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Bichromatic Classifications of Points Using Strips

Authors Jaegun Lee , Chaeyoon Chung , Hee-Kap Ahn

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LIPIcs.SWAT.2026.29.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Bichromatic Classification
  • Separation
  • Strip
  • Duality

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Abstract

Given a set of n points in the plane, each colored either blue or red, we study the problem of finding a strip that separates the blue points from the red points. Specifically, we consider the following two variants: (1) locating a strip that contains no red points while maximizing the number of blue points within the strip, and (2) locating a strip that contains all blue points while minimizing the number of red points within the strip. For variant (1), we present an O(n²)-time algorithm, improving upon the previously best O(n²log n)-time result. We also show that this running time is optimal under the standard 3SUM conjecture. We also give an output-sensitive algorithm with running time O(k_{opt} n log n) that returns a strip, where k_{opt} is the number of blue points not contained within the strip in an optimal solution. We extend our results to the case of up to t parallel strips, obtaining an O(n²log n)-time algorithm. For variant (2), an optimal Θ(nlog n)-time algorithm is known for t = 1. We show 3SUM-hardness for t = 2 and give an O(n²)-time algorithm. For any t ≥ 3, we present an O(n²log n)-time algorithm.

Cite As Get BibTex

Jaegun Lee, Chaeyoon Chung, and Hee-Kap Ahn. Bichromatic Classifications of Points Using Strips. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SWAT.2026.29

Author Details

Jaegun Lee
  • Department of Computer Science and Engineering, Pohang University of Science and Technology, Republic of Korea
Chaeyoon Chung
  • Department of Computer Science and Engineering, Pohang University of Science and Technology, Republic of Korea
Hee-Kap Ahn
  • Department of Computer Science and Engineering, Graduate School of Artificial Intelligence, Pohang University of Science and Technology, Republic of Korea

Funding

This work was partly supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2023-00219980), and the Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No. RS-2019-II191906, Artificial Intelligence Graduate School Program (POSTECH)) and (No. 2017-0-00905, Software Star Lab (Optimal Data Structure and Algorithmic Applications in Dynamic Geometric Environment)).

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