New results on the coarseness of bicolored point sets

Let $S$ be a 2-colored (red and blue) set of $n$ points in the plane. A subset $I$ of $S$ is an island if there exits a convex set $C$ such that $I=C\cap S$. The discrepancy of an island is the absolute value of the number of red minus the number of blue points it contains. A convex partition of $S$ is a partition of $S$ into islands with pairwise disjoint convex hulls. The discrepancy of a convex partition is the discrepancy of its island of minimum discrepancy. The coarseness of $S$ is the discrepancy of the convex partition of $S$ with maximum discrepancy. This concept was recently defined by Bereg et al. [CGTA 2013]. In this paper we study the following problem: Given a set $S$ of $n$ points in general position in the plane, how to color each of them (red or blue) such that the resulting 2-colored point set has small coarseness? We prove that every $n$-point set $S$ can be colored such that its coarseness is $O(n^{1/4}\sqrt{\log n})$. This bound is almost tight since there exist $n$-point sets such that every 2-coloring gives coarseness at least $\Omega(n^{1/4})$. Additionally, we show that there exists an approximation algorithm for computing the coarseness of a 2-colored point set, whose ratio is between $1/128$ and $1/64$, solving an open problem posted by Bereg et al. [CGTA 2013]. All our results consider $k$-separable islands of $S$, for some $k$, which are those resulting from intersecting $S$ with at most $k$ halfplanes.Comment: Presented at the Mexican Conference on Discrete Mathematics and Computational Geometry 2013, Oaxaca, Mexic

On the Coarseness of Bicolored Point Sets

Let R be a set of red points and B a set of blue points on the plane. In this paper we introduce a new concept, which we call coarseness, for measuring how blended the elements of S = R ∪ B are. For X ⊆ S, let ∇(X) = ||X ∩ R | − |X ∩ B| | be the bichromatic discrepancy of X. We say that a partition Π = {S1, S2,...,Sk} of S is convex if the convex hulls of its members are pairwise disjoint. The discrepancy of a convex partition Π of S is the minimum ∇(Si) over the elements of Π. The coarseness of S is the discrepancy of the convex partition of S with maximum discrepancy. We study the coarseness of bicolored point sets, and relate it to well blended point sets. In particular, we show combinatorial results on the coarseness of general configurations and give efficient algorithms for computing the coarseness of two specific cases, namely when the set of points is in convex position and when the measure is restricted to convex partitions with two elements.

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On the coarseness of bicolored point sets. (English summary) Comput. Geom. 46 (2013), no. 1, 65–77. Summary: “Let R be a set of red points and B a set of blue points on the plane. In this paper we introduce a new concept, which we call coarseness, for measuring how blended the elements of S = R ∪ B are. For X ⊆ S, let ∇(X) = ||X ∩ R | − |X ∩ B| | be the bichromatic discrepancy of X. We say that a partition Π = {S1, S2,..., Sk} of S is convex if the convex hulls of its members are pairwise disjoint. The discrepancy of a convex partition Π of S is the minimum ∇(Si) over the elements of Π. The coarseness of S is the discrepancy of the convex partition of S with maximum discrepancy. We study the coarseness of bicolored point sets, and relate it to well blended point sets. In particular, we show combinatorial results on the coarseness of general configurations and give efficient algorithms for computing the coarseness of two specific cases, namely when the set of points is in convex position and when the measure is restricted to convex partitions with tw

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On the coarseness of bicolored point sets. (English summary) Comput. Geom. 46 (2013), no. 1, 65–77. Summary: “Let R be a set of red points and B a set of blue points on the plane. In this paper we introduce a new concept, which we call coarseness, for measuring how blended the elements of S = R ∪ B are. For X ⊆ S, let ∇(X) = ||X ∩ R | − |X ∩ B| | be the bichromatic discrepancy of X. We say that a partition Π = {S1, S2,..., Sk} of S is convex if the convex hulls of its members are pairwise disjoint. The discrepancy of a convex partition Π of S is the minimum ∇(Si) over the elements of Π. The coarseness of S is the discrepancy of the convex partition of S with maximum discrepancy. We study the coarseness of bicolored point sets, and relate it to well blended point sets. In particular, we show combinatorial results on the coarseness of general configurations and give efficient algorithms for computing the coarseness of two specific cases, namely when the set of points is in convex position and when the measure is restricted to convex partitions with tw

Previous Up Next Article Citations From References: 1 From Reviews: 0

On the coarseness of bicolored point sets. (English summary) Comput. Geom. 46 (2013), no. 1, 65–77. Summary: “Let R be a set of red points and B a set of blue points on the plane. In this paper we introduce a new concept, which we call coarseness, for measuring how blended the elements of S = R ∪ B are. For X ⊆ S, let ∇(X) = ||X ∩ R | − |X ∩ B| | be the bichromatic discrepancy of X. We say that a partition Π = {S1, S2,..., Sk} of S is convex if the convex hulls of its members are pairwise disjoint. The discrepancy of a convex partition Π of S is the minimum ∇(Si) over the elements of Π. The coarseness of S is the discrepancy of the convex partition of S with maximum discrepancy. We study the coarseness of bicolored point sets, and relate it to well blended point sets. In particular, we show combinatorial results on the coarseness of general configurations and give efficient algorithms for computing the coarseness of two specific cases, namely when the set of points is in convex position and when the measure is restricted to convex partitions with tw