An efficient algorithm for identifying rainbow ortho-convex 4-sets in k-colored point sets

TL;DR

This work addresses the Rainbow Ortho-Convex Positive Area 4-Set Problem for a $k$-colored point set $P$ of size $n$ by exploiting an equivalence to a 4-colored cross: a center $c$ whose four open quadrants each contain a point from $P$ of a distinct color. The authors develop an $O(n\log n)$-time algorithm for the related 4-Colored Cross problem that is independent of $k$, based on a horizontal separator set derived from $y$-coordinates and small candidate point sets per separator, and prove an $\Omega(n\log n)$ lower bound in the algebraic computation tree model using Ben-Or’s framework and a chain of linear-time reductions (2COUG -> 2CNS -> 4CC). The results establish tight complexity bounds for the problem without requiring general-position assumptions and lay groundwork for extensions to separator configurations with fixed orientations and variable angles, potentially impacting related rainbow-structure problems in computational geometry.

Abstract

Let $P$ be a $k$-colored set of $n$ points in the plane, $4 \leq k \leq n$. We study the problem of deciding if $P$ contains a subset of four points of different colors such that its Rectilinear Convex Hull has positive area. We show this problem to be equivalent to deciding if there exists a point $c$ in the plane such that each of the open quadrants defined by $c$ contains a point of $P$, each of them having a different color. We provide an $O(n \log n)$-time algorithm for this problem, where the hidden constant does not depend on $k$; then, we prove that this problem has time complexity $Ω(n \log n)$ in the algebraic computation tree model. No general position assumptions for $P$ are required.

Figures (4)

• Figure 1: A point set, its convex hull, and its $\mathcal{O}$-convex hull for two distinct sets of two orientations.
• Figure 2: The rectilinear convex hull of three sets of four points.
• Figure 3: An $(n-2)$-colored point set of size $n$ containing no subset of four points with different colors whose rectilinear convex hull has positive area. Each point is labeled with its color.
• Figure 4: Illustration of the proof of Lemma \ref{['lem:2couip_2cns']}.

Theorems & Definitions (21)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof