A Linear Time Algorithm for the Maximum Overlap of Two Convex Polygons Under Translation
This solves a long-standing open problem in computational geometry, providing an optimal algorithm for a fundamental geometric optimization task.
The paper presents the first linear-time randomized algorithm for computing the maximum overlap area of two convex polygons under translation, improving upon the previous O((n+m) log(n+m)) algorithm from 1998.
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.