Graph algorithms and NP-completeness
Graph algorithms and NP-completeness
Matrix multiplication via arithmetic progressions
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
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Complexity of parallel matrix computations
Theoretical Computer Science
The multi-tree approach to reliability in distributed networks
Information and Computation
Processor efficient parallel solution of linear systems over an abstract field
SPAA '91 Proceedings of the third annual ACM symposium on Parallel algorithms and architectures
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SIAM Journal on Computing
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Graph Algorithms
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Extremal Graph Theory
Random Weighted Laplacians, Lovász minimum digraphs and finding minimum separators
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
Faster dynamic matchings and vertex connectivity
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Dominators, directed bipolar orders, and independent spanning trees
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
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Let G = (V, E) be a directed graph and n denote |V|. We show that G is k-vertex connected iff for every subset X of V with |X| = k, there is an embedding of G in the (k-1)-dimensional space Rk-1, f : V &rarr:Rk-1, such that no hyperplane contains k points of {f(v) | v &egr; V}, and for each v &egr; V - X, f(v) is in the convex hull of {f(w) | (v, w) &egr; E}. This result generalizes to directed graphs the notion of convex embeddings of undirected graphs introduced by Linial, Lova´sz and Wigderson in “Rubber bands, convex embeddings and graph connectivity,” Combinatorica 8 (1988), 91-102.Using this characterization, a directed graph can be tested for k-vertex connectivity by a Monte Carlo algorithm in time O((M(n) + nM(k)).(log n)) with error probability n, and by a Las Vegas algorithm in expected time O((M(n)+nM(k)).k), where M(n) denotes the number of arithmetic steps for multiplying two n x n matrices (M(n) = O(n2.3755)). Our Monte Carlo algorithm improves on the best previous deterministic and randomized time complexities for k n0.19; e.g., for k = (n0.5, the factor of improvement is n0.62. Both algorithms have processor efficient parallel versions that run in O((log n)2) time on the EREW PRAM model of computation, using a number of processors equal to (log n) times the respective sequential time complexities. Our Monte Carlo parallel algorithm improves on the number of processors used by the best previous (Monte Carlo) parallel algorithm by a factor of at least (n2/(log n)3) while having the same running time.Generalizing the notion of s-t numberings, we give a combinatorial construction of a directed s-t numbering for any 2-vertex connected directed graph.