## Abstract

We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algorithm for this problem that returns in time 2^{O(d2)}m^{2}n ^{2} log^{2}(mn) the Fréchet distance lower bound between two imprecise polygonal curves with n and m vertices, respectively. We give an improved algorithm for the planar case with running time O(mn log ^{2}(mn) + (m^{2} + n^{2})log(mn)). In the d-dimensional orthogonal case, where points are modelled as axis-parallel boxes, and we use the L_{∞} distance, we give an O(dmn log(dmn))-time algorithm. We also give efficient O(dmn)-time algorithms to approximate the Fréchet distance upper bound, as well as the smallest possible Fréchet distance lower/upper bound that can be achieved between two imprecise point sequences when one is allowed to translate them. These algorithms achieve constant factor approximation ratios in "realistic" settings (such as when the radii of the balls modelling the imprecise points are roughly of the same size).

Original language | English (US) |
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Title of host publication | Algorithms and Computation - 21st International Symposium, ISAAC 2010, Proceedings |

Pages | 422-433 |

Number of pages | 12 |

Edition | PART 2 |

DOIs | |

State | Published - 2010 |

Event | 21st Annual International Symposium on Algorithms and Computations, ISAAC 2010 - Jeju Island, Korea, Republic of Duration: Dec 15 2010 → Dec 17 2010 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Number | PART 2 |

Volume | 6507 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 21st Annual International Symposium on Algorithms and Computations, ISAAC 2010 |
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Country/Territory | Korea, Republic of |

City | Jeju Island |

Period | 12/15/10 → 12/17/10 |

### Bibliographical note

Funding Information:Work by Ahn was supported by the Korea Research Foundation Grant funded by the Korean Government(KRF-2008-614-D00008). Work by Knauer and Scher-fenberg was supported by the German Science Foundation (DFG) under grant Al 253/5-3. Work by Schlipf was supported by the Deutsche Forschungsgemein-schaft within the research training group ’Methods for Discrete Structures’(GRK 1408).

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science