Convex Polygon Containment: Improving Quadratic to Near Linear Time

TL;DR

This work advances convex polygon containment by breaking the long-standing quadratic barrier: it achieves near-linear time for finding the largest similar copy of a triangle inside a convex polygon and extends the framework to general constant k with $O(k^{O(1/\varepsilon)} n^{1+\varepsilon})$ time for any $\varepsilon>0$. The key ideas center on exploiting 3-contact (and 4-contact) contact patterns, partitioning the boundary of Q into arcs, and using monotone pairings plus ellipse-range data structures to enable near-linear or subquadratic subproblems without resorting to parametric search. A notable contribution is a new near-linear bound $O(k^{O(1)} n polylog n)$ on the number of 4-contact copies, which disproves a prior conjecture and informs the enumeration strategy. Overall, the paper provides a direct, scalable approach to maximizing contained copies and reveals deep structural properties of contact configurations that enable substantial time savings in polygon containment problems.

Abstract

We revisit a standard polygon containment problem: given a convex $k$-gon $P$ and a convex $n$-gon $Q$ in the plane, find a placement of $P$ inside $Q$ under translation and rotation (if it exists), or more generally, find the largest copy of $P$ inside $Q$ under translation, rotation, and scaling. Previous algorithms by Chazelle (1983), Sharir and Toledo (1994), and Agarwal, Amenta, and Sharir (1998) all required $Ω(n^2)$ time, even in the simplest $k=3$ case. We present a significantly faster new algorithm for $k=3$ achieving $O(n$polylog $n)$ running time. Moreover, we extend the result for general $k$, achieving $O(k^{O(1/\varepsilon)}n^{1+\varepsilon})$ running time for any $\varepsilon>0$. Along the way, we also prove a new $O(k^{O(1)}n$polylog $n)$ bound on the number of similar copies of $P$ inside $Q$ that have 4 vertices of $P$ in contact with the boundary of $Q$ (assuming general position input), disproving a conjecture by Agarwal, Amenta, and Sharir (1998).

Figures (6)

• Figure 1: The construction from Lemma \ref{['lem:int']}. Each dashed triangle is a similar copy of $\triangle p_1p_2p_3$, and the purple arc is $f_{v_1}(\Gamma_3)$. (We draw $\Gamma_2$ and $\Gamma_3$ instead of $\overleftrightarrow{\Gamma_2}$ and $\overleftrightarrow{\Gamma_3}$ for visual clarity.)
• Figure 2: (Left) Monotonically increasing pairing. (Right) Monotonically decreasing pairing.
• Figure 3: An example of a pairing. Each dashed triangle is a similar copy of $\triangle p_1p_2p_3$.
• Figure 4: Partitioning each $\Gamma_i(S)$ into $\Gamma_i(S^+)$ and $\Gamma_i(S^-)$, for $i \in \{1,2,3\}$.
• Figure 5: Example where both sub-arcs constituting $\Gamma_3 \setminus \textrm{clip}_3(\Gamma_3, \gamma_2)$ (denoted $(\Gamma_3 \setminus \textrm{clip}_3(\Gamma_3, \gamma_2))^+$ and $(\Gamma_3 \setminus \textrm{clip}_3(\Gamma_3, \gamma_2))^-$) have a similarity transformation that places $p_1$ at a vertex $v_1$ of $\Gamma_1$, $p_2$ on $\gamma_2$, and $p_3$ on $\Gamma_3 \setminus \textrm{clip}_3(\Gamma_3, \gamma_2)$. $X^+(v_1)$ and $X^-(v_2)$ can be found rapidly via Lemma \ref{['lem:int']} since $(\Lambda(\Gamma_3 \setminus \textrm{clip}_3(\Gamma_3, \gamma_2)) + \theta_{p_1p_2}) \cap (\Lambda(\gamma_2)+\theta_{p_1p_3}) = \emptyset$ (mod $\pi)$. Dashed triangles are similar to $\triangle p_1p_2p_3$.

Theorems & Definitions (26)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof
• Lemma 5
• proof