WADS 2003: Algorithms and Data Structures pp 329-339

An Improved Bound on Boolean Matrix Multiplication for Highly Clustered Data

• Leszek Gąsieniec
• Andrzej Lingas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2748)

Abstract

We consider the problem of computing the product of two n ×n Boolean matrices A and B.

For two 0-1 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 1 and u j =u 1 for j=1,...,l. For any n ×n Boolean matrix C, let G C 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 G C . We show that the product of A and B as well as the so called witnesses of the product can be computed in time Õ(n(n + 1 min{MWT(A), MWT(B t )})).

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 . 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 1s 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 .

Keywords

Span Tree Search Tree Minimum Span Tree Combinatorial Algorithm Boolean Matrix
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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