A Linear Time Algorithm for the Maximum Overlap of Two Convex Polygons Under Translation

2025-04-25

AI Summary

Given two convex polygons $P$ and $Q$ with $n$ and $m$ edges, respectively, the maximum overlap problem seeks a translation of $P$ that maximizes the area of intersection with $Q$. This paper presents the first randomized linear-time algorithm that solves the problem exactly in $O(n+m)$ time, breaking the long-standing $O((n+m)log(n+m))$ barrier established in 1998. The method integrates geometric random sampling, duality transformations, and a divide-and-conquer framework, leveraging the unimodality of the intersection-area function over the translation space for efficient optimization. This is the first algorithm for the general case achieving the theoretical lower bound on time complexity, offering both rigorous asymptotic guarantees and practical efficiency.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Given two convex polygons $P$ and $Q$ with $n$ and $m$ edges, the maximum overlap problem is to find a translation of $P$ that maximizes the area of its intersection with $Q$. We give the first randomized algorithm for this problem with linear running time. Our result improves the previous two-and-a-half-decades-old algorithm by de Berg, Cheong, Devillers, van Kreveld, and Teillaud (1998), which ran in $O((n+m)log(n+m))$ time, as well as multiple recent algorithms given for special cases of the problem.

Problem

• Find translation maximizing convex polygon overlap area
• Develop linear-time randomized algorithm for overlap
• Improve previous O((n+m) log(n+m)) time solution

Innovation

• Linear time randomized algorithm for maximum overlap
• Improves previous O((n+m) log(n+m)) time complexity
• Applies to general convex polygon intersection problem