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Maximum rectilinear convex subsets

Hernán González-Aguilar, David Orden, Pablo Pérez-Lantero, David Rappaport, Carlos Seara, Javier Tejel, Jorge Urrutia

TL;DR

This work provides efficient $O(n^3)$-time, $O(n^2)$-space algorithms for maximizing the size, area, or weight of rectilinear convex hulls induced by a subset of a point set under rectilinear (orthoconvex) constraints. It introduces a cohesive framework built on four-separator geometry and staircase decompositions to solve MaxRCH and its variants, and extends these techniques to related problems such as MaxOrthoconvexPolygon and MaxStaircasePolygon with matching asymptotic performance. The results unify multiple Erdős-Szekeres-type problems in the rectilinear setting, offering simpler, interchangeable dynamic-programming formulations and practical preprocessing for constant-time rectangle queries. The methods have potential applications in VLSI layout, visibility, and related geometric optimization tasks where axis-aligned convexity governs feasibility and performance.

Abstract

Let $P$ be a set of $n$ points in the plane. We consider a variation of the classical Erdős-Szekeres problem, presenting efficient algorithms with $O(n^3)$ running time and $O(n^2)$ space complexity that compute: (1) A subset $S$ of $P$ such that the boundary of the rectilinear convex hull of $S$ has the maximum number of points from $P$, (2) a subset $S$ of $P$ such that the boundary of the rectilinear convex hull of $S$ has the maximum number of points from $P$ and its interior contains no element of $P$, (3) a subset $S$ of $P$ such that the rectilinear convex hull of $S$ has maximum area and its interior contains no element of $P$, and (4) when each point of $P$ is assigned a weight, positive or negative, a subset $S$ of $P$ that maximizes the total weight of the points in the rectilinear convex hull of $S$. We also revisit the problems of computing a maximum-area orthoconvex polygon and computing a maximum-area staircase polygon, amidst a point set in a rectangular domain. We obtain new and simpler algorithms to solve both problems with the same complexity as in the state of the art.

Maximum rectilinear convex subsets

TL;DR

This work provides efficient -time, -space algorithms for maximizing the size, area, or weight of rectilinear convex hulls induced by a subset of a point set under rectilinear (orthoconvex) constraints. It introduces a cohesive framework built on four-separator geometry and staircase decompositions to solve MaxRCH and its variants, and extends these techniques to related problems such as MaxOrthoconvexPolygon and MaxStaircasePolygon with matching asymptotic performance. The results unify multiple Erdős-Szekeres-type problems in the rectilinear setting, offering simpler, interchangeable dynamic-programming formulations and practical preprocessing for constant-time rectangle queries. The methods have potential applications in VLSI layout, visibility, and related geometric optimization tasks where axis-aligned convexity governs feasibility and performance.

Abstract

Let be a set of points in the plane. We consider a variation of the classical Erdős-Szekeres problem, presenting efficient algorithms with running time and space complexity that compute: (1) A subset of such that the boundary of the rectilinear convex hull of has the maximum number of points from , (2) a subset of such that the boundary of the rectilinear convex hull of has the maximum number of points from and its interior contains no element of , (3) a subset of such that the rectilinear convex hull of has maximum area and its interior contains no element of , and (4) when each point of is assigned a weight, positive or negative, a subset of that maximizes the total weight of the points in the rectilinear convex hull of . We also revisit the problems of computing a maximum-area orthoconvex polygon and computing a maximum-area staircase polygon, amidst a point set in a rectangular domain. We obtain new and simpler algorithms to solve both problems with the same complexity as in the state of the art.

Paper Structure

This paper contains 9 sections, 13 theorems, 34 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Some notation and definitions
  3. Rectilinear convex hull of maximum size
  4. Maximum size/area empty rectilinear convex hulls and maximum weight rectilinear convex hull
  5. Maximum size empty rectilinear convex hull
  6. Maximum area empty rectilinear convex hull
  7. Maximum weight rectilinear convex hull
  8. Maximum area orthoconvex polygon
  9. Empty staircase polygon with the largest area

Key Result

Lemma 2.11

Let $P$ be a point set in the plane. For all $S\subseteq P$, $|S|\ge 2$, either $RCH(S)$ or $RCH(S')$ is vertically separable.

Figures (15)

  • Figure 1: (a) A point set with a connected rectilinear convex hull. (b) A point set whose rectilinear convex hull is disconnected, two of its components are pinched points.
  • Figure 2: (a) A $1$-staircase. (b) The definition of the sets $Q_i(p)$. (c) A 7-point set $P$ and the set $M_1(P)$. The vertices of the boundary of $M_1(P)$ in $P$ are the 1-extremal points of $P$. The thick polygonal line is the $1$-staircase associated with $P$.
  • Figure 3: Left: Sets $R_{p\setminus q}$, $R_{q\setminus p}$, $R_{p,q}$, $R'_{p\setminus q}$, $R'_{q\setminus p}$ and $R'_{p,q}$. Right: Example of $\mathcal{C}_{p,q}$.
  • Figure 4: Examples of triple $1$-staircases $\mathcal{T}_{p,q}$.
  • Figure 5: Cases in the recursive computation of $T_{p,q}$.
  • ...and 10 more figures

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.9
  • Definition 2.10
  • Lemma 2.11
  • ...and 29 more