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

Timothy M. Chan, Isaac M. Hair

TL;DR

This work addresses the maximum overlap problem for translating one convex polygon against another, achieving the first linear-time exact algorithm in 2D by a randomized approach. The authors introduce a novel block-structure prune-and-search framework that reduces oracle costs as the search space tightens, integrated with a configuration-space formulation, epsilon-cuttings, and area-prefix sums. By decomposing inputs into angle-based blocks and iteratively refining the search region, they obtain a geometric-series time bound and perform a multidimensional parametric search to reach $O(n+m)$ expected time. The result significantly improves the long-standing $O((n+m)\log(n+m))$ bound and provides a new methodological template for similar geometric optimization problems, with extensions and higher-dimensional considerations discussed in the remarks.

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.

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

TL;DR

This work addresses the maximum overlap problem for translating one convex polygon against another, achieving the first linear-time exact algorithm in 2D by a randomized approach. The authors introduce a novel block-structure prune-and-search framework that reduces oracle costs as the search space tightens, integrated with a configuration-space formulation, epsilon-cuttings, and area-prefix sums. By decomposing inputs into angle-based blocks and iteratively refining the search region, they obtain a geometric-series time bound and perform a multidimensional parametric search to reach expected time. The result significantly improves the long-standing bound and provides a new methodological template for similar geometric optimization problems, with extensions and higher-dimensional considerations discussed in the remarks.

Abstract

Given two convex polygons and with and edges, the maximum overlap problem is to find a translation of that maximizes the area of its intersection with . 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 time, as well as multiple recent algorithms given for special cases of the problem.

Paper Structure

This paper contains 20 sections, 13 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation and Background.
  3. A Superlinear Barrier?
  4. Our Result.
  5. A New Kind of Prune-and-Search?
  6. Remarks.
  7. Preliminaries
  8. The Objective Function.
  9. Cuttings.
  10. Area Prefix Sums.
  11. The Configuration Space.
  12. Angles and Angle Ranges.
  13. Blocking Scheme
  14. A Data Structure Using Blocks
  15. The Linear Time Algorithm
  16. ...and 5 more sections

Key Result

Lemma 2.1

$\sqrt{\textnormal{Area}((P+t)\cap Q)}$ is downward concave over all values of $t$ that yield nonzero overlap of $P+t$ with $Q$.

Figures (2)

  • Figure 1: A partition of example polygons $P$ and $Q$ into blocks, with $b = 4$. Note that the spatial sizes of blocks may vary greatly, and the number of edges in each $P_i$ and $Q_j$ may differ. In fact, for some indices $i$, one of $P_i$ or $Q_j$ may contain just the origin.
  • Figure 2: A series of triangles $\mathcal{T}_0$, $\mathcal{T}_1$, and $\mathcal{T}_2$ as produced by the algorithm. The (unknown) optimal placement $t^*$ is denoted by an asterisk. The red curves indicate contour lines of the objective function, with darker circles indicating larger values of $\textnormal{Area}((P+t) \cap Q)$.

Theorems & Definitions (24)

  • Lemma 2.1
  • Definition 2.2: $\varepsilon$-Cuttings
  • Lemma 2.3
  • Definition 3.1: $(\mu, b, \mathcal{T})$-Block Structure
  • Lemma 4.0: Point Oracle
  • Lemma 4.0: Cascading Subroutine
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 14 more