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On the Minimum Consistent Subset Problem

  • Ahmad BiniazEmail author
  • Sergio Cabello
  • Paz Carmi
  • Jean-Lou De Carufel
  • Anil Maheshwari
  • Saeed Mehrabi
  • Michiel Smid
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11646)

Abstract

Let P be a set of n colored points in the plane. Introduced by Hart [7], a consistent subset of P, is a set \(S\subseteq P\) such that for every point p in \(P\setminus S\), the closest point of p in S has the same color as p. The consistent subset problem is to find a consistent subset of P with minimum cardinality. This problem is known to be NP-complete even for two-colored point sets. Since the initial presentation of this problem, aside from the hardness results, there has not been significant progress from the algorithmic point of view. In this paper we present the following algorithmic results:
  1. 1.

    The first subexponential-time algorithm for the consistent subset problem.

     
  2. 2.

    An \(O(n\log n)\)-time algorithm that finds a consistent subset of size two in two-colored point sets (if such a subset exists). Towards our proof of this running time we present a deterministic \(O(n \log n)\)-time algorithm for computing a variant of the compact Voronoi diagram; this improves the previously claimed expected running time.

     
  3. 3.

    An \(O(n\log ^2 n)\)-time algorithm that finds a minimum consistent subset in two-colored point sets where one color class contains exactly one point; this improves the previous best known \(O(n^2)\) running time which is due to Wilfong (SoCG 1991).

     
  4. 4.

    An O(n)-time algorithm for the consistent subset problem on collinear points that are given from left to right; this improves the previous best known \(O(n^2)\) running time.

     
  5. 5.

    A non-trivial \(O(n^6)\)-time dynamic programming algorithm for the consistent subset problem on points arranged on two parallel lines.

     

To obtain these results, we combine tools from paraboloid lifting, planar separators, additively-weighted Voronoi diagrams with respect to convex distance functions, point location in farthest-point Voronoi diagrams, range trees, minimum covering of a circle with arcs, and several geometric transformations.

Keywords

Consistent subset Colored points Planar separator Voronoi diagram Paraboloid lifting Range tree Circle covering 

Notes

Acknowledgement

This work initiated at the Sixth Annual Workshop on Geometry and Graphs, March 11–16, 2018, at the Bellairs Research Institute of McGill University, Barbados. The authors are grateful to the organizers and to the participants of this workshop.

Ahmad Biniaz was supported by NSERC Postdoctoral Fellowship. Sergio Cabello was supported by the Slovenian Research Agency, program P1-0297 and projects J1-8130, J1-8155. Paz Carmi was supported by grant 2016116 from the United States – Israel Binational Science Foundation. Jean-Lou De Carufel, Anil Maheshwari, and Michiel Smid were supported by NSERC. Saeed Mehrabi was supported by NSERC and by Carleton-Fields Postdoctoral Fellowship.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ahmad Biniaz
    • 1
    Email author
  • Sergio Cabello
    • 2
  • Paz Carmi
    • 3
  • Jean-Lou De Carufel
    • 4
  • Anil Maheshwari
    • 5
  • Saeed Mehrabi
    • 5
  • Michiel Smid
    • 5
  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Department of Mathematics, IMFM and FMFUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Department of Computer ScienceBen-Gurion University of the NegevBeer-ShevaIsrael
  4. 4.School of Electrical Engineering and Computer ScienceUniversity of OttawaOttawaCanada
  5. 5.School of Computer ScienceCarleton UniversityOttawaCanada

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