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
Randomized algorithms
Handbook of combinatorics (vol. 1)
Fast estimation of diameter and shortest paths (without matrix multiplication)
Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
On Diameter Verification and Boolean Matrix Multiplication.
On Diameter Verification and Boolean Matrix Multiplication.
Witnesses for Boolean matrix multiplication and for shortest paths
SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
Fast Boolean Matrix Multiplication for Highly Clustered Data
WADS '01 Proceedings of the 7th International Workshop on Algorithms and Data Structures
A Geometric Approach to Boolean Matrix Multiplication
ISAAC '02 Proceedings of the 13th International Symposium on Algorithms and Computation
Improved output-sensitive quantum algorithms for Boolean matrix multiplication
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
A relation-algebraic view on evolutionary algorithms for some graph problems
EvoCOP'06 Proceedings of the 6th European conference on Evolutionary Computation in Combinatorial Optimization
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We describe almost optimal (on the average) combinatorial algorithms for the following algorithmic problems : (i) computing the boolean matrix product, (ii) finding witnesses for boolean matrix multiplication and (iii) computing the diameter and all-pairs-shortest-paths of a given (unweighted) graph/digraph. For each of these problems, we assume that the input instances are drawn from suitable distributions. A random boolean matrix (graph/digraph) is one in which each entry (edge/arc) is set to 1 or 0 (included) independently with probability p. Even though fast algorithms have been proposed earlier, they are based on algebraic approaches which are complex and difficult to implement. Our algorithms are purely combinatorial in nature and are much simpler and easier to implement. They are based on a simple combinatorial approach to multiply boolean matrices. Using this approach, we design fast algorithms for (a) computing product and witnesses when A and B both are random boolean matrices or when A is random and B is arbitrary but fixed (or vice versa) and (b) computing diameter, distances and shortest paths between all pairs in the given random graph/digraph. Our algorithms run in O(n2(log n)) time with O(n-3) failure probability thereby yielding algorithms with expected running times within the same bounds. Our algorithms work for all values of p.