Efficient sequential and parallel algorithms for the negative cycle problem

Kavvadias, Dimitrios; Pantziou, Grammati; Spirakis, Paul G.; Zaroliagis, Christos

doi:10.1007/3-540-58325-4_190

Cited by 10 publications

(5 citation statements)

References 15 publications

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“…This generalizes the results in [15] for outerplanar digraphs. To the best of our knowledge, this is the most general class of graphs for which the complexity of computing a shortest path tree matches that of finding a negative cycle.…”

Section: Introductionsupporting

confidence: 64%

“…If the digraph contains negative real edge weights, then one needs O(nm) time to either construct a shortest path tree, or find a negative weight cycle [2]. For outerplanar digraphs, in O(n) time, a shortest path tree can be constructed [10], [11], or a negative cycle can be found [15]. For planar digraphs with nonnegative real edge weights, an O(n) time algorithm for computing a shortest path tree is given in [16]; with negative but integer edge weights larger than −L (L ≥ 1), the same paper gives an O(n 4/3 log(nL)) time algorithm which constructs a shortest path tree, or finds a negative cycle.…”

Section: Introductionmentioning

confidence: 99%

“…For planar digraphs with nonnegative real edge weights, an O(n) time algorithm for computing a shortest path tree is given in [16]; with negative but integer edge weights larger than −L (L ≥ 1), the same paper gives an O(n 4/3 log(nL)) time algorithm which constructs a shortest path tree, or finds a negative cycle. The best algorithm to construct a shortest path tree, or find a negative cycle in a planar digraph with real edge weights takes (in the worst case) O(n 1.5 log n) time [15].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Shortest Paths in Digraphs of Small Treewidth. Part I: Sequential Algorithms

Chaudhuri¹,

Zaroliagis²

2000

Algorithmica

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We consider the problem of preprocessing an n-vertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give algorithms that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O(α(n)) time after O(n) preprocessing. This improves upon previously known results for the same problem. We also give a dynamic algorithm which, after a change in an edge weight, updates the data structure in time O(n β ), for any constant 0 < β < 1. Furthermore, an algorithm of independent interest is given: computing a shortest path tree, or finding a negative cycle in linear time.

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Section: Introductionsupporting

confidence: 64%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Shortest Paths in Digraphs of Small Treewidth. Part I: Sequential Algorithms

Chaudhuri¹,

Zaroliagis²

2000

Algorithmica

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“…Also, our algorithms seem to be very efficient for the class of all appropriately sparse graphs. As has been established in [15] and [29] random G n, p graphs with threshold function 1/n are planar with probability one and have expected value for q equal to O (1).…”

Section: Previous Resultsmentioning

confidence: 88%

“…Since G op can be tested for a negative cycle in O(n) time [29], we can assume without loss of generality that G op does not have such a cycle. Preprocess G op using algorithm Pre Outerplanar (Section 3.…”

Section: Single-source Shortest Paths In Outerplanar Digraphsmentioning

confidence: 99%

Improved Algorithms for Dynamic Shortest Paths

Djidjev¹,

Pantziou²,

Zaroliagis³

2000

Algorithmica

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Abstract. We describe algorithms for finding shortest paths and distances in outerplanar and planar digraphs that exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. In the case of outerplanar digraphs, our data structures can be updated after any such change in only logarithmic time. A distance query is also answered in logarithmic time. In the case of planar digraphs, we give an interesting tradeoff between preprocessing, query, and update times depending on the value of a certain topological parameter of the graph. Our results can be extended to n-vertex digraphs of genus O(n 1−ε ) for any ε > 0.

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Hammock-on-ears decomposition: A technique for the efficient parallel solution of shortest paths and other problems

Kavvadias

Pantziou

Spirakis

et al. 1994

Mathematical Foundations of Computer Science 1994

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We s h o w h o w to decompose e ciently in parallel any graph int o a n umber,~ , o f outerplanar subgraphs (called hammocks) satisfying certain separator properties. Our work combines and extends the sequential hammock decomposition technique introduced by G. F rederickson and the parallel ear decomposition technique, thus we call it the hammock-on-ears decomposition. W e m e n tion that hammock-on-ears decomposition also draws from techniques in computational geometry and that an embedding of the graph does not need to be provided with the input. We a c hieve this decomposition in O(log n log log n) time using O(n + m) CREW PRAM processors, for an n-vertex, m-edge graph or digraph. The hammock-on-ears decomposition implies a general framework for solving graph problems e ciently. Its value is demonstrated by a v ariety o f applications on a signi cant class of (di)graphs, namely that of sparse (di)graphs. T h i s class consists of all (di)graphs which h a ve ã between 1 and (n), and includes planar graphs and graphs with genus o(n). We i m p r o ve previous bounds for certain instances of shortest paths and related problems, in this class of graphs. These problems include all pairs shortest paths, all pairs reachability, and detection of a negative cycle.

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Efficient sequential and parallel algorithms for the negative cycle problem

Cited by 10 publications

References 15 publications

Shortest Paths in Digraphs of Small Treewidth. Part I: Sequential Algorithms

Shortest Paths in Digraphs of Small Treewidth. Part I: Sequential Algorithms

Improved Algorithms for Dynamic Shortest Paths

Hammock-on-ears decomposition: A technique for the efficient parallel solution of shortest paths and other problems

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