## Abstract

In the (1 + ε, r)-approximate near-neighbor problem for curves (ANNC) under some similarity measure δ, the goal is to construct a data structure for a given set C of curves that supports approximate near-neighbor queries: Given a query curve Q, if there exists a curve C ∈ C such that δ(Q, C) ≤ r, then return a curve C^{0} ∈ C with δ(Q, C^{0}) ≤ (1 + ε)r. There exists an efficient reduction from the (1 + ε)-approximate nearest-neighbor problem to ANNC, where in the former problem the answer to a query is a curve C ∈ C with δ(Q, C) ≤ (1 + ε) · δ(Q, C^{∗}), where C^{∗} is the curve of C most similar to Q. Given a set C of n curves, each consisting of m points in d dimensions, we construct a data structure for ANNC that uses n · O(^{1}_{ε} )^{md} storage space and has O(md) query time (for a query curve of length m), where the similarity measure between two curves is their discrete Fréchet or dynamic time warping distance. Our method is simple to implement, deterministic, and results in an exponential improvement in both query time and storage space compared to all previous bounds. Further, we also consider the asymmetric version of ANNC, where the length of the query curves is k m, and obtain essentially the same storage and query bounds as above, except that m is replaced by k. Finally, we apply our method to a version of approximate range counting for curves and achieve similar bounds.

Original language | English |
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Title of host publication | 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020 |

Editors | Artur Czumaj, Anuj Dawar, Emanuela Merelli |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771382 |

DOIs | |

State | Published - 1 Jun 2020 |

Externally published | Yes |

Event | 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020 - Virtual, Online, Germany Duration: 8 Jul 2020 → 11 Jul 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 168 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020 |
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Country/Territory | Germany |

City | Virtual, Online |

Period | 8/07/20 → 11/07/20 |

### Bibliographical note

Funding Information:Funding Arnold Filtser: Supported by the Simons Foundation. Omrit Filtser: Supported by the Eric and Wendy Schmidt Fund for Strategic Innovation, by the Council for Higher Education of Israel, and by Ben-Gurion University of the Negev. Matthew J. Katz: Supported by grant 1884/16 from the Israel Science Foundation.

Publisher Copyright:

© Arnold Filtser, Omrit Filtser, and Matthew J. Katz; licensed under Creative Commons License CC-BY 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020).

## Keywords

- (asymmetric) approximate nearest neighbor
- Approximation algorithms
- Dynamic time warping
- Fréchet distance
- Polygonal curves
- Range counting