V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proc. 40th IEEE Symposium on Foundations of Computer Science (FOCS'99), 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....insertion or deletion. This bound has been improved by Fakcharoemphol and Rao in [6] who designed a fully dynamic algorithm for single source shortest paths in planar directed graphs that supports both queries and edge weight updates in O(n 4=5 13=5 n) amortized time per edge operation. King [16] presented a fully dynamic algorithm for maintaining all pairs shortest paths in directed graphs with positive integer weights less than C: the running time of her algorithm is n) per update. The space required is O(n ) a simple method for reducing space to O(n ) is shown in [17] ....

....update in O(S Delta n n) amortized time and each query in O(1) worst case time. The best known bounds for dynamic all pairs shortest paths on general graphs are thus O(1) worst case time per distance query and ) amortized time per update in case of integer weights in the range [0; C] [16], or O(S Delta n ) amortized time per update in case of arbitrary real valued weights (with at most S different values per edge) 4] Note that update bounds of [4] are slightly worse than the update bounds of [16] when restricted to positive integer edge weights, and that in both cases ....

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V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proc. 40th IEEE Symposium on Foundations of Computer Science (FOCS'99), pages 81--99, 1999.

....and Rao in [7] who designed a fully dynamic algorithm for single source shortest paths in planar directed graphs that supports both queries and edge weight updates in O(n 4=5 13=5 time per edge operation. The rst big step on general graphs and integer weights was made by King [13], who presented a fully dynamic algorithm for maintaining all pairs shortest paths in directed graphs with positive integer weights less than C: the running time of her algorithm is C log n ) per update. In previous work [3, 4] we have solved fully dynamic APSP on general directed graphs with ....

....the following time bounds: it supports any sequence of operations in ) amortized time per update and one look up in the worst case per distance query; it can also report shortest paths in optimal worst case time. We remark that our algorithm improves substantially over previous bounds [3, 4, 13]. Furthermore, and di erently from all the previous approaches, it solves fully dynamic APSP in its generality. Indeed, it runs on non negative real weights, and each weight has no limit on the number of di erent values it can assume. Despite few decades of research on dynamic shortest paths, we ....

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V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proc. 40th IEEE Symposium on Foundations of Computer Science (FOCS'99), pages 81-99, 1999.

....achieving constant query time. The answer to any query may be incorrect in the sense that it may not report a path when there exists one. All pairs shortest paths : The previous best known decremental algorithm for maintaining exact all pairs shortest paths in unweighted graph was given by King [8] with O(n ) update time. For shortest distance query, we present an algorithm that requires O( n) amortized update time per edge deletion and O(1) query time w.h.p. in the sense that it may not report the shortest path with probability less than n . If the query is to report the ....

....all pairs shortest paths under deletion of edges with error factor of 1 for any 0. The data structure answers each query (distance or path reporting) in optimal time and takes O(n log log 1 n) amortized time per edge deletion. The previous best known algorithm due to King [8] maintained exact all pairs shortest paths with O(n ) update time. Our algorithm signi cantly reduces the update time at the expense of introducing (1 ) approximation in accuracy of the answer. 9 6 Maintaining all pairs shortest paths We maintain all pairs shortest paths by keeping a ....

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Valerie King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. FOCS, 40:325-334, 1999.

....other hand, if the shortest paths in P are independent, exclude d can perform the same computation in O(m n log n) worst case time. 3 Existing Algorithms A straight forward approach to solving the distance sensitivity problem is to use a recent dynamic all pairs shortest paths (APSP) algorithm [DI01, K99, KT01] and delete a speci c failed link. The time required is quite high ( amortized) even for unweighted graphs though after that the queries can be answered in O(1) time assuming failure of that speci c link. In contrast, the distance sensitivity problem asks for a preprocessing of the graph ....

V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proc. of the 40th IEEE Annual Symposium on Foundations of Computer Science (FOCS'99), pp. 81-99, 1999. 10

.... weights is O(Sn 2 5 log 3 n) amortized time, where S is the maximum number of different real weights that each edge can assume throughout the sequence of updates [1] Even in the special case of integer edge weights less than C, maintaining APSP requires O(n 2 5 C lv ) amortized time per update [4]. Thus, if the goal is to answer queries as quickly as possible, this might be better than just answering each query with a single source shortest path (SSSP) computation taking, e.g. O(n log n m) time using Fredman and Tarjan s Fibonacci heaps [3] However, our main target here is to answer ....

V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transi- tive closure in digraphs. In Proc. doth IEEE Symposium on Foundations of Computer Science (FOCS'99), pages 81-99, 1999.

....log(nC) per operation. This bound has been improved by Fakcharoemphol and Rao in [11] who designed a fully dynamic algorithm for single source shortest paths in planar directed graphs that supports both queries and edge weight updates in 4 5 13 5 n) amortized time per edge operation. King [24] presented a fully dynamic algorithm for maintaining all pairs shortest paths in directed graphs with positive integer weights less than C: the running time of her algorithm is # C log n ) per update. The space required is O(n ) a simple method for reducing space to O(n ) is shown in ....

....fully dynamic algorithms for APSP on graphs with arbitrary real weights. The fastest asymptotic static algorithm is by Takaoka [36] and thus our algorithm is faster than recomputing from scratch by a factor of n S ) We can also support within the same time bounds the generalized updates of [24], i.e. decreasing the weight of a set of edges incident to the same vertex and increasing the weight of an arbitrary subset of edges. In the special case where edge weights can only be increased, we show how to support updates faster in O(S n) amortized time. This algorithm is randomized ....

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V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proc. 40th IEEE Symposium on Foundations of Computer Science (FOCS'99), pages 81--99, 1999. 25

....Very recently, King and Sagert [16] showed how to support queries in O(1) time and updates in O(n 2:26 ) time for general directed graphs and O(n 2 ) time for directed acyclic graphs; their algorithm is randomized with one side error. The bounds of King and Sagert were further improved by King [15], who exhibited a deterministic algorithm on general digraphs with O(1) query time and O(n 2 log n) amortized time per update operations, where updates are insertions of a set of edges incident to the same vertex and deletions of an arbitrary subset of edges. We remark that all these algorithms ....

....a deterministic algorithm on general digraphs with O(1) query time and O(n 2 log n) amortized time per update operations, where updates are insertions of a set of edges incident to the same vertex and deletions of an arbitrary subset of edges. We remark that all these algorithms (except [15]) use fast matrix multiplication as a subroutine. We observe that fully dynamic transitive closure algorithms with O(1) query time maintain explicitly the transitive closure of the input graph, in order to answer each query with exactly one lookup (on its adjacency matrix) Since an update may ....

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V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proc. 40th IEEE Symposium on Foundations of Computer Science (FOCS'99), 1999.

....having positive integer weights less than C: the amortized running time of their algorithm is O(Cn log n) per edge insertion. Henzinger et al. 13] designed a fully dynamic algorithm for APSP on planar graphs with integer weights, with a running time of O(n 9=7 log(nC) per operation. King [14] presented a fully dynamic algorithm for maintaining all pairs shortest paths in directed graphs with positive integer weights less than C: the running time of her algorithm is O(n 2:5 p C log n ) per update. It seems thus quite natural to ask whether one can eciently solve fully dynamic ....

....APSP on graphs with arbitrary real weights. The fastest asymptotic static algorithm is by Takaoka [24] and thus our algorithm is faster than recomputing from scratch by a factor of O( p n log log n = S log 3:5 n) We can also support within the same time bounds the generalized updates of [14], i.e. decreasing the weight of a set of edges incident to the same vertex and increasing the weight of an arbitrary subset of edges. In the special case where edge weights can only be increased, we show how to support updates faster in O(S n log 3 n) amortized time. This algorithm is ....

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V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proc. 40th IEEE Symposium on Foundations of Computer Science (FOCS'99), pages 81-99, 1999.

.... a much harder problem and much of the research so far was concentrated on the design of partially dynamic algorithms (see e.g. 4, 7, 8, 11, 26, 27, 28, 32] Only recently, fully dynamic algorithms have started to appear for maintenance of shortest path trees [18, 19, 30] and transitive closure [9, 23, 24, 25]. However, despite the number of interesting theoretical results achieved, very little has been done so far with respect to implementations even for the most fundamental problems (the only implementation e ort known to us is concerned with the maintenance of shortest path trees [10, 17] In this ....

....inputs and on an input motivated by a real world graph. We have shown with experimental data that there are several cases where some of the dynamic algorithms can be quite fast in practice. We plan to continue this experimental work by implementing the recent fully dynamic algorithms in [9, 24, 25]. The ecient implementation of these algorithms, however, may be very time consuming. Nevertheless, we believe that once all these implementations are available, an extensive experimental study of all algorithms may shed new light in the development of better dynamic algorithms for transitive ....

V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proc. 40th IEEE Symposium on Foundations of Computer Science, pp. 81-91, 1999.

....Very recently, King and Sagert [12] showed how to support queries in O(1) time and updates in O(n 2:26 ) time for general directed graphs and O(n 2 ) time for directed acyclic graphs; their algorithm is randomized with one side error. The bounds of King and Sagert were further improved by King [11], who exhibited a deterministic algorithm on general digraphs with O(1) query time and O(n 2 log n) amortized time per update operations, where updates are insertions of a set of edges incident to the same vertex and deletions of an arbitrary subset of edges. We remark that all these algorithms ....

....a deterministic algorithm on general digraphs with O(1) query time and O(n 2 log n) amortized time per update operations, where updates are insertions of a set of edges incident to the same vertex and deletions of an arbitrary subset of edges. We remark that all these algorithms (except [11]) use fast matrix multiplication as a subroutine. We observe that fully dynamic transitive closure algorithms with O(1) query time maintain explicitly the transitive closure of the input graph, in order to answer each query with exactly one lookup (on its adjacency matrix) Since an update may ....

[Article contains additional citation context not shown here]

V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proc. 40th IEEE Symposium on Foundations of Computer Science (FOCS'99), 1999.

....dynamic graph reachability, the graph is directed and the query operation is Path(u, v) which returns a Boolean indicating whether there is a path form u to v in the graph. The upper bounds known for directed graph connectivity are much worse than the bounds known for undirected graph connectivity [44]. Interestingly, no better lower bounds are known for the directed case. An interesting special case of directed dynamic graph reachability is the case of upward planar source sink graphs. There, the dynamic reachability can be solved in time O(log n) per operation [55] Husfeldt, Rauhe and Skyum ....

V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proc. 40th Annual Symposium on Foundations of Computer Science (FOCS'99), pages 81--89, 1999.

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V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proc. 40th IEEE Symposium on Foundations of Computer Science (FOCS'99), 1999.

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V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proc. of 40th FOCS, pages 81--91, 1999.

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King, V., Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs, in: Proc. 40th IEEE Symposium on Foundations of Computer Science (FOCS'99), 1999, pp. 81--91.

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V. King. Fully dynamic algorithms for maintaining allpairs shortest paths and transitive closure in digraphs. In ##### #### #### ######### ## ########### ## ######## #######. IEEE, October 1999.

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V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proceedings of FOCS'99, pages 81--91, 1999.

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V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, New York, New York, pages 81--91, 1999.

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V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, New York, New York, pages 81-89, 1999.

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V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proc. 40th IEEE Symposium on Foundations of Computer Science (FOCS'99), pages 81--99, 1999.

No context found.

V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proc. 40th IEEE Symposium on Foundations of Computer Science (FOCS'99), pages 81--99, 1999.

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Valerie King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In 40th Annual Symposium on Foundations of Computer Science, pages 81--91. IEEE Computer Society, 1999.

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