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Faster Algorithms for Incremental Topological Ordering

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Automata, Languages and Programming (ICALP 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5125))

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Abstract

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.

This paper combines the best results of [12] and [7]. The former gives two incremental topological ordering algorithms, with amortized time bounds per arc addition of O(m 1/2 + (nlogn)/m 1/2) and O(n 2.5/m). The latter, which was written with knowledge of the former, gives an algorithm with a bound of O(m 1/2).

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Author information

Authors and Affiliations

  1. Princeton University, Princeton, NJ 08544

    Bernhard Haeupler, Siddhartha Sen & Robert E. Tarjan

  2. Indian Institute of Science, Bangalore, India

    Telikepalli Kavitha & Rogers Mathew

  3. HP Laboratories, Palo Alto, CA 94304

    Robert E. Tarjan

Authors
  1. Bernhard Haeupler
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  2. Telikepalli Kavitha
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  3. Rogers Mathew
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  4. Siddhartha Sen
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  5. Robert E. Tarjan
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Editor information

Editors and Affiliations

  1. School of Computer Science, Reykjavik University, Kringlan 1, 103, Reykjavík, Iceland

    Luca Aceto

  2. Department of Computer Science, IT-Parken, University of Aarhus, Ã…bogade 34, 8200, Ã…rhusN, Denmark

    Ivan Damgård

  3. Department of Computer Science, University of Liverpool, Ashton Building, L69 3BX, Liverpool, UK

    Leslie Ann Goldberg

  4. School of Computer Science, Reykjavik University, Kringlan 1, 103, Reykjavik, Iceland

    Magnús M. Halldórsson

  5. Department of Computer Science, Reykjavik University, Kringlan 1, 103, Reykjavík, Iceland

    Anna Ingólfsdóttir

  6. Université de Bordeaux-1, LaBRI, 351, Cours de la Libération, 33405, Talence cedex, France

    Igor Walukiewicz

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© 2008 Springer-Verlag Berlin Heidelberg

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Haeupler, B., Kavitha, T., Mathew, R., Sen, S., Tarjan, R.E. (2008). Faster Algorithms for Incremental Topological Ordering. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70575-8_35

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  • DOI: https://doi.org/10.1007/978-3-540-70575-8_35

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-70574-1

  • Online ISBN: 978-3-540-70575-8

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