On the Minimum Consistent Subset Problem

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 subset of 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(n log 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 R-3, in O(n log n) time one can decide whether or not there is a point in a cone. (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) 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) 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, 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.