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Faster Algorithms for Incremental Topological Ordering.Robert Tarjan

Haeupler, Bernhard and Kavitha, Telikepalli and Mathew, Rogers and Sen, Siddhartha (2008) Faster Algorithms for Incremental Topological Ordering.Robert Tarjan. In: ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I, Berlin, Heidelberg.

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Official URL: http://www.springerlink.com/content/8537660vr7j15m...


We present two online algorithms for maintaining a topological order of a directed acyclic graph as arcs are added, and detecting a cycle when one is created. Our first algorithm takes O(m 1/2) amortized time per arc and our second algorithm takes O(n 2.5/m) amortized time per arc, where n is the number of vertices and m is the total number of arcs. For sparse graphs, our O(m 1/2) bound improves the best previous bound by a factor of logn and is tight to within a constant factor for a natural class of algorithms that includes all the existing ones. Our main insight is that the two-way search method of previous algorithms does not require an ordered search, but can be more general, allowing us to avoid the use of heaps (priority queues). Instead, the deterministic version of our algorithm uses (approximate) median-finding; the randomized version of our algorithm uses uniform random sampling. For dense graphs, our O(n 2.5/m) bound improves the best previously published bound by a factor of n 1/4 and a recent bound obtained independently of our work by a factor of logn. Our main insight is that graph search is wasteful when the graph is dense and can be avoided by searching the topological order space instead. Our algorithms extend to the maintenance of strong components, in the same asymptotic time bounds.

Item Type: Conference Paper
Publisher: Springer-Verlag
Additional Information: Copyright of this article belongs to Springer-Verlag.
Department/Centre: Division of Electrical Sciences > Computer Science & Automation
Date Deposited: 23 Sep 2011 09:22
Last Modified: 23 Sep 2011 09:22
URI: http://eprints.iisc.ac.in/id/eprint/40702

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