## Abstract

Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe a randomized algorithm to preprocess the graph in O(gn log n) time with high probability, so that the shortest-path distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)-time algorithm of Klein [Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, 2005] for multiple-source shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortest-path tree as its source point moves continuously around the boundary of f. As an application of our algorithm, we describe algorithms to compute a shortest noncontractible or nonseparating cycle in embedded, undirected graphs in O(g^{2}n log n) time with high probability. Our high-probability time bounds hold in the worst case for generic edge weights or with an additional O(log n) factor for arbitrary edge weights.

Original language | English (US) |
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Pages (from-to) | 1542-1571 |

Number of pages | 30 |

Journal | SIAM Journal on Computing |

Volume | 42 |

Issue number | 4 |

DOIs | |

State | Published - 2013 |

## Keywords

- Computational topology
- Dynamic data structures
- Parametric shortest paths
- Topological graph theory

## ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)