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Improved Decremental Algorithms for Maintaining Transitive Closure and Allpairs Shortest Paths

Baswana, Surender and Sen, Sandeep and Hariharan, Ramesh (2002) Improved Decremental Algorithms for Maintaining Transitive Closure and Allpairs Shortest Paths. In: of the thiry-fourth annual ACM symposium on Theory of computing, May 19-21, 2002, Montreal, Quebec, Canada, pp. 117-123.

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Abstract

We present improved algorithms for maintaining transitive closure and all-pairs shortest paths/distances in a digraph under deletion of edges.(MATH) For the problem of transitive closure, the previous best known algorithms, for achieving O(1) query time, require O(\min(m, \frac{n^3}{m}))$ amortized update time, implying an upper bound of O(n^{\frac{3}{2}})$ on update time per edge-deletion. We present an algorithm that achieves $O(1)$ query time and O(n \log^2n + \frac{n^2}{\sqrt{m}}{\sqrt{\log n}})$ update time per edge-deletion, thus improving the upper bound to O(n^{\frac{4}{3}}\sqrt[3]{\log n})$.(MATH) For the problem of maintaining all-pairs shortest distances in unweighted digraph under deletion of edges, we present an algorithm that requires O(\frac{n^3}{m} \log^2 n)$ amortized update time and answers a distance query in O(1) time. This improves the previous best known update bound by a factor of log n. For maintaining all-pairs shortest paths, we present an algorithm that achieves O(\min(n^{\frac{3}{2}} \sqrt{\log n}, \frac{n^3}{m} \log ^2n))$ amortized update time and reports a shortest path in optimal time (proportional to the length of the path). For the latter problem we improve the worst amortized update time bound by a factor of O(\sqrt{\frac{n}{\log n}})$.(MATH) We also present the first decremental algorithm for maintaining all-pairs (1+&egr;) approximate shortest paths/distances, for any &egr; > 0, that achieves a sub-quadratic update time of O(n log2n + \frac{n^2}{\sqrt{\epsilon m}}\sqrt{\log n})$ and optimal query time.Our algorithms are randomized and have one-sided error for query (with probability O(1/nc) for any constant c).

Item Type: Conference Paper
Additional Information: ©ACM,2002.This is the author's version of the work.It is posted here by permission of ACM for your personal use.Not for redistribution.The definitive version was published in Proceedings of the thiry-fourth annual ACM symposium on Theory of computing,2002 http://doi.acm.org/10.1145/509907.509928
Keywords: BFS tree;dynamic;graph;transitive closure;shortest path
Department/Centre: Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation)
Date Deposited: 15 Oct 2007
Last Modified: 19 Sep 2010 04:12
URI: http://eprints.iisc.ernet.in/id/eprint/253

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