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Chromatic k-Nearest Neighbor Queries

Authors Thijs van der Horst, Maarten Löffler, Frank Staals

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Author Details

Thijs van der Horst
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Maarten Löffler
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Frank Staals
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands

Funding

  • Löffler, Maarten: Partially supported by the Dutch Research Council (NWO) under the project numbers 614.001.504 and 628.011.005.

Acknowledgements

We would like to thank an anonymous reviewer for the randomized solution presented in Section 3.2.1, which led to our current solution for finding 𝒟^k_2(q) in Section 3.2.2.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.ESA.2022.67

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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • data structure
  • nearest neighbor
  • classification

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