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DOI: 10.4230/LIPIcs.SoCG.2024.63

Faster Fréchet Distance Approximation Through Truncated Smoothing

Authors: Thijs van der Horst and Tim Ophelders

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract

The Fréchet distance is a popular distance measure for curves. Computing the Fréchet distance between two polygonal curves of n vertices takes roughly quadratic time, and conditional lower bounds suggest that even approximating to within a factor 3 cannot be done in strongly-subquadratic time, even in one dimension. The current best approximation algorithms present trade-offs between approximation quality and running time. Recently, van der Horst et al. (SODA, 2023) presented an O((n²/α) log³ n) time α-approximate algorithm for curves in arbitrary dimensions, for any α ∈ [1, n]. Our main contribution is an approximation algorithm for curves in one dimension, with a significantly faster running time of O(n log³ n + (n²/α³) log²n log log n). Additionally, we give an algorithm for curves in arbitrary dimensions that improves upon the state-of-the-art running time by a logarithmic factor, to O((n²/α) log² n). Both of our algorithms rely on a linear-time simplification procedure that in one dimension reduces the complexity of the reachable free space to O(n²/α) without making sacrifices in the asymptotic approximation factor.

Cite as

Thijs van der Horst and Tim Ophelders. Faster Fréchet Distance Approximation Through Truncated Smoothing. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 63:1-63:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{vanderhorst_et_al:LIPIcs.SoCG.2024.63,
  author =	{van der Horst, Thijs and Ophelders, Tim},
  title =	{{Faster Fr\'{e}chet Distance Approximation Through Truncated Smoothing}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{63:1--63:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.63},
  URN =		{urn:nbn:de:0030-drops-200083},
  doi =		{10.4230/LIPIcs.SoCG.2024.63},
  annote =	{Keywords: Fr\'{e}cht distance, approximation algorithms, simplification}
}

Document

DOI: 10.4230/LIPIcs.ESA.2022.67

Chromatic k-Nearest Neighbor Queries

Authors: Thijs van der Horst, Maarten Löffler, and Frank Staals

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract

Let P be a set of n colored points. We develop efficient data structures that store P and can answer chromatic k-nearest neighbor (k-NN) queries. Such a query consists of a query point q and a number k, and asks for the color that appears most frequently among the k points in P closest to q. Answering such queries efficiently is the key to obtain fast k-NN classifiers. Our main aim is to obtain query times that are independent of k while using near-linear space. We show that this is possible using a combination of two data structures. The first data structure allow us to compute a region containing exactly the k-nearest neighbors of a query point q, and the second data structure can then report the most frequent color in such a region. This leads to linear space data structures with query times of O(n^{1/2} log n) for points in ℝ¹, and with query times varying between O(n^{2/3}log^{2/3} n) and O(n^{5/6} polylog n), depending on the distance measure used, for points in ℝ². These results can be extended to work in higher dimensions as well. Since the query times are still fairly large we also consider approximations. If we are allowed to report a color that appears at least (1-ε)f^* times, where f^* is the frequency of the most frequent color, we obtain a query time of O(log n + log log_{1/(1-ε)} n) in ℝ¹ and expected query times ranging between Õ(n^{1/2}ε^{-3/2}) and Õ(n^{1/2}ε^{-5/2}) in ℝ² using near-linear space (ignoring polylogarithmic factors).

Cite as

Thijs van der Horst, Maarten Löffler, and Frank Staals. Chromatic k-Nearest Neighbor Queries. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 67:1-67:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{vanderhorst_et_al:LIPIcs.ESA.2022.67,
  author =	{van der Horst, Thijs and L\"{o}ffler, Maarten and Staals, Frank},
  title =	{{Chromatic k-Nearest Neighbor Queries}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{67:1--67:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.67},
  URN =		{urn:nbn:de:0030-drops-170055},
  doi =		{10.4230/LIPIcs.ESA.2022.67},
  annote =	{Keywords: data structure, nearest neighbor, classification}
}

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Documents authored by van der Horst, Thijs

Faster Fréchet Distance Approximation Through Truncated Smoothing

Abstract

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Chromatic k-Nearest Neighbor Queries

Abstract

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