On the Minimum Consistent Subset Problem
The first subexponential-time algorithm for the consistent subset problem.
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.
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).
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.
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.
KeywordsConsistent subset Colored points Planar separator Voronoi diagram Paraboloid lifting Range tree Circle covering
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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