## Abstract

We present a data structure for answering approximate shortest path queries in a planar subdivision from a fixed source. Let p ≤ 1 be a real number. Distances in each face of this subdivision are measured by a possibly asymmetric convex distance function whose unit disk is contained in a concentric unit Euclidean disk and contains a concentric Euclidean disk with radius 1/p. Different convex distance functions may be used for different faces, and obstacles are allowed. Let e be any number strictly between 0 and 1. Our data structure returns a (1 + e) approximation of the shortest path cost from the fixed source to a query destination in 0(log) time. Afterwards, a (1 + ε)-approximate shortest path can be reported in 0(log n) time plus the complexity of the path. The data structure uses 0(log)space and can be built in 0((log))^{2} time- Our time and space bounds do not depend on any other parameter; in particular, they do not depend on any geometric parameter of the subdivision such as the minimum angle.

Original language | English (US) |
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Pages (from-to) | 1888-1918 |

Number of pages | 31 |

Journal | SIAM Journal on Computing |

Volume | 39 |

Issue number | 5 |

DOIs | |

State | Published - Feb 25 2010 |

## Keywords

- Anisotropic regions
- Approximation algorithms
- Convex distance functions
- Shortest path

## ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)