An improved bound on Boolean matrix multiplication for highly clustered data
Author
Summary, in English
We consider the problem of computing the product of two n x n Boolean matrices A and B. For two 0 - I strings s = s(1)s(2) .... s(m) and u = u(1)u(2) ... u(m), an extended Hamming distance, eh(s, u), between the strings, is defined by a recursive equation eh(s, u) = eh(s(l+1) ... s(m), u(l+1) ... u(m)) + (s(1) + u(1) mod 2), where l is the maximum number, s.t., s(j) = s, and u(j) = u(1) for j = For any n x n Boolean matrix C, let GC be a complete weighted graph on the rows of C, where the weight of an edge between two rows is equal to its extended Hamming distance. Next, let MWT(C) be the weight of a minimum weight spanning, tree of GC. WE! show that the product of A and B as well as the so called witnesses of the product can be computed in time (O) over bar (n(n + min{MWT(A), MWT(B-t)}))(1). Since the extended Hamming distance between two strings never exceeds the standard Hamming distance between them, our result subsumes an earlier similar result on the Boolean matrix product in terms of the Hamming distance due to Bjorklund and Lingas [4]. We also observe that min{MWT(A),MWT(B-t)} = O(min{r(A),r(B)}), where r(A) and r(B) reflect the minimum number of rectangles required to cover ls in A and B, respectively. Hence, our result also generalizes the recent upper bound on the Boolean matrix product in terms of r(A) and r(B), due to Lingas [12].
Department/s
- Computer Science
Publishing year
2003
Language
English
Pages
329-339
Publication/Series
Lecture Notes in Computer Science
Volume
2748
Document type
Conference paper
Publisher
Springer
Topic
- Computer Science
Conference name
8th International Workshop, WADS 2003
Conference date
2003-07-30 - 2003-08-01
Conference place
Ottawa, Ontario, Canada
Status
Published
Project
- VR 2002-4049
ISBN/ISSN/Other
- ISSN: 0302-9743
- ISSN: 1611-3349
- ISBN: 978-3-540-40545-0