## Abstract

The subcubic (O(n^{w}) for w(3) algorithms to multiply Boolean matrices do not provide the witnesses; namely, they compute C=A.B but if C_{ij}=1 they do not find an index k (a witness) such that A_{ik}=B_{kj}=1. The authors design a deterministic algorithm for computing the matrix of witnesses that runs in O(n^{w}) time, where here O(n^{w}) denotes O(n^{w}(log n)/sup O(1)/). The subcubic methods to compute the shortest distances between all pairs of vertices also do not provide for witnesses; namely they compute the shortest distances but do not generate information for computing quickly the paths themselves. A witness for a shortest path from v_{i} to v_{j} is an index k such that v_{k} is the first vertex on such a path. They describe subcubic methods to compute such witnesses for several versions of the all pairs shortest paths problem. As a result, they derive shortest paths algorithms that provide characterization of the shortest paths in addition to the shortest distances in the same time (up to a polylogarithmic factor) needed for computing the distances; namely O(n/sup (3+w)/2/) time in the directed case and O(n^{w}) time in the undirected case. They also design an algorithm that computes witnesses for the transitive closure in the same time needed to compute witnesses for Boolean matrix multiplication.

Original language | English |
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Title of host publication | Proceedings - 33rd Annual Symposium on Foundations of Computer Science, FOCS 1992 |

Publisher | IEEE Computer Society |

Pages | 417-426 |

Number of pages | 10 |

Volume | 1992-October |

ISBN (Electronic) | 0818629002 |

DOIs | |

State | Published - 1 Jan 1992 |

Externally published | Yes |

Event | 33rd Annual Symposium on Foundations of Computer Science, FOCS 1992 - Pittsburgh, United States Duration: 24 Oct 1992 → 27 Oct 1992 |

### Conference

Conference | 33rd Annual Symposium on Foundations of Computer Science, FOCS 1992 |
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Country/Territory | United States |

City | Pittsburgh |

Period | 24/10/92 → 27/10/92 |