## Abstract

We study two fundamental problems dealing with curves in the plane, namely, the nearest-neighbor problem and the center problem. Let C be a set of n polygonal curves, each of size m. In the nearest-neighbor problem, the goal is to construct a compact data structure over C, such that, given a query curve Q, one can efficiently find the curve in C closest to Q. In the center problem, the goal is to find a curve Q, such that the maximum distance between Q and the curves in C is minimized. We use the well-known discrete Fréchet distance function, both under L_{∞} and under L_{2}, to measure the distance between two curves. For the nearest-neighbor problem, despite discouraging previous results, we identify two important cases for which it is possible to obtain practical bounds, even when m and n are large. In these cases, either Q is a line segment or C consists of line segments, and the bounds on the size of the data structure and query time are nearly linear in the size of the input and query curve, respectively. The returned answer is either exact under L_{∞}, or approximated to within a factor of 1 + ε under L_{2}. We also consider the variants in which the location of the input curves is only fixed up to translation, and obtain similar bounds, under L_{∞}. As for the center problem, we study the case where the center is a line segment, i.e., we seek the line segment that represents the given set as well as possible. We present near-linear time exact algorithms under L_{∞}, even when the location of the input curves is only fixed up to translation. Under L_{2}, we present a roughly O(n^{2}m^{3}) -time exact algorithm.

Original language | English |
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Title of host publication | Algorithms and Data Structures - 16th International Symposium, WADS 2019, Proceedings |

Editors | Zachary Friggstad, Mohammad R. Salavatipour, Jörg-Rüdiger Sack |

Publisher | Springer Verlag |

Pages | 28-42 |

Number of pages | 15 |

ISBN (Print) | 9783030247652 |

DOIs | |

State | Published - 2019 |

Externally published | Yes |

Event | 16th International Symposium on Algorithms and Data Structures, WADS 2019 - Edmonton, Canada Duration: 5 Aug 2019 → 7 Aug 2019 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11646 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 16th International Symposium on Algorithms and Data Structures, WADS 2019 |
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Country/Territory | Canada |

City | Edmonton |

Period | 5/08/19 → 7/08/19 |

### Bibliographical note

Funding Information:B. Aronov was supported by NSF grants CCF-12-18791 and CCF-15-40656, and by grant 2014/170 from the US-Israel Binational Science Foundation. O. Filtser was supported by the Israeli Ministry of Science, Technology & Space, and by grant 2014/170 from the US-Israel Binational Science Foundation. Most of the work on this project by M. Horton was performed while visiting the Department of Computer Science and Engineering at the Tandon School of Engineering, New York University in the spring/summer of 2018, partially supported by NSF grant CCF-12-18791. M. Katz was supported by grant 1884/16 from the Israel Science Foundation and by grant 2014/170 from the US-Israel Binational Science Foundation. Part of the work on this project by M. Katz was performed while visiting the Department of Computer Science and Engineering at the Tandon School of Engineering, New York University in the spring of 2018, partially supported by NSF grants CCF-12-18791 and CCF-15-40656. Work of K. Sheikhan on this paper was performed while at the Tandon School of Engineering, New York University, supported by NSF grant CCF-12-18791.

Funding Information:

Acknowledgements. B. Aronov was supported by NSF grants CCF-12-18791 and CCF-15-40656, and by grant 2014/170 from the US-Israel Binational Science Foundation. O. Filtser was supported by the Israeli Ministry of Science, Technology & Space, and by grant 2014/170 from the US-Israel Binational Science Foundation. Most of the work on this project by M. Horton was performed while visiting the Department of Computer Science and Engineering at the Tandon School of Engineering, New York University in the spring/summer of 2018, partially supported by NSF grant CCF-12-18791. M. Katz was supported by grant 1884/16 from the Israel Science Foundation and by grant 2014/170 from the US-Israel Binational Science Foundation. Part of the work on this project by M. Katz was performed while visiting the Department of Computer Science and Engineering at the Tandon School of Engineering, New York University in the spring of 2018, partially supported by NSF grants CCF-12-18791 and CCF-15-40656. Work of K. Sheikhan on this paper was performed while at the Tandon School of Engineering, New York University, supported by NSF grant CCF-12-18791.

Publisher Copyright:

© Springer Nature Switzerland AG 2019.

## Keywords

- (Approximation) algorithms
- Clustering
- Data structures
- Fréchet distance
- Nearest-neighbor queries
- Polygonal curves