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

Abstract

Let P be a k-colored set of n points in the plane, $4\le k\le n$. 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 $\Omega (n\log n)$ in the algebraic computation tree model. No general position assumptions for P are required.