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 \(S\subseteq P\) 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 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 \(\mathbb {R}^3\), 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.

令P为d维欧几里得空间中n个有色点的集合。哈特（1968），一个引入一致子集的P，是一组\（S \ subseteq P \） ，使得对于每一个点p在\（P {\ setminus}Š\） ，的最近点p在小号具有与p相同的颜色。一致子集问题是找到P的一致子集用最小的基数。即使对于两个颜色的点集，此问题也是已知的NP完全问题。自从最初提出这个问题以来，除了硬度结果外，从算法的角度来看还没有取得重大进展。在本文中，我们针对飞机上的一致性子集问题给出以下算法结果：（1）一致性子集问题的第一个次指数时间算法。（2）一种\（O（n（log n）\） -时间算法，该算法在双色点集中找到大小为2的一致子集（如果存在这样的子集）。在此过程中，我们证明了以下具有独立利益的结果：给定一个圆锥体的n个平移（定义为n个半空间的交集）和n个在点\（\ mathbb {R} ^ 3 \） ，在\（O（N \ log n）的\）每次一个可以决定是否不存在锥形的点。（3）一种\（O（n \ log ^ 2 n）\）时间算法，该算法在两种颜色的点集中找到最小一致子集，其中一种颜色类别恰好包含一个点；这缩短了先前由于Wilfong（SoCG 1991）而最出名的\（O（n ^ 2）\）运行时间。（4）O（n）-时间算法，用于从左至右给出的共线点上的一致子集问题；这样可以缩短以前最著名的\（O（n ^ 2）\）运行时间。（5）非平凡\（O（n ^ 6）\）两条平行线上的点上的一致子集问题的实时动态规划算法。为了获得这些结果，我们结合了以下工具：平面分离器，抛物面提升，关于凸距离函数的加重加权Voronoi图，最远Voronoi图中的点位置，范围树，圆弧的最小覆盖度以及几个几何转变。