A Fast Output-Sensitive Algorithm for Boolean Matrix Multiplication - Lingas, Andrzej (Algorithmica, 2011)
We use randomness to exploit the potential sparsity of the Boolean matrix product in order to speed up the computation of the product. Our new fast output-sensitive algorithm for Boolean matrix product and its witnesses is randomized and provides the Boolean product and its witnesses almost certainly. Its worst-case time performance is expressed in terms of the input size and the number of non-zero entries of the product matrix. It runs in time , where the input matrices have size n × n , the number of non-zero entries in the product matrix is at most s , ω is the exponent of the fast matrix multiplication and denotes O ( f ( n )log d n ) for some constant d . By the currently best bound on ω , its running time can be also expressed as . Our algorithm is substantially faster than the output-sensitive column-row method for Boolean matrix product for s larger than n 1.232 and it is never slower than the fast -time algorithm for this problem. By applying the fast rectangular matrix multiplication, we can refine our upper bound further to the form , where ω ( p , q , t ) is the exponent of the fast multiplication of an n p × n q matrix by an n q × n t matrix. We also present a partial derandomization of our algorithm as well as its generalization to include the Boolean product of rectangular Boolean matrices. Finally, we show several applications of our output-sensitive algorithms.