Amortized efficiency of a path retrieval data structure
Theoretical Computer Science
Finding paths and deleting edges in directed acyclic graphs
Information Processing Letters
Maintenance of transitive closures and transitive reductions of graphs
Proceedings of the International Workshop WG '87 on Graph-theoretic concepts in computer science
A fully dynamic algorithm for maintaining the transitive closure
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Improved decremental algorithms for maintaining transitive closure and all-pairs shortest paths
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Introduction to Algorithms
Improved Dynamic Reachability Algorithms for Directed Graphs
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Fully Dynamic Algorithms for Maintaining All-Pairs Shortest Paths and Transitive Closure in Digraphs
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Fully dynamic transitive closure: breaking through the O(n/sup 2/) barrier
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
An Experimental Study of Dynamic Algorithms for Transitive Closure
Journal of Experimental Algorithmics (JEA)
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We obtain a new fully dynamic algorithm for maintaining the transitive closure of a directed graph. Our algorithm maintains the transitive closure matrix in a total running time of O(mn + (ins + del) · n2), where ins (del) is the number of insert (delete) operations performed. Here n is the number of vertices in the graph and m is the initial number of edges in the graph. Obviously, reachability queries can be answered in constant time. The algorithm uses only O(n2) time which is essentially optimal for maintaining the transitive closure matrix. Our algorithm can also support path queries. If v is reachable from u, the algorithm can produce a path from u to v in time proportional to the length of the path. The best previously known algorithm for the problem is due to Demetrescu and Italiano [2000]. Their algorithm has a total running time of O(n3 + (ins + del) · n2). The query time is also constant. In addition, we also present a simple algorithm for directed acyclic graphs (DAGs) with a total running time of O(mn + ins · n2 + del). Our algorithms are obtained by combining some new ideas with techniques of Italiano [1986, 1988], King [1999], King and Thorup [2001] and Frigioni et al. [2001]. We also note that our algorithms are extremely simple and can be easily implemented.