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In search of the Giant Convex Quadrilateral hidden in the Mountains

Nandana Ghosh, Rakesh Gupta, Ankush Acharyya

TL;DR

The paper investigates inner-approximation optimization for a $1.5$D terrain by seeking the maximum-area convex quadrilateral $Q^*$ inscribed in the terrain. It develops an exact $O(n^2\log n)$ algorithm for computing $Q^*$ by exploiting structural properties that force a base on the terrain base $\Pi$ and by organizing edge candidates through extremal chords and butterfly configurations. In addition, it provides an $O(n\log n)$ method to compute the maximum-area axis-parallel rectangle ${\bigbox}^*$ inside the terrain and proves that ${\bigbox}^*$ yields a $\frac{1}{2}$-approximation to $Q^*$, offering a fast, provable surrogate when exact optimization is expensive. The results combine geometric insight with butterfly theory and shortest-path structures to achieve both exact and approximate solutions, advancing inner-approximation techniques for $1.5$D terrains with practical implications for pattern recognition and GIS-inspired computations.

Abstract

A $1.5$D terrain is a simple polygon bounded by a line segment $\ell$ and a polygonal chain monotone with respect to the line segment $\ell$. Usually, $\ell$ is chosen aligned to the $x$-axis, and is called the base of the terrain. In this paper, we consider the problem of finding a convex quadrilateral of maximum area inside a $1.5$D terrain in $I\!\!R^2$. We present an $O(n^2\log n)$ time algorithm for this problem, where $n$ is the number of vertices of the terrain. Finally, we show that the maximum area axis-parallel rectangle inside the terrain yields a $\frac{1}{2}$ factor approximation result to the maximum area convex quadrilateral problem.

In search of the Giant Convex Quadrilateral hidden in the Mountains

TL;DR

The paper investigates inner-approximation optimization for a D terrain by seeking the maximum-area convex quadrilateral inscribed in the terrain. It develops an exact algorithm for computing by exploiting structural properties that force a base on the terrain base and by organizing edge candidates through extremal chords and butterfly configurations. In addition, it provides an method to compute the maximum-area axis-parallel rectangle inside the terrain and proves that yields a -approximation to , offering a fast, provable surrogate when exact optimization is expensive. The results combine geometric insight with butterfly theory and shortest-path structures to achieve both exact and approximate solutions, advancing inner-approximation techniques for D terrains with practical implications for pattern recognition and GIS-inspired computations.

Abstract

A D terrain is a simple polygon bounded by a line segment and a polygonal chain monotone with respect to the line segment . Usually, is chosen aligned to the -axis, and is called the base of the terrain. In this paper, we consider the problem of finding a convex quadrilateral of maximum area inside a D terrain in . We present an time algorithm for this problem, where is the number of vertices of the terrain. Finally, we show that the maximum area axis-parallel rectangle inside the terrain yields a factor approximation result to the maximum area convex quadrilateral problem.

Paper Structure

This paper contains 11 sections, 11 theorems, 1 equation, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Computation of $Q^*$
  4. Properties of $Q^*$
  5. Algorithm to compute $Q^*$
  6. Left edge is supported by an extremal chord
  7. Left edge is a balanced chord
  8. An approximation
  9. Computation of $\bigbox^*$
  10. $\bigbox^*$ gives a $\frac{1}{2}$-approximation of $Q^*$
  11. Acknowledgements

Key Result

lemma thmcounterlemma

There exists an optimum solution $Q^*$ whose one side coincides with $\Pi$.

Figures (9)

  • Figure 1: $Q=\Diamond \alpha\beta\gamma\delta$ is a maximal area convex quadrilateral inscribed in a terrain.
  • Figure 2: There exists $Q^*$ whose base $\cal B$ always touches the base $\Pi$.
  • Figure 3: $\cal T$ has positive slope and $\cal R$ passes through two vertices of $\mathfrak{T}$.
  • Figure 4: $\cal T$ has negative slope and $\cal R$ passes through two vertices of $\mathfrak{T}$ or it is bisected at a vertex of $\mathfrak{T}$.
  • Figure 5: Butterfly structures
  • ...and 4 more figures

Theorems & Definitions (18)

  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • ...and 8 more