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Rainbow polygons for colored point sets in the plane

David Flores-Peñaloza, Mikio Kano, Leonardo Martínez-Sandoval, David Orden, Javier Tejel, Csaba D. Tóth, Jorge Urrutia, Birgit Vogtenhuber

TL;DR

The values of $\operatorname{rb-index}(k)$ are determined and it is proved that for a $k$-colored set of points in the plane in general position, a perfect rainbow polygon with at most $10 \lfloor\frac{k}{7}\rfloor + 11$ vertices can be computed in $O(n\log n)$ time.

Abstract

Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let $\operatorname{rb-index}(S)$ denote the smallest size of a perfect rainbow polygon for a colored point set $S$, and let $\operatorname{rb-index}(k)$ be the maximum of $\operatorname{rb-index}(S)$ over all $k$-colored point sets in general position; that is, every $k$-colored point set $S$ has a perfect rainbow polygon with at most $\operatorname{rb-index}(k)$ vertices. In this paper, we determine the values of $\operatorname{rb-index}(k)$ up to $k=7$, which is the first case where $\operatorname{rb-index}(k)\neq k$, and we prove that for $k\ge 5$, \[ \frac{40\lfloor (k-1)/2 \rfloor -8}{19} %Birgit: \leq\operatorname{rb-index}(k)\leq 10 \bigg\lfloor\frac{k}{7}\bigg\rfloor + 11. \] Furthermore, for a $k$-colored set of $n$ points in the plane in general position, a perfect rainbow polygon with at most $10 \lfloor\frac{k}{7}\rfloor + 11$ vertices can be computed in $O(n\log n)$ time.

Rainbow polygons for colored point sets in the plane

TL;DR

The values of are determined and it is proved that for a -colored set of points in the plane in general position, a perfect rainbow polygon with at most vertices can be computed in time.

Abstract

Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let denote the smallest size of a perfect rainbow polygon for a colored point set , and let be the maximum of over all -colored point sets in general position; that is, every -colored point set has a perfect rainbow polygon with at most vertices. In this paper, we determine the values of up to , which is the first case where , and we prove that for , Furthermore, for a -colored set of points in the plane in general position, a perfect rainbow polygon with at most vertices can be computed in time.

Paper Structure

This paper contains 12 sections, 21 theorems, 6 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Covering trees versus perfect rainbow polygons
  3. Rainbow indexes of k = 3, 4, 5, 6, 7
  4. The strip $ST$ contains points of all $k-3$ other colors.
  5. The only yellow point $w$ in $\overline{ST}$ is on $\ell_2$.
  6. Upper bound for rainbow indexes
  7. The lower arc of $\mathrm{conv}(S)$ has 3 vertices.
  8. The lower arc of $\operatorname{conv}(S)$ has 4 or more vertices.
  9. Lower bound for rainbow indexes
  10. Construction.
  11. Analysis.
  12. Conclusions

Key Result

Lemma 1

Let $T$ be a noncrossing covering tree and $\mathcal{M}$ a partition of the edges into the minimum number of pairwise noncrossing segments. If $s\geq 2$ and $t\geq 0$, then for every $\varepsilon>0$, there exists a simple polygon $P$ with $2s+t$ vertices such that $\operatorname{area}(P)\leq \vareps

Figures (11)

  • Figure 1: (a) Thickening a tree to obtain a perfect rainbow polygon. Different colors are represented by different geometric objects. (b) A noncrossing covering tree for the eight black points that can be partitioned into five segments, $s_1=u_1u_8$, $s_2=u_2u_3$, $s_3=u_2u_4$, $s_4=u_2u_5$, and $s_5=u_6u_7$; and two forks, $u_6$ with multiplicity 1 and $u_2$ with multiplicity 2.
  • Figure 2: (a) A polygon $\overline{P}$ with 28 vertices and zero area in $\overline{\mathbb{R}^2}$. The dashed circle indicates the circle at infinity. (b) A modified polygon $P$, where all vertices are in the plane.
  • Figure 3: Illustrating the proof of Lemma \ref{['lem:strip']}. The points of $S_1$ and $S_3$ are drawn as red squares and blue circles, respectively.
  • Figure 4: Illustrating the proof of Proposition \ref{['prop:rb4']}.
  • Figure 5: Illustrating the proof of Lemma \ref{['lem-2']}.
  • ...and 6 more figures

Theorems & Definitions (38)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • Lemma 6
  • ...and 28 more