# Encyclopedia of Algorithms

2008 Edition
| Editors: Ming-Yang Kao

# Fully Dynamic Transitive Closure

1999; King
• Valerie King
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30162-4_158

## 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 $$O(n^2 \log n)$$

This is a preview of subscription content, log in to check access.

1. 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)
2. 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)
3. 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)Google Scholar
4. 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)Google Scholar
5. 5.
King, V., Sagert, G.: A fully dynamic algorithm for maintaining the transitive closure, (presented in FOCS 1999). JCCS 65(1), 150–167 (2002)
6. 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)Google Scholar
7. 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)Google Scholar
8. 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)Google Scholar
9. 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. 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)