Years and Authors of Summarized Original Work
1999; King
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 n is number of nodes in the graph. The basic technique is extended to give fully dynamic algorithms for approximate and exact all-pairs shortest paths problems.
The basic building block of these algorithms is a method of maintaining...
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Recommended Reading
Demestrescu C, Italiano GF (2005) Trade-offs for fully dynamic transitive closure on DAG's: breaking through the O(n2) barrier, (presented in FOCS 2000). J ACM 52(2):147–156
Demestrescu C, Italiano GF (2004) A new approach to dynamic all pairs shortest paths, (presented in STOC 2003). J ACM 51(6):968–992
Frigioni D, Miller T, Nanni U, Zaroliagis CD (2001) An experimental study of dynamic algorithms for transitive closure. ACM J Exp Algorithms 6(9)
King V (1999) 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). IEEE Computer Society, New York, pp 81–91
King V, Sagert G (2002) A fully dynamic algorithm for maintaining the transitive closure (presented in FOCS 1999). JCCS 65(1):150–167
King V, Thorup M (2001) A space saving trick for dynamic transitive closure and shortest path algorithms. In: Proceedings of the 7th annual international conference of computing and cominatorics (COCOON). Lecture notes computer science, vol 2108/2001. Springer, Heidelberg, pp 269–277
Roditty L (2003) A faster and simpler fully dynamic transitive closure. In: Proceedings of the 14th annual ACM-SIAM symposium on discrete algorithms (ACMIEEE SODA). ACM, Baltimore, pp 404–412
Roditty L, Zwick U (2002) Improved dynamic reachability algorithms for directed graphs. In: Proceedings of the 43rd annual symposium on foundation of computer science (IEEE FOCS). IEEE Computer Society, Vancouver, pp 679–688
Roditty L, Zwick U (2004) 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 (ACMSTOC). ACM, Chicago, pp 184–191
Sankowski S (2004) Dynamic transitive closure via dynamic matrix inverse. In: Proceedings of the 45th annual symposium on foundations of computer science (IEEE FOCS). IEEE Computer Society, Rome, pp 509–517
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King, V. (2016). Fully Dynamic Transitive Closure. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_158
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