Paper Info
Title
On the Minimum Consistent Subset Problem.
Abstract
Let $P$ be a set of $n$ colored points in the plane. Introduced by Hart (1968), a consistent subset of $P$, is a set $Ssubseteq P$ such that for every point $p$ in $Psetminus S$, 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 a significant progress from the algorithmic point of view. In this paper we present the following algorithmic results: 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). 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. An $O(nlog^2 n)$-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. An $O(n)$-time algorithm for the consistent subset problem on collinear points; this improves the previous best known $O(n^2)$ running time. 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 planar separators, additively-weighted Voronoi diagrams with respect to convex distance functions, point location in farthest-point Voronoi diagrams, range trees, paraboloid lifting, minimum covering of a circle with arcs, and several geometric transformations.
Year
Venue
Field
2018
arXiv: Computational Geometry
Discrete mathematics,Dynamic programming,Binary logarithm,Combinatorics,Point location,Transformation geometry,Cardinality,Regular polygon,Parallel,Voronoi diagram,Mathematics
DocType
Volume
Citations 
Journal
abs/1810.09232
0
PageRank 
References 
Authors
0.34
0
7
Name
Order
Citations
PageRank
Ahmad Biniaz14420.67
Sergio Cabello2334.91
Anil Maheshwari3869104.47
Paz Carmi432143.14
Saeed Mehrabi503.38
Jean-Lou De Carufel67622.63
Michiel H. M. Smid755093.58