## Samenvatting

We consider the problem of computing the Fréchet distance between two curves for which the exact locations of the vertices are unknown. Each vertex may be placed in a given uncertainty region for that vertex, and the objective is to place vertices so as to minimise the Fréchet distance. This problem was recently shown to be NP-hard in 2D, and it is unclear how to compute an optimal vertex placement at all.

We give a polynomial-time algorithm for 1D curves with intervals as uncertainty regions. In contrast, we show that the problem is NP-hard in 1D in the case that vertices are placed to maximise the Fréchet distance.

We also study the weak Fréchet distance between uncertain curves. While finding the optimal placement of vertices seems more difficult than for the regular Fréchet distance—and indeed we can easily prove that the problem is NP-hard in 2D—the optimal placement of vertices in 1D can be computed in polynomial time. Finally, we investigate the discrete weak Fréchet distance, for which, somewhat surprisingly, the problem is NP-hard already in 1D.

We give a polynomial-time algorithm for 1D curves with intervals as uncertainty regions. In contrast, we show that the problem is NP-hard in 1D in the case that vertices are placed to maximise the Fréchet distance.

We also study the weak Fréchet distance between uncertain curves. While finding the optimal placement of vertices seems more difficult than for the regular Fréchet distance—and indeed we can easily prove that the problem is NP-hard in 2D—the optimal placement of vertices in 1D can be computed in polynomial time. Finally, we investigate the discrete weak Fréchet distance, for which, somewhat surprisingly, the problem is NP-hard already in 1D.

Originele taal-2 | Engels |
---|---|

Titel | Algorithms and Data Structures |

Subtitel | 17th International Symposium, WADS 2021, Virtual Event, August 9–11, 2021, Proceedings |

Redacteuren | Anna Lubiw, Mohammad Salavatipour |

Plaats van productie | Cham, Switzerland |

Uitgeverij | Springer Nature |

Pagina's | 243–257 |

Aantal pagina's | 15 |

ISBN van elektronische versie | 978-3-030-83508-8 |

ISBN van geprinte versie | 978-3-030-83507-1 |

DOI's | |

Status | Gepubliceerd - 11 aug 2021 |

Evenement | 17th Algorithms and Data Structures Symposium - Online, Halifax, Canada Duur: 9 aug 2021 → 11 aug 2021 Congresnummer: 17 https://projects.cs.dal.ca/wads2021/ |

### Publicatie series

Naam | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 12808 LNCS |

ISSN van geprinte versie | 0302-9743 |

ISSN van elektronische versie | 1611-3349 |

### Congres

Congres | 17th Algorithms and Data Structures Symposium |
---|---|

Verkorte titel | WADS 2021 |

Land/Regio | Canada |

Stad | Halifax |

Periode | 9/08/21 → 11/08/21 |

Internet adres |