Given two 'n' X 'n' Boolean matrices A and B (i.e., each element of A and B is either 0 or 1), their Boolean Product Witness Matrix (BPWM)is an 'n' X 'n' integer matrix C with C(i,j) = some 'k' such that A(i,k) = B(k,j) = 1 and C(i,j) = 0 if no such 'k' exists. The best known algorithms to date for the BPWM problem have a worst-case time-complexity that is log(n) times the time required to multiply two 'n' X 'n' integer matrices. Currently, the best known algorithm for integer matrix multiplication is due to Coppersmith and Winograd and requires O(n ** 2.376) steps. In this paper, we describe a randomized algorithm for solving the BPWM problem with an expected time-complexity of O(log(n) * n ** 2) on mutually independent input matrices. This algorithm indirectly provides methods to find transitive closures of graphs and to solve the all-pairs-shortest-path problem for unweighted, undirected graphs with an expected run time of O(log(n) * n ** 2). Our algorithm is practical and simpler to implement than currently best known algorithms for this problem, which use
Coppersmith and Winograd's algorithm as a prime component. Although, instances of the BPWM problem can be constructed for which our algorithm does not terminate in O(log(n) * n ** 2) steps, we show that such
instances are extremely rare.
By: Anshul Gupta, Pankaj Rohatgi, Ramesh Agarwal
Published in: RC21224 in 1998
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