## Abstract

The Fréchet distance is a well studied similarity measure between curves. The discrete Fréchet distance is an analogous similarity measure, defined for two sequences of m and n points, where the points are usually sampled from input curves. We consider a variant, called the discrete Fréchet distance with shortcuts, which captures the similarity between (sampled) curves in the presence of outliers. When shortcuts are allowed only in one noise-containing curve, we give a randomized algorithm that runs in O((m+n)6/5+ϵ) expected time, for any ϵ > 0. When shortcuts are allowed in both curves, we give an O((m2/3n2/3 + m + n) log3(m + n))-time deterministic algorithm.

We also consider the semi-continuous Fréchet distance with one-sided shortcuts, where we have a sequence of m points and a polygonal curve of n edges, and shortcuts are allowed only in the sequence. We show that this problem can be solved in randomized expected time O((m + n)2/3 m2/3n1/3 log(m + n)).

Our techniques are novel and may find further applications. One of the main new technical results is: Given two sets of points A and B in the plane and an interval I, we develop an algorithm that decides whether the number of pairs (x, y) ∈ A × B whose distance dist(x, y) is in I, is less than some given threshold L. The running time of this algorithm decreases as L increases. In case there are more than L pairs of points whose distance is in I, we can get a small sample of pairs that contains a pair at approximate median distance (i.e., we can approximately "bisect" I). We combine this procedure with additional ideas to search, with a small overhead, for the optimal one-sided Fréchet distance with shortcuts, using a very fast decision procedure. We also show how to apply this technique for approximate distance selection (with respect to rank), and a somewhat more involved variant of this technique is used in the solution of the semicontinuous Fréchet distance with one-sided shortcuts. In general, the new technique can apply to optimization problems for which the decision procedure is very fast but standard techniques like parametric search make the optimization algorithm substantially slower.

We also consider the semi-continuous Fréchet distance with one-sided shortcuts, where we have a sequence of m points and a polygonal curve of n edges, and shortcuts are allowed only in the sequence. We show that this problem can be solved in randomized expected time O((m + n)2/3 m2/3n1/3 log(m + n)).

Our techniques are novel and may find further applications. One of the main new technical results is: Given two sets of points A and B in the plane and an interval I, we develop an algorithm that decides whether the number of pairs (x, y) ∈ A × B whose distance dist(x, y) is in I, is less than some given threshold L. The running time of this algorithm decreases as L increases. In case there are more than L pairs of points whose distance is in I, we can get a small sample of pairs that contains a pair at approximate median distance (i.e., we can approximately "bisect" I). We combine this procedure with additional ideas to search, with a small overhead, for the optimal one-sided Fréchet distance with shortcuts, using a very fast decision procedure. We also show how to apply this technique for approximate distance selection (with respect to rank), and a somewhat more involved variant of this technique is used in the solution of the semicontinuous Fréchet distance with one-sided shortcuts. In general, the new technique can apply to optimization problems for which the decision procedure is very fast but standard techniques like parametric search make the optimization algorithm substantially slower.

Original language | English |
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Title of host publication | Proceedings of the thirtieth annual symposium on Computational geometry (SOCG'14) |

Place of Publication | New York, NY |

Publisher | Association for Computing Machinery (ACM) |

Pages | 377-386 |

Number of pages | 10 |

ISBN (Electronic) | 9781450325943 |

ISBN (Print) | 9781450325943 |

DOIs | |

State | Published - 8 Jun 2014 |

Externally published | Yes |

Event | 30th Annual Symposium on Computational Geometry, SoCG 2014 - Kyoto, Japan Duration: 8 Jun 2014 → 11 Jun 2014 |

### Publication series

Name | Proceedings of the Annual Symposium on Computational Geometry |
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### Conference

Conference | 30th Annual Symposium on Computational Geometry, SoCG 2014 |
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Country/Territory | Japan |

City | Kyoto |

Period | 8/06/14 → 11/06/14 |

### Bibliographical note

DBLP's bibliographic metadata records provided through http://dblp.org/search/publ/api are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.## Keywords

- Approximate distance selection and counting
- Curve matching
- Discrete Fréchet distance
- Geometric optimization
- Outliers
- Shortcuts