## Abstract

Let G = (V,E,w) be a directed graph, where w : V → ℝ is a weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u,v the capacity from u to v, denoted by c(u,v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem the task is to find the capacities for all ordered pairs of vertices. Our main result is an O(n ^{2.575}) time algorithm for APBP. The exponent is derived from the exponent of fast matrix multiplication. A variant of our algorithm computes shortest paths of maximum bottleneck weight. Let d(u,v) denote the (unweighted) distance from u to v, and let sc(u,v) denote the maximum bottleneck weight of a path from u to v having length d(u,v). The All-Pairs Bottleneck Shortest Paths (APBSP) problem is to compute sc(u,v) for all ordered pairs of vertices. We present an algorithm for APBSP whose running time is O(n ^{2.86}).

Original language | English |
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Pages (from-to) | 621-633 |

Number of pages | 13 |

Journal | Algorithmica |

Volume | 59 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2011 |

## Keywords

- Bottleneck paths
- Directed weighted graphs
- Shortest paths

## ASJC Scopus subject areas

- Computer Science (all)
- Computer Science Applications
- Applied Mathematics