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Greedy Monochromatic Island Partitions

Steven van den Broek, Wouter Meulemans, Bettina Speckmann

TL;DR

We study partitioning a set $S$ of $n$ planar points, colored with $k \ge 2$ colors, into the fewest possible monochromatic islands, where an island $I$ satisfies $\mathcal{CH}(I) \cap S = I$ and partitions require pairwise-disjoint convex hulls. The authors introduce three greedy algorithms—disjoint-greedy, overlap-greedy, and line-greedy—and analyze their approximation with respect to $\text{Opt}_{\text{P}}$, the minimum island-partition size. Key results include a lower-bound of $\Omega\left(\frac{n}{\log^2 n}\right)$ for disjoint-greedy, an $O(\log n)$-approximation for overlap-greedy on island covers (with a barrier and a constructive $O\left(\text{Opt}_{\text{P}}^2 \log^2 n\right)$-algorithm to obtain a partition), and an $O\left(\text{Opt}_{\text{P}} \log^2 n\right)$-approximation via line-based separation. The work also connects island partitions to geometric line-separation problems, illuminating the trade-offs between greedy island construction and partition feasibility in geometric settings.

Abstract

Constructing partitions of colored points is a well-studied problem in discrete and computational geometry. We study the problem of creating a minimum-cardinality partition into monochromatic islands. Our input is a set $S$ of $n$ points in the plane where each point has one of $k \geq 2$ colors. A set of points is monochromatic if it contains points of only one color. An island $I$ is a subset of $S$ such that $\mathcal{CH}(I) \cap S = I$, where $\mathcal{CH}(I)$ denotes the convex hull of $I$. We identify an island with its convex hull; therefore, a partition into islands has the additional requirement that the convex hulls of the islands are pairwise-disjoint. We present three greedy algorithms for constructing island partitions and analyze their approximation ratios.

Greedy Monochromatic Island Partitions

TL;DR

We study partitioning a set of planar points, colored with colors, into the fewest possible monochromatic islands, where an island satisfies and partitions require pairwise-disjoint convex hulls. The authors introduce three greedy algorithms—disjoint-greedy, overlap-greedy, and line-greedy—and analyze their approximation with respect to , the minimum island-partition size. Key results include a lower-bound of for disjoint-greedy, an -approximation for overlap-greedy on island covers (with a barrier and a constructive -algorithm to obtain a partition), and an -approximation via line-based separation. The work also connects island partitions to geometric line-separation problems, illuminating the trade-offs between greedy island construction and partition feasibility in geometric settings.

Abstract

Constructing partitions of colored points is a well-studied problem in discrete and computational geometry. We study the problem of creating a minimum-cardinality partition into monochromatic islands. Our input is a set of points in the plane where each point has one of colors. A set of points is monochromatic if it contains points of only one color. An island is a subset of such that , where denotes the convex hull of . We identify an island with its convex hull; therefore, a partition into islands has the additional requirement that the convex hulls of the islands are pairwise-disjoint. We present three greedy algorithms for constructing island partitions and analyze their approximation ratios.
Paper Structure (4 sections, 1 theorem, 4 equations, 4 figures)

This paper contains 4 sections, 1 theorem, 4 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related work.
  3. Overview.
  4. Disjoint-Greedy

Key Result

Lemma 1

For any $\ell \in \mathbb{N}_{\geq 1}$, there exist an $\varepsilon > 0$ and a $0 < \delta < \varepsilon$ such that for input $\textsf{\upshape FlatTree}$ disjoint-greedy returns a partition $\mathcal{P}$ that is a superset of $\,V$.

Figures (4)

  • Figure 1: Left: optimal island partition; middle-left: disjoint-greedy island partition; middle-right: overlap-greedy island cover; right: line-greedy separating lines.
  • Figure 2: The problem instance for $\ell = 5$. The lines in the figure are not part of the problem instance, but illustrate its structure. The purple squares represent red and blue points lying close together inside a square. The red disk inside the square represents many red points placed together inside a disk. The centers of the purple squares lie within the strip bounded by the two dashed lines.
  • Figure 3: The solution returned by disjoint-greedy.
  • Figure 4: An alternative solution, serving as an upper bound for the optimal solution.

Theorems & Definitions (1)

  • Lemma 1