A dynamization of the all pairs least cost path problem
Proceedings on STACS 85 2nd annual symposium on theoretical aspects of computer science
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Theoretical Computer Science
Incremental algorithms for minimal length paths
Journal of Algorithms
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Network flows: theory, algorithms, and applications
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Journal of Algorithms
Faster shortest-path algorithms for planar graphs
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
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Theoretical Computer Science
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Bounded Incremental Computation
Bounded Incremental Computation
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SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
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SAT'06 Proceedings of the 9th international conference on Theory and Applications of Satisfiability Testing
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FLOPS'12 Proceedings of the 11th international conference on Functional and Logic Programming
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We study the problem of maintaining the distances and the shortest paths from a source node in a directed graph with arbitrary arc weights, when weight updates of arcs are performed. We propose algorithms that work for any graph and require linear space and optimal query time. If a negative{length cycle is added during weight-decrease operations it is detected by the algorithms. The algorithms explicitly deal with zero{length cycles. We show that, if the graph has a k-bounded accounting function (as in the case of graphs with genus, arboricity, degree, treewidth or pagenumber bounded by k, and k-inductive graphs) the algorithms require O(k ċ n ċ log n) worst case time. In the case of graphs with n nodes and m arcs k = O(√m); this gives O(√m ċ n ċ log n) worst case time per operation, which is better for a factor of O(√m log n) than recomputing everything from scratch after each update. If we perform also insertions and deletions of arcs, then the above bounds become amortized.