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Fully Dynamic Transitive Closure

1999; King

  • Reference work entry
Encyclopedia of Algorithms

Keywords and Synonyms

Incremental algorithms for digraphs; Fully dynamic graph algorithm for maintaining transitive closure; All-pairs dynamic reachability        

Problem Definition

Design a data structure for a directed graph with a fixed set of node which can process queries of the form “Is there a path from i to j ?” and updates of the form: “Insert edge (i, j)”; “Delete edge (i, j)”. The goal is to minimize update and query times, over the worst case sequence of queries and updates. Algorithms to solve this problem are called “fully dynamic” as opposed to “partially dynamic” since both insertions and deletions are allowed.

Key Results

This work [4] gives the first deterministic fully dynamic graph algorithm for maintaining the transitive closure in a directed graph. It uses \( O(n^2 \log n) \) amortized time per update and O(1) worst case query time where nis number of nodes in the graph. The basic technique is extended to give fully dynamic algorithms for approximate and exact...

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Recommended Reading

  1. Demestrescu, C., Italiano, G.F.: Trade-offs for fully dynamic transitive closure on DAG's: breaking through the \( O(n^2) \) barrier, (presented in FOCS 2000). J. ACM 52(2), 147–156 (2005)

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  2. Demestrescu, C., Italiano, G.F.: A new approach to dynamic all pairs shortest paths, (presented in STOC 2003). J. ACM 51(6), 968–992 (2004)

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  3. Frigioni, D., Miller, T., Nanni, U., Zaroliagis, C.D.: An experimental study of dynamic algorithms for transitive closure. ACM J Exp. Algorithms 6(9) (2001)

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  4. King, V.: Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In: Proceedings of the 40th Annual IEEE Symposium on Foundation of Computer Science. ComiIEEE FOCS pp. 81–91. IEEE Computer Society, New York (1999)

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  5. King, V., Sagert, G.: A fully dynamic algorithm for maintaining the transitive closure, (presented in FOCS 1999). JCCS 65(1), 150–167 (2002)

    MathSciNet  MATH  Google Scholar 

  6. King, V., Thorup, M.: A space saving trick for dynamic transitive closure and shortest path algorithms. In: Proceedings of the 7th Annual International Conference of Computing and Cominatorics, vol. 2108/2001, pp. 269–277. Lect. Notes Comp. Sci. COCOON Springer, Heidelberg (2001)

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  7. Roditty, L.: A faster and simpler fully dynamic transitive closure. In: Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms. ACM IEEE SODA, pp. 404–412. ACM, Baltimore (2003)

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  8. Roditty, L., Zwick, U.: Improved dynamic reachability algorithms for directed graphs. In: Proceedings of the 43rd Annual Symposium on Foundation of Computer Science. IEEE FOCS, pp. 679–688 IEEE Computer Society, Vancouver, Canada (2002)

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  9. Roditty, L., Zwick, U.: A fully dynamic reachability algorithm for directed graphs with an almost linear update time. In: Proceedings of the 36th ACM Symposium on Theory of Computing. ACM STOC, pp. 184–191 ACM, Chicago (2004)

    Google Scholar 

  10. Sankowski, S.: Dynamic transitive closure via dynamic matrix inverse. In: Proceedings of the 45th Annual Symposium on Foundations of Computer Science. IEEE FOCS, 509–517, IEEE Computer Society, Rome, Italy (2004)

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Author information

Authors and Affiliations

  1. Department of Computer Science Department, University of Victoria, Victoria, BC, Canada

    Valerie King

Authors
  1. Valerie King
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

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© 2008 Springer-Verlag

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King, V. (2008). Fully Dynamic Transitive Closure. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_158

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