TY - CHAP

T1 - An improved bound on Boolean matrix multiplication for highly clustered data

AU - Ga̧sieniec, Leszek

AU - Lingas, Andrzej

PY - 2003/1/1

Y1 - 2003/1/1

N2 - We consider the problem of computing the product of two n x n Boolean matrices A and B. For two 0-1 strings s = s1s2....sm and u = u1u2...um, an extended Hamming distance, eh(s, u), between the strings, is defined by a recursive equation eh(s,u) = eh(sl+1...sm,ul+1...um) + (s1 + u1 mod 2), where l is the maximum number, s.t., sj = s1 and uj = u1 for j = 1,...,l. 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 Ō(n(n + min{MWT(A), MWT(Bt)})) 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 Björklund and Lingas [4]. We also observe that min{MWT(A), MWT(Bt)} = O(min{rA,rB}), where rA and TB reflect the minimum number of rectangles required to cover 1s in A and B, respectively. Hence, our result also generalizes the recent upper bound on the Boolean matrix product in terms of rA and rB, due to Lingas [12].

AB - We consider the problem of computing the product of two n x n Boolean matrices A and B. For two 0-1 strings s = s1s2....sm and u = u1u2...um, an extended Hamming distance, eh(s, u), between the strings, is defined by a recursive equation eh(s,u) = eh(sl+1...sm,ul+1...um) + (s1 + u1 mod 2), where l is the maximum number, s.t., sj = s1 and uj = u1 for j = 1,...,l. 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 Ō(n(n + min{MWT(A), MWT(Bt)})) 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 Björklund and Lingas [4]. We also observe that min{MWT(A), MWT(Bt)} = O(min{rA,rB}), where rA and TB reflect the minimum number of rectangles required to cover 1s in A and B, respectively. Hence, our result also generalizes the recent upper bound on the Boolean matrix product in terms of rA and rB, due to Lingas [12].

UR - http://www.scopus.com/inward/record.url?scp=35248863004&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35248863004&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-45078-8_29

DO - 10.1007/978-3-540-45078-8_29

M3 - Chapter

AN - SCOPUS:35248863004

SN - 3540405453

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 329

EP - 339

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

A2 - Dehne, Frank

A2 - Sack, Jorg-Rudiger

A2 - Smid, Michiel

PB - Springer Verlag

ER -