## 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^{′}∈ C with δ(Q, C^{′}) ≤ (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(1ε)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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Pages (from-to) | 1490-1519 |

Number of pages | 30 |

Journal | Algorithmica |

Volume | 85 |

Issue number | 5 |

DOIs | |

State | Published - 2023 |

### Bibliographical note

Funding Information:Arnold Filtser was partially supported by Grant 1042/22 from the Israel Science Foundation. Omrit Filtser was 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 was partially supported by Grant 1884/16 from the Israel Science Foundation.

Publisher Copyright:

© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

## Keywords

- Dynamic time warping
- Fréchet distance
- Nearest neighbor search
- Polygonal curves