Drawing the Almost Convex Set in an Integer Grid of Minimum Size

TL;DR

This paper characterizes nested almost convex sets of points, proves their uniqueness up to order type, and provides efficient algorithms for constructing and recognizing such sets with integer coordinates.

Contribution

It introduces a new characterization of nested almost convex sets, leading to algorithms for their construction and recognition with optimal time complexities.

Findings

Unique (up to order type) nested almost convex set of n points exists.

Linear time algorithm constructs nested almost convex sets with bounded integer coordinates.

O(n log n) algorithm determines if a set is a nested almost convex set.

Abstract

In 2001, K\'arolyi, Pach and T\'oth introduced a family of point sets to solve an Erd\H{o}s-Szekeres type problem; which have been used to solve several other Ed\H{o}s-Szekeres type problems. In this paper we refer to these sets as nested almost convex sets. A nested almost convex set $\mathcal{X}$ has the property that the interior of every triangle determined by three points in the same convex layer of $\mathcal{X}$, contains exactly one point of $\mathcal{X}$. In this paper, we introduce a characterization of nested almost convex sets. Our characterization implies that there exists at most one (up to order type) nested almost convex set of $n$ points. We use our characterization to obtain a linear time algorithm to construct nested almost convex sets of $n$ points, with integer coordinates of absolute values at most $O(n^{\log_2 5})$. Finally, we use our characterization to obtain an $O(n\log n)$-time algorithm to determine whether a set of points is a nested almost convex set.