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Robust Classification of Dynamic Bichromatic Point Sets in R²

Authors Erwin Glazenburg , Marc van Kreveld , Frank Staals

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Author Details

Erwin Glazenburg
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Marc van Kreveld
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Frank Staals
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands

Funding

  • Glazenburg, Erwin: Supported by the Dutch Research Council (NWO) under project OCENW.M20.135.

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Erwin Glazenburg, Marc van Kreveld, and Frank Staals. Robust Classification of Dynamic Bichromatic Point Sets in R². In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.34

Abstract

Let R ∪ B be a set of n points in R², and let k ∈ 1..n. Our goal is to compute a line that "best" separates the "red" points R from the "blue" points B with at most k outliers. We present an efficient semi-online dynamic data structure that can maintain whether such a separator exists ("semi-online" meaning that when a point is inserted, we know when it will be deleted). Furthermore, we present efficient exact and approximation algorithms that compute a linear separator that is guaranteed to misclassify at most k, points and minimizes the distance to the farthest outlier. Our exact algorithm runs in O(nk + n log n) time, and our (1+ε)-approximation algorithm runs in O(ε^(-1/2)((n + k²) log n)) time. Based on our (1+ε)-approximation algorithm we then also obtain a semi-online data structure to maintain such a separator efficiently.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • classification
  • duality
  • data structures
  • dynamic
  • linear programming

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