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An improved bound on Boolean matrix multiplication for highly clustered data

Gasieniec, L and Lingas, Andrzej LU (2003) 8th International Workshop, WADS 2003 In Lecture Notes in Computer Science 2748. p.329-339
Abstract
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... (More)
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]. (Less)
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author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
in
Lecture Notes in Computer Science
volume
2748
pages
329 - 339
publisher
Springer
conference name
8th International Workshop, WADS 2003
external identifiers
  • wos:000185605300029
  • scopus:35248863004
ISSN
1611-3349
0302-9743
ISBN
978-3-540-40545-0
DOI
10.1007/978-3-540-45078-8_29
project
VR 2002-4049
language
English
LU publication?
yes
id
fb5e93c3-2e7c-4641-b5af-514ef4020281 (old id 299947)
date added to LUP
2007-08-07 14:10:46
date last changed
2018-01-14 03:35:16
@inproceedings{fb5e93c3-2e7c-4641-b5af-514ef4020281,
  abstract     = {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].},
  author       = {Gasieniec, L and Lingas, Andrzej},
  booktitle    = {Lecture Notes in Computer Science},
  isbn         = {978-3-540-40545-0},
  issn         = {1611-3349},
  language     = {eng},
  pages        = {329--339},
  publisher    = {Springer},
  title        = {An improved bound on Boolean matrix multiplication for highly clustered data},
  url          = {http://dx.doi.org/10.1007/978-3-540-45078-8_29},
  volume       = {2748},
  year         = {2003},
}