Average-case complexity of shortest-paths problems in the vertex-potential model

Cooper, Colin; Frieze, Alan; Mehlhorn, Kurt; Priebe, Volker

doi:10.1002/(sici)1098-2418(200001)16:1<33::aid-rsa3>3.0.co;2-0

Cited by 10 publications

(5 citation statements)

References 21 publications

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“…Next, since both the Tree and Hourglass algorithms allow negative arc weights, it would be interesting to analyze their expected-case running time complexity for a model that permits negative arcs such as the vertex potential model [5,6].…”

Section: Discussionmentioning

confidence: 99%

“…In the uniform model, arc weights are drawn at random, independently of each other, according to a common probability distribution. A more general model is the endpoint-independent model [3,24], where, for each vertex v, a sequence of n − 1 non-negative arc weights is generated by a deterministic or stochastic process and then randomly permuted and assigned to the outgoing arcs of v. In the vertex potential model [5,6], arc weights can be both positive and negative. This is a probability model with arbitrary real arc weights but without negative cycles.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, Moffat and Takaoka [18] and Mehlhorn and Priebe [17] improved the running time to O(n 2 log n). In the vertex potential model, Cooper et al [6] gave an algorithm with an expected-case running time O(n 2 log n). All the above algorithms use Dijkstra-like approach.…”

Section: Introductionmentioning

confidence: 99%

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Modifications of the Floyd-Warshall algorithm with nearly quadratic expected-time

2021

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The paper describes two relatively simple modifications of the well-known Floyd-Warshall algorithm for computing all-pairs shortest paths. A fundamental difference of both modifications in comparison to the Floyd-Warshall algorithm is that the relaxation is done in a smart way. We show that the expected-case time complexity of both algorithms is O(n 2 log 2 n) for the class of complete directed graphs on n vertices with arc weights selected independently at random from the uniform distribution on [0, 1]. Theoretically best known algorithms for this class of graphs are all based on Dijkstra's algorithm and obtain a better expected-case bound. However, by conducting an empirical evaluation we prove that our algorithms are at least competitive in practice with best know algorithms and, moreover, outperform most of them. The reason for the practical efficiency of the presented algorithms is the absence of use of priority queue. * A preliminary version of this work has been published in Shortest Path Solvers: From Software to Wetware, volume 32 of Emergence, Complexity and Computation (2018). The authors would like to thank the reviewer for excellent comments that substantially improved the quality of the paper.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Modifications of the Floyd-Warshall algorithm with nearly quadratic expected-time

2021

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“…The Dijkstra's algorith m and Bellman-Ford algorith m are imp lemented in Lin k State Routing protocols and Distance Vector routing protocols respectively [12]. The convergence time co mplexity of Dijkstra's algorith m is found to be O(|E| log |V|) [13] and that of Bellman-Ford algorith m is O(|E||V|) [14] when configured with |V| vertices and |E| edges. The computational time co mplexity increases for larger networks and they do not handle graphs having negative weighted edges and cycles [15].…”

Section: Related Workmentioning

confidence: 99%

Reliable Mobile Ad-Hoc Network Routing Using Firefly Algorithm

Persis¹,

Robert²

2016

IJISA

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Routing in Mobile Ad-hoc NETwork (MANET) is a contemporary graph problem that is solved using various shortest path search techniques. The routing algorith ms employed in modern routers use deterministic algorith ms that extract an exact nondominated set of solutions from the search space. The search efficiency of these algorithms is found to have an exponential time co mp lexity in the worst case. Moreover this problem is a mult i-objective optimization problem in nature for MA NET and it is required to consider changing topology layout. This study attempts to employ a formulat ion incorporating objectives viz., delay, hopdistance, load, cost and reliability that has significant impact on network performance. Simu lation with different random topologies has been carried out to illustrate the imp lementation of an exhaustive search algorith m and it is observed that the algorithm could handle small-scale networks limited to 15 nodes. A random search meta-heuristic that adopts the nature of firefly swarm has been proposed for larger networks to yield an appro ximated non-dominated path set. Firefly Algorith m is found to perform better than the exact algorithm in terms of scalability and computational time.

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“…A different, yet, related subject that admits a plethora of literature is average case analysis for graph algorithms (e.g. [15,25,11,29]). There it is assumed that the edge weights in the input of some graph algorithm are drawn from a specified probability distribution and the goal is to analyze the expected run-time of the algorithm with respect to that distribution; refer to [16] for a survey.…”

Section: Related Workmentioning

confidence: 99%

Approximating the Statistics of various Properties in Randomly Weighted Graphs

Emek

Korman

Shavitt

2011

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

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Consider the setting of randomly weighted graphs, namely, graphs whose edge weights are chosen independently according to probability distributions with finite support over the non-negative reals. Under this setting, weighted graph properties such as the diameter, the radius (with respect to a designated vertex), and the weight of a minimum spanning tree become random variables and we are interested in computing their expectation. Unfortunately, this turns out to be #P-hard. In this paper, we define a family of weighted graph properties (that includes the above three) and show that for each property in this family, the problem of computing the k th moment (and in particular, the expectation) of the corresponding random variable admits a fully polynomial-time randomized approximation scheme (FPRAS) for every fixed k.

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Average-case complexity of shortest-paths problems in the vertex-potential model

Cited by 10 publications

References 21 publications

Modifications of the Floyd-Warshall algorithm with nearly quadratic expected-time

Modifications of the Floyd-Warshall algorithm with nearly quadratic expected-time

Reliable Mobile Ad-Hoc Network Routing Using Firefly Algorithm

Approximating the Statistics of various Properties in Randomly Weighted Graphs

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