A New Algorithm for Finding All Shortest Paths in a Graph of Positive Arcs in Average Time $O(n^2 \log ^2 n)$

Spira, Philip M.

doi:10.1137/0202004

Cited by 94 publications

(30 citation statements)

References 3 publications

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“…Average-case analysis of shortest-paths algorithms mainly focused on the All-Pairs Shortest-Paths (APSP) problem on either the complete graph or random graphs with Ω(n · log n) edges and random edge weights [9,26,40,49,63]. Average-case running times of O(n 2 · log n) as compared to worst-case cubic bounds are obtained by virtue of an initial pruning step: if L denotes a bound on the maximum shortest-path weight, then the algorithms discard insignificant edges of weight larger than L; they will not be part of the final solution.…”

Section: Random Edge Weightsmentioning

confidence: 99%

Average-case complexity of single-source shortest-paths algorithms: lower and upper bounds

Meyer

2003

Journal of Algorithms

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Section: Random Edge Weightsmentioning

confidence: 99%

Average-case complexity of single-source shortest-paths algorithms: lower and upper bounds

Meyer

2003

Journal of Algorithms

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“…Hassin and Zemel [HZ85] and Frieze and Grimmett [FG85] gave simple algorithms that solve the APSP problem, when the input graph is drawn from K n (Exp(1)), in O(n 2 log n) expected time. Spira [Spi73] initiated the study of the expected running time of SSSP and APSP algorithms in a much more general probabilistic model, now referred to as the end-point independent model. The input graph in this model is a complete directed graph on n vertices.…”

Section: Average Case Resultsmentioning

confidence: 99%

“…Spira [Spi73] analyzed his algorithm in the end-point independent model mentioned in Section 1.2. Note that K n (Exp(1)) is clearly an end-point independent model.…”

Section: Spira's Algorithmmentioning

confidence: 99%

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A Forward-Backward Single-Source Shortest Paths Algorithm

Wilson

Zwick

2013

2013 IEEE 54th Annual Symposium on Foundations of Computer Science

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We describe a new forward-backward variant of Dijkstra's and Spira's Single-Source Shortest Paths (SSSP) algorithms. While essentially all SSSP algorithm only scan edges forward, the new algorithm scans some edges backward. The new algorithm assumes that edges in the outgoing and incoming adjacency lists of the vertices appear in nondecreasing order of weight. (Spira's algorithm makes the same assumption about the outgoing adjacency lists, but does not use incoming adjacency lists.) The running time of the algorithm on a complete directed graph on n vertices with independent exponential edge weights is O(n), with very high probability. This improves on the previously best result of O(n log n), which is best possible if only forward scans are allowed, exhibiting an interesting separation between forward-only and forward-backward SSSP algorithms. As a consequence, we also get a new all-pairs shortest paths algorithm. The expected running time of the algorithm on complete graphs with independent exponential edge weights is O(n 2 ), matching a recent algorithm of Demetrescu and Italiano as analyzed by Peres et al. Furthermore, the probability that the new algorithm requires more than O(n 2 ) time is exponentially small, improving on the O(n −1/26 ) probability bound obtained by Peres et al.

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“…For example, the weighted graph itself is not a reasonable encoding of the solution. Most commonly used shortest-path algorithms, in particular Dijkstra's and Bellman-Ford's, fit into this class; this is the case also with many other algorithms, e.g., [12], [13]. However, some algorithms are not path-comparison-based, e.g., [14], as it adds weights of edges that do not form a single path.…”

Section: Lower Bound For Additive Weightsmentioning

confidence: 99%

Computing shortest paths for any number of hops

Guérin

Orda

2002

IEEE/ACM Trans. Networking

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In this paper, we introduce and investigate a "new" path optimization problem that we denote the all hops optimal path (AHOP) problem. The problem involves identifying, for all hop counts, the optimal, i.e., minimum weight, path(s) between a given source and destination(s). The AHOP problem arises naturally in the context of quality-of-service (QoS) routing in networks, where routes (paths) need to be computed that provide services guarantees, e.g., delay or bandwidth, at the minimum possible "cost" (amount of resources required) to the network. Because service guarantees are typically provided through some form of resource allocation on the path (links) computed for a new request, the hop count, which captures the number of links over which resources are allocated, is a commonly used cost measure. As a result, a standard approach for determining the cheapest path available that meets a desired level of service guarantees is to compute a minimum hop shortest (optimal) path. Furthermore, for efficiency purposes, it is desirable to precompute such optimal minimum hop paths for all possible service requests. Providing this information gives rise to solving the AHOP problem. The paper's contributions are to investigate the computational complexity of solving the AHOP problem for two of the most prevalent cost functions (path weights) in networks, namely, additive and bottleneck weights. In particular, we establish that a solution based on the Bellman-Ford algorithm is optimal for additive weights, but show that this does not hold for bottleneck weights for which a lower complexity solution exists. KeywordsHop-restricted shortest paths, maximum bandwidth, minimum delay, networks, quality-of-service routing This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This journal article is available at ScholarlyCommons: http://repository.upenn.edu/ese_papers/28 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 10, NO. 5, OCTOBER 2002 613 Computing Shortest Paths for Any Number of Hops Roch Guérin, Fellow, IEEE, and Ariel Orda, Senior Member, IEEE Abstract-In this paper, we introduce and investigate a "new" path optimization problem that we denote the all hops optimal path (AHOP) problem. The problem involves identifying, for all hop counts, the optimal, i.e., minimum weight, path(s) between a given source and destination(s). The AHOP problem arises naturally in the context of quality-of-service (QoS) routing in networks, where routes (paths) need to be computed that provide services guarantees, e.g., ...

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A New Algorithm for Finding All Shortest Paths in a Graph of Positive Arcs in Average Time $O(n^2 \log ^2 n)$

Cited by 94 publications

References 3 publications

Average-case complexity of single-source shortest-paths algorithms: lower and upper bounds

Average-case complexity of single-source shortest-paths algorithms: lower and upper bounds

A Forward-Backward Single-Source Shortest Paths Algorithm

Computing shortest paths for any number of hops

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