most efficient general-purpose constructions for arbitrary computations (streaming or otherwise), due to Goldwasser, Kalai, and Rothblum [19]. This requires several new insights and enhancements to move the methodology from a theoretical result to a practical possibility. Our main contribution is in achieving a prover that runs in time O(S(n) log S(n)), where S(n) is the size of an arithmetic circuit computing the function of interest; this compares favorably to the poly(S(n)) runtime for the prover promised in [19]. Our experimental results demonstrate that a practical general-purpose protocol for verifiable computation may be significantly closer to reality than previously realized.Second, we describe a set of techniques that achieve genuine scalability for protocols fine-tuned for specific important problems in streaming and database processing. Focusing in particular on noninteractive protocols for problems ranging from matrix-vector multiplication to bipartite perfect matching, we build on prior work [8,15] to achieve a prover that runs in nearly linear-time, while obtaining optimal tradeoffs between communication cost and the user's working memory. Existing techniques required (substantially) superlinear time for the prover. Finally, we develop improved interactive protocols for specific problems based on a linearization technique originally due to Shen [34]. We argue that even if general-purpose methods improve, fine-tuned protocols will remain valuable in real-world settings for key problems, and hence special attention to specific problems is warranted.

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“…It uses a single source shortest paths algorithm, which bypasses the sorting bottleneck of Dijkstra's algorithm [9] and runs in O(m) time. [12,13], and Seidel [21] have shown that if matrix multiplication can be performed in O(M(n)) time, then the all pairs shortest paths problem for unweighted directed graphs can be solved in Õ(√n 3 M(n)) time and the all pairs shortest paths problem for unweighted undirected graphs can be solved in Õ(M(n)) time. Here Õ( ) is to hide the polylogarithmic factor i.e.…”

Section: Previous Work: All Pairs Shortest Path Algorithmsmentioning

confidence: 99%

An Efficient Algorithm for All Pair Shortest Paths

Singh¹,

Kumar²,

Pandey³

2010

IJCEE

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-In this paper, an all pairs optimized shortest path algorithm is presented for an unweighted and undirected graph with some additive error of at most 2.This algorithm can be extended for weighted graph also but it will not work for directed graph due to absence of commutative property. The run time of the algorithm is of order Ο(n5/2), where n is the number of vertices present in the graph. This algorithm is much simpler than the existing algorithms. A study of upper bounds on the size of a maximal independent set of such graphs has also been discussed.Index Terms-Additive Error, Commutative, Multi-plicative Error, Optimized. I. INTRODUCTIONThe single source shortest paths algorithm gives the shortest paths from a source vertex to every other vertex of the graphs. One such classical algorithm for unweighted graphs is breadth-first search (BFS) algorithm [8]. Some algorithms do not use single source shortest paths algorithms as subroutine, like one of the most classical algorithm for directed and weighted graph by Floyd and Warshall [8]. An all pairs optimized shortest paths algorithm does not report the exact shortest paths for every pair of vertices and distance reported may have some error.Almost every algorithm for the all pairs shortest paths problem, except those based on fast matrix multiplication, has running time of O(n3) in worst case. There exist algorithms based on fast matrix multiplication for the all pairs shortest paths problem that achieve sub-cubic running time, but these fast matrix multiplication algorithms are better than the naive O(n3) time algorithm only for very large values of n. This is where the need of approximation algorithms arises. Though optimization algorithms do not give precise output, but are faster. Many sub-cubic running time and simple algorithms have been designed for all pairs optimized shortest path problem. These algorithms achieve sub-cubic running time, but the distance reported has some multiplicative or additive error.Definition 1: An algorithm is said to compute all pairs α-optimized shortest paths or all pairs optimized shortest paths of stretch α for some α ≥ 1, if for every pair of vertices u, v ∈ V , the distance reported is bounded by α.δ (u, v), Rajendra Kumar is with Computer Science and engineering Department, Vidya College of Engineering, Meerut (Uttar Pradesh), India (phone: +91-9412002322, e-mail: rajendra04@gmail.com, website: http://www.rkronline.in).Vijay Shankar Pandey is with the Board of Apprenticeship Training (SR), Chennai, India, (e-mail: vsp125@gmail.com). where δ (u, v) is the actual shortest distance between u and v.Definition 2: An algorithm is said to compute all pairs shortest paths with an additive one-sided error of at most β or all pairs optimized shortest paths of surplus β for some β ≥ 0, if for every pair of vertices u, v ∈V, the distance reported is bounded by δ(u, v) + β, where δ(u, v) is the actual shortest distance between u and v.An interesting property of the shortest path is that a shortest path between two vertices...

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On the All-Pairs-Shortest-Path Problem in Unweighted Undirected Graphs

Cited by 229 publications

References 0 publications

All-Pairs Bottleneck Paths in Vertex Weighted Graphs

All-Pairs Bottleneck Paths in Vertex Weighted Graphs

Practical verified computation with streaming interactive proofs

An Efficient Algorithm for All Pair Shortest Paths

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