### Abstract

We consider the problem of computing the product of two n x 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 x 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 Ō(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 Björklund and Lingas [4]. We also observe that min{MWT(A), MWT(B^{t})} = O(min{r_{A},r_{B}}), where r_{A} and T_{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 [12].

Original language | English (US) |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Editors | Frank Dehne, Jorg-Rudiger Sack, Michiel Smid |

Publisher | Springer Verlag |

Pages | 329-339 |

Number of pages | 11 |

ISBN (Print) | 3540405453 |

DOIs | |

State | Published - Jan 1 2003 |

Externally published | Yes |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 2748 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(pp. 329-339). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2748). Springer Verlag. https://doi.org/10.1007/978-3-540-45078-8_29