An improved bound on Boolean matrix multiplication for highly clustered data
(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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/299947
- author
- Gasieniec, L and Lingas, Andrzej LU
- organization
- publishing date
- 2003
- type
- Chapter in Book/Report/Conference proceeding
- publication status
- published
- subject
- host publication
- Algorithms and Data Structures
- series title
- Lecture Notes in Computer Science
- volume
- 2748
- pages
- 329 - 339
- publisher
- Springer
- conference name
- 8th International Workshop, WADS 2003
- conference location
- Ottawa, Ontario, Canada
- conference dates
- 2003-07-30 - 2003-08-01
- 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
- 2016-04-01 12:32:56
- date last changed
- 2024-06-19 01:24:32
@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 = {{Algorithms and Data Structures}}, isbn = {{978-3-540-40545-0}}, issn = {{1611-3349}}, language = {{eng}}, pages = {{329--339}}, publisher = {{Springer}}, series = {{Lecture Notes in Computer Science}}, 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}}, doi = {{10.1007/978-3-540-45078-8_29}}, volume = {{2748}}, year = {{2003}}, }