## Abstract

We present an algorithm that finds, for each vertex of an undirected graph, a shortest cycle containing it. While for directed graphs this problem reduces to the All-Pairs Shortest Paths problem, this is not known to be the case for undirected graphs. We present a truly sub-cubic randomized algorithm for the undirected case. Given an undirected graph with n vertices and integer weights in 1,...,M, it runs in Õ(Mn(^{ω+3)/2}) time where ω<2.376 is the exponent of matrix multiplication. As a by-product, our algorithm can be used to determine which vertices lie on cycles of length at most t in Õ(M^{nω}t) time. For the case of bounded real edge weights, a variant of our algorithm solves the problem up to an additive error of ∈ in Õ(n(^{ω+6)/3}) time.

Original language | American English |
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Pages (from-to) | 1057-1061 |

Number of pages | 5 |

Journal | Information Processing Letters |

Volume | 111 |

Issue number | 21-22 |

DOIs | |

State | Published - 15 Dec 2011 |

## Keywords

- Girth
- Graph algorithms
- Shortest path
- Undirected graph

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Signal Processing
- Information Systems
- Computer Science Applications