A practical algorithm for Boolean matrix multiplication
Information Processing Letters
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
On the all-pairs-shortest-path problem
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Fast randomized algorithms for distributed edge coloring
PODC '92 Proceedings of the eleventh annual ACM symposium on Principles of distributed computing
Witnesses for Boolean matrix multiplication and for transitive closure
Journal of Complexity
Randomized algorithms
Approximating matrix multiplication for pattern recognition tasks
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
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Given two n × n Boolean matrices A and B, their Boolean Product Witness Matrix (BPWM) is an n × n integer matrix W with wij = some k such that aik = bkj = 1, (1 ≤ i, j, k ≤ n) and wij = 0 if no such k exists. The best known algorithms to date for solving the BPWM problem have a worst-case complexity of O(n&ohgr; log n), where O(n&ohgr;) is the time required to multiply two n × n integer matrices. The theoretically best known value of &ohgr; is 2.376, but is higher for algorithms that are practical to implement for reasonable problem sizes. In this paper, we present a relatively simple and practical randomized algorithm for solving the BPWM problem. The expected completion time of our algorithms is O(n2 log n) for a large class of Boolean matrices, including but not limited to all mutually-independent matrices. We show that the probability of the algorithm requiring more than O(n2 log n) is vanishingly small for most input Boolean matrices, irrespective of the density of 1's or 0's in them.