@inproceedings{2f32c049295f4de2b1dc20685a70b30d,
title = "A geometric approach to Boolean matrix multiplication",
abstract = "For a Boolean matrix D, let r(D) be the minimum number of rectangles sufficient to cover exactly the rectilinear region formed by the 1-entries in D. Next, let m(D) be the minimum of the number of 0-entries and the number of 1-entries in D. Suppose that the rectilinear regions formed by the 1-entries in two n x n Boolean matrices A and B totally with q edges are given. We show that in time (O) over tilde (q + min{r(A)r(B), n(n + r(A)), n(n + r(B))})(1) one can construct a data structure which for any entry of the Boolean product of A and B reports whether or not it is equal to 1, and if so, reports also the so called witness of the entry, in time 0 (log q). As a corollary, we infer that if the matrices A and B are given as input, their product and the witnesses of the product can be computed in time (O) over tilde (n(n + min{r(A), r(B)})). This implies in particular that the product of A and B and its witnesses can be computed in time (O) over tilde (n(n + min{m(A),m(B)})). In contrast to the known sub-cubic algorithms for Boolean matrix multiplication based on arithmetic 0 - 1-matrix multiplication, our algorithms do not involve large hidden constants in their running time and are easy to implement.",
keywords = "sweep-line method, running time, asymptotic upper bounds, spanning tree, time probability, monotone circuits, combinatorial algorithm, subcubic approximation, hidden constants, arithmetic (0 - 1)-matrix multiplication, subcubic algorithms, input matrices, entry witness, Boolean product, data structure, Boolean matrices, rectilinear region, rectangles, minimum number, geometric approach, Boolean matrix multiplication",
author = "Andrzej Lingas",
year = "2002",
language = "English",
volume = "2518",
publisher = "Springer",
pages = "501--510",
booktitle = "Algorithms and Computation : 13th International Symposium, ISAAC 2002, Vancouver, BC, Canada, November 21-23, 2002. Proceedings (Lecture Notes in Computer Science)",
address = "Germany",
note = "Algorithms and Computation. 13th International Symposium, ISSAC 2002. ; Conference date: 21-11-2002 Through 23-11-2002",
}