C. Demetrescu and G. F. Italiano. Fully dynamic transitive closure: breaking through the O(n  Home/Search   Document Details and Download   Summary   Related Articles   Check

....edges with O(1) query time. The rst algorithm was given by Poutre and Leeuwen [10] and it achieved O(m) amortized update time. The algorithm is thus good for sparse graph, but for dense graphs the update time could be as high as O(n ) The second algorithm was given by Demetrescu and Italiano [3] and it achieves O( amortized update time [2] It can be seen that a combination these two algorithms together yields an upper bound of ) on the update time. In this paper, we present an ecient algorithm which achieves O(1) query time with high probability and O(n log log n) ....

....and O(n log deletion. Previously there were two algorithms for maintaining transitive closure under deletion of edges with O(1) query time. The rst algorithm due to Poutre and Leeuwen [10] achieved O(m) amortized update time while the second algorithm due to Demetrescu and Italiano [3] achieved O( amortized update time. These two algorithms together establish an upper bound of O(n ) on the update time. Combining our algorithm with these existing algorithms, we can state the following Corollary. Corollary 4.1.1 There exists an algorithm for maintaining transitive ....

Camil Demetrescu and G.F. Italiano. Fully dynamic transitive closure : Breaking through the o(n ) barrier. FOCS, 41:381-389, 2000.

....is striking in two respects: i) the independence of the asymptotic bounds on the dimension d, and (ii) the usage of fast matrix multiplication, which is rare among algorithms in computational geometry. Applications of fast matrix multiplication are more common in dynamic graph algorithms, e.g. [10, 23], but the sublinearity of our update bound is still unusual. In Section 7, we partially explain why a polylogarithmic solution is unlikely given the current state of the art, and why points (i) and (ii) might be inherent to the problem itself, at least for d # 3. Previous geometric work. ....

....in a single update. Alternatively, one can reduce the problem to reachability in a dynamic directed graph, so that a vertex update causes only a constant number of edge changes to the graph (hint: create two copies of each vertex and each edge) However, dynamic directed graph reachability [10] is computationally more demanding (the goal there was an o(n 2 ) update bound for dense graphs) Previous graph work. The dynamic subgraph connectivity problem has indeed been proposed before, as the author has learned afterwards, in a paper by Frigioni and Italiano [16] The motivation there ....

C. Demetrescu and G. F. Italiano. Fully dynamic transitive closure: breaking through the O(n 2 ) barrier. In Proc. 41st IEEE Sympos. Found. Comput. Sci., pages 381--389, 2000.

.... 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 ....

C. Demetrescu and G. F. Italiano. Fully dynamic transitive closure: Breaking through the O(n 2 ) barrier. In Proc. 41st IEEE Symp. on Foundations of Computer Science, to appear, 2000.

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C. Demetrescu and G.F. Italiano. Fully dynamic transitive closure: Breaking through the O(n ) barrier. In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science (FOCS'00), pages 381--389, 2000.

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C. Demetrescu and G.F. Italiano. Fully dynamic transitive closure: Breaking through the O(n ) barrier. In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science (FOCS'00), pages 381--389, 2000.

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C. Demetrescu and G.F. Italiano. Fully dynamic transitive closure: Breaking through the O(n ) barrier. In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science (FOCS'00), pages 381--389, 2000.

.... Omega Gamma n) time, then one might as well afford to slow down the distance queries in order to speed the updates up. We remark that, despite many decades of research in the area of dynamic graph problems, there seem to be very few query update trade offs available in the literature (see e.g. [3, 14]) Our Results. In this paper, we present a new fully dynamic shortest path algorithm for directed graphs and arbitrary real weights (with at most S different values per edge weight) which achieves O(S n) amortized time per update and O(1) worstcase time per query. This improves over [4] ....

....shortest paths with at most 2 i edges. The main idea is to keep log k polynomials P i (Y ) Y , 1 i log k, over the fmin; g semiring with instances of the data structure presented in [4] The degree 3 for the polynomial P i (Y ) Y is used in a dynamic setting exactly as described in [3] for the simpler problem of fully dynamic transitive closure. This yields the following bounds: Theorem 1 Any k Decrease and k Increase operation requires O(S Delta k Delta n n) amortized time and any k Query is answered in O(1) worst case time. For paths with more than k edges, we ....

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C. Demetrescu and G.F. Italiano. Fully dynamic transitive closure: Breaking through the O(n ) barrier. In Proc. of the 41st IEEE Annual Symposium on Foundations of Computer Science (FOCS'00), pages 381--389, 2000. Full paper available at the URL: http://arXiv.org/abs/cs.DS/0104001.

....shortest path realizing that distance. Since reporting a shortest path might require as much as # n) time, then one might as well a#ord to slow down the distance queries in order to speed the updates up. Note that there are only a few query update trade o#s available in the literature (see e.g. [7, 19] for dynamic transitive closure) Our Results. In this paper we a#rmatively answer both questions. Our first contribution is a fully dynamic algorithm for maintaining APSP on directed graphs with arbitrary real weights. Given a directed graph G, subject to dynamic operations, and such that each ....

....their space usage is O(n ) We remark that our algorithms use simple data structures, and thus seem amenable to e#cient implementations. Techniques. To achieve our bounds, we manage to extend to dynamic shortest paths the algebraic framework and the lazy evaluation techniques developed in [7] for the simpler problem of dynamic transitive closure. In particular, we show how to cast fully dynamic all pairs shortest paths into the problem of maintaining polynomials of matrices over the semiring. The equivalence between APSP and matrix multiplication on the semiring is well known ....

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C. Demetrescu and G.F. Italiano. Fully dynamic transitive closure: Breaking through the O(n ) barrier. In Proc. of the 41st IEEE Annual Symposium on Foundations of Computer Science (FOCS'00), pages 381--389, 2000. Full paper available at the URL: http://arXiv.org/abs/cs.DS/0104001.

....Large Data Sets: Science and Engineering and by CNR, the Italian National Research Council under contract n. 00.00346.CT26. This work is based on the first author s PhD Thesis [4] and a preliminary version has been presented at the 41st Annual Symp. on Foundations of Computer Science (FOCS 2000) [5]. y Email: demetres dis.uniroma1.it. URL: http: www.dis.uniroma1.it demetres. Part of this work has been done while visiting AT T Shannon Laboratory, Florham Park, NJ. z Email: italiano info.uniroma2.it. URL: http: www.info.uniroma2.it italiano. Part of this work has been done while ....

C. Demetrescu and G.F. Italiano. Fully dynamic transitive closure: Breaking through the O(n 2 ) barrier. In Proc. of the 41st IEEE Annual Symposium on Foundations of Computer Science (FOCS'00), pages 381--389, 2000.

....less than C and runs in O(C n 2 ) amortized time. Our algorithms are based on simple data structures and thus seem amenable to ecient implementations. To achieve our bounds, we manage to extend to dynamic shortest paths the algebraic framework and the lazy evaluation techniques developed in [5] for the simpler problem of dynamic transitive closure. In particular, we show how to cast fully dynamic all pairs shortest paths into the problem of maintaining polynomials of matrices over the fmin, g semiring. The equivalence between APSP and matrix multiplication on the fmin, g semiring is ....

....to extend the results of this section to the case where the same variable may occur in polynomial P more than once. Our Data Structure. We consider only the case of polynomials of degree k 2. The case k 2 can be easily supported by using a technique similar to the one presented in [5] for maintaining polynomials over Boolean matrices, i.e. by considering an equivalent representation b P for P such that b P has degree 2. We maintain polynomials of degree k 2 with the following elementary data structures: 2h matrices X a 1 ; X a 2 2 M n that store the current value ....

C. Demetrescu and G.F. Italiano. Fully dynamic transitive closure: Breaking through the O(n 2 ) barrier. In Proc. of the 41st IEEE Annual Symposium on Foundations of Computer Science (FOCS'00), pages 381-389, 2000. Full paper available at the URL: http://arXiv.org/abs/cs.DS/0104001.

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C. Demetrescu and G. F. Italiano. Fully dynamic transitive closure: breaking through the O(n

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C. Demetrescu and G. Italiano. Fully dynamic transitive closure: Breaking through the O(n

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C. Demetrescu and G.F. Italiano. Fully dynamic transitive closure: Breaking through the O(n ) barrier. In Proceedings of FOCS'00, pages 381--389, 2000.

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C. Demetrescu and G. Italiano. Fully dynamic transitive closure: Breaking through the O(n ) barrier. In Proceedings of the 41th Annual IEEE Symposium on Foundations of Computer Science, Redondo Beach, California, pages 381--389, 2000.

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C. Demetrescu and G. F. Italiano. Fully dynamic transitive closure: breaking through the O(n ) barrier. In Proceedings of the 41th Annual IEEE Symposium on Foundations of Computer Science, Redondo Beach, California, pages 381-389, 2000.

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C. Demetrescu and G. F. Italiano. Fully dynamic transitive closure: breaking through the O(n ) barrier. In Proc. of 41st FOCS, pages 381--389, 2000.

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C. Demetrescu and G. F. Italiano. Fully dynamic transitive closure: breaking through the O(n ) barrier. In Proc. IEEE Symposium on Foundations of Computer Science, pages 381--389, 2000.

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C. Demetrescu and G. F. Italiano. Fully dynamic transitive closure: breaking through the O(n ) barrier. In Proc. IEEE Symposium on Foundations of Computer Science, pages 381--389, 2000.

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