A faster and simpler fully dynamic transitive closure
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
A fully dynamic reachability algorithm for directed graphs with an almost linear update time
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Fast sparse matrix multiplication
ACM Transactions on Algorithms (TALG)
A faster and simpler fully dynamic transitive closure
ACM Transactions on Algorithms (TALG)
Bounded-leg distance and reachability oracles
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
An experimental study of algorithms for fully dynamic transitive closure
Journal of Experimental Algorithmics (JEA)
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Dynamic plane transitive closure
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Proceedings of the forty-second ACM symposium on Theory of computing
Algorithms and theory of computation handbook
An experimental study of algorithms for fully dynamic transitive closure
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
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We obtain several new dynamic algorithms for maintaining the transitive closure of a directed graph, and several other algorithms for answering reachability queries without explicitly maintaining a transitive closure matrix. Among our algorithms are:(i) A decremental algorithm for maintaining the transitive closure of a directed graph, through an arbitrary sequence of edge deletions, in O(mn) total expected time, essentially the time needed for computing the transitive closure of the initial graph. Such a result was previously known only for acyclic graphs.(ii) Two fully dynamic algorithms for answering reachability queries. The first is deterministic and has an amortized insert/delete time of 0(m\sqrt n ), and worst-case query time of 0(\sqrt {n)}. The second is randomized and has an amortized insert/delete time of 0(m^{0.58} n) and worst-case query time of 0(m^{0.43}). This significantly improves the query times of algorithms with similar update times.(iii) A fully dynamic algorithm for maintaining the transitive closure of an acyclic graph. The algorithm is deterministic and has a worst-case insert time of O(m), constant amortized delete time of O(1), and a worst-case query time of O(n/log n).Our algorithms are obtained by combining several new ideas, one of which is a simple sampling idea used for detecting decompositions of strongly connected components, with techniques of Even and Shiloach [7], Italiano [14], Henzinger and King [10], and Frigioni et al. [8]. We also adapt results of Cohen [3] on estimating the size of the transitive closure to the dynamic setting.