In search of the Giant Convex Quadrilateral hidden in the Mountains

2025-11-17

AI Summary

This paper addresses the exact computation of the maximum-area convex quadrilateral inscribed in a 1.5D terrain. We propose the first deterministic algorithm for this geometric optimization problem, running in $O(n^2 log n)$ time. Our method leverages monotone chain decomposition, dynamic convex hull maintenance, extremal vertex enumeration, and plane-sweep techniques. Crucially, it exploits the $y$-monotonicity and $x$-ordering inherent in 1.5D terrains to design a structure-aware enumeration strategy that avoids brute-force search. Theoretically, we prove that the area of any maximum-area axis-aligned rectangle inscribed in the terrain is at least half the area of the optimal convex quadrilateral—establishing a constant-factor (1/2) approximation guarantee. Experimental evaluation confirms the algorithm’s efficiency and practicality. To our knowledge, this is the first tight-complexity solution for terrain-aware geometric optimization of convex quadrilaterals.

Technology Category

• Search and Optimization: Non-convex Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $O(n^2log n)$ 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.

Problem

• Finding maximum area convex quadrilateral in 1.5D terrain
• Developing O(n² log n) algorithm for quadrilateral detection
• Establishing approximation ratio for axis-parallel rectangles

Innovation

• Algorithm finds maximum area convex quadrilateral
• Uses O(n^2 log n) time complexity approach
• Employs axis-parallel rectangle as approximation method