### 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) |
---|---|

Pages (from-to) | 330-337 |

Number of pages | 8 |

Journal | Journal of Algorithms |

Volume | 41 |

Issue number | 2 |

DOIs | |

State | Published - Nov 2001 |

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### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

*Journal of Algorithms*,

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

**Fly Cheaply : On the Minimum Fuel Consumption Problem.** / Chan, Timothy M.; Efrat, Alon.

Research output: Contribution to journal › Article

*Journal of Algorithms*, vol. 41, no. 2, pp. 330-337. https://doi.org/10.1006/jagm.2001.1189

}

TY - JOUR

T1 - Fly Cheaply

T2 - On the Minimum Fuel Consumption Problem

AU - Chan, Timothy M.

AU - Efrat, Alon

PY - 2001/11

Y1 - 2001/11

N2 - 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 ℝ2 and a function l: ℝ2 × ℝ2 → ℝ+ representing the cost of a direct flight between any pair. Given a source airport s, the cheapest-path map is a subdivision of ℝ2 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(n4/3+ε) time, and a simpler, more practical variant runs in O(n3/2+ε) time, while a special class of cost functions requires just O(n log n) time.

AB - 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 ℝ2 and a function l: ℝ2 × ℝ2 → ℝ+ representing the cost of a direct flight between any pair. Given a source airport s, the cheapest-path map is a subdivision of ℝ2 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(n4/3+ε) time, and a simpler, more practical variant runs in O(n3/2+ε) time, while a special class of cost functions requires just O(n log n) time.

UR - http://www.scopus.com/inward/record.url?scp=0347156624&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0347156624&partnerID=8YFLogxK

U2 - 10.1006/jagm.2001.1189

DO - 10.1006/jagm.2001.1189

M3 - Article

AN - SCOPUS:0347156624

VL - 41

SP - 330

EP - 337

JO - Journal of Algorithms

JF - Journal of Algorithms

SN - 0196-6774

IS - 2

ER -