### Abstract

In planning a flight, stops at intermediate airports are sometimes necessary to minimize fuel consumption, even if a direct flight is available. We investigate the problem of finding the cheapest path from one airport to another, given a set of n airports in ℝ^{2} and a function l: ℝ^{2} × ℝ^{2} → ℝ^{+} representing the cost of a direct flight between any pair. Given a source airport s, the cheapest-path map is a subdivision of ℝ^{2} where two points lie in the same region iff their cheapest paths from s use the same sequence of intermediate airports. We show a quadratic lower bound on the combinatorial complexity of this map for a class of cost functions. Nevertheless, we are able to obtain subquadratic algorithms to find the cheapest path from s to all other airports for any well-behaved cost function l: our general algorithm runs in O(n^{4/3+ε}) time, and a simpler, more practical variant runs in O(n^{3/2+ε}) time, while a special class of cost functions requires just O(n log n) time.

Original language | English (US) |
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Pages (from-to) | 330-337 |

Number of pages | 8 |

Journal | Journal of Algorithms |

Volume | 41 |

Issue number | 2 |

DOIs | |

State | Published - Nov 2001 |

### ASJC Scopus subject areas

- Control and Optimization
- Computational Mathematics
- Computational Theory and Mathematics

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## Cite this

*Journal of Algorithms*,

*41*(2), 330-337. https://doi.org/10.1006/jagm.2001.1189