TY - JOUR

T1 - On the Minimum Consistent Subset Problem

AU - Biniaz, Ahmad

AU - Cabello, Sergio

AU - Carmi, Paz

AU - De Carufel, Jean Lou

AU - Maheshwari, Anil

AU - Mehrabi, Saeed

AU - Smid, Michiel

N1 - Funding Information:
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. We are grateful to the organizers and to the participants of this workshop. We are also grateful to Otfried Cheong for helpful comments. We thank anonymous reviewers whose comments improved the readability of the paper. In particular the open problems mentioned in Sect. 7 are borrowed from these comments. 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.
Funding Information:
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. We are grateful to the organizers and to the participants of this workshop. We are also grateful to Otfried Cheong for helpful comments. We thank anonymous reviewers whose comments improved the readability of the paper. In particular the open problems mentioned in Sect. are borrowed from these comments. 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.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2021/7/1

Y1 - 2021/7/1

N2 - Let P be a set of n colored points in the d-dimensional Euclidean space. Introduced by Hart (1968), a consistent subset of P, is a set S⊆ P such that for every point p in P\ 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 for the consistent subset problem in the plane: (1) The first subexponential-time algorithm for the consistent subset problem. (2) An O(nlog n) -time algorithm that finds a consistent subset of size two in two-colored point sets (if such a subset exists). Along the way we prove the following result which is of an independent interest: given n translations of a cone (defined as the intersection of n halfspaces) and n points in R3, in O(nlog n) time one can decide whether or not there is a point in a cone. (3) An O(nlog 2n) -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(n2) running time which is due to Wilfong (SoCG 1991). (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(n2) running time. (5) A non-trivial O(n6) -time dynamic programming algorithm for the consistent subset problem on points arranged on two parallel lines. To obtain these results, we combine tools from planar separators, paraboloid lifting, 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.

AB - Let P be a set of n colored points in the d-dimensional Euclidean space. Introduced by Hart (1968), a consistent subset of P, is a set S⊆ P such that for every point p in P\ 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 for the consistent subset problem in the plane: (1) The first subexponential-time algorithm for the consistent subset problem. (2) An O(nlog n) -time algorithm that finds a consistent subset of size two in two-colored point sets (if such a subset exists). Along the way we prove the following result which is of an independent interest: given n translations of a cone (defined as the intersection of n halfspaces) and n points in R3, in O(nlog n) time one can decide whether or not there is a point in a cone. (3) An O(nlog 2n) -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(n2) running time which is due to Wilfong (SoCG 1991). (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(n2) running time. (5) A non-trivial O(n6) -time dynamic programming algorithm for the consistent subset problem on points arranged on two parallel lines. To obtain these results, we combine tools from planar separators, paraboloid lifting, 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.

KW - Additively-weighted Voronoi diagrams

KW - Consistent subset

KW - Farthest-point Voronoi diagrams

KW - Paraboloid lifting

KW - Planar separators

KW - Point location

KW - Range trees

UR - http://www.scopus.com/inward/record.url?scp=85103874547&partnerID=8YFLogxK

U2 - 10.1007/s00453-021-00825-8

DO - 10.1007/s00453-021-00825-8

M3 - Article

AN - SCOPUS:85103874547

SN - 0178-4617

VL - 83

SP - 2273

EP - 2302

JO - Algorithmica

JF - Algorithmica

IS - 7

ER -