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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.13292 |
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| _version_ | 1866929282671968256 |
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| author | Chan, Timothy M. Hair, Isaac M. |
| author_facet | Chan, Timothy M. Hair, Isaac M. |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13292 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convex Polygon Containment: Improving Quadratic to Near Linear Time Chan, Timothy M. Hair, Isaac M. Computational Geometry 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). |
| title | Convex Polygon Containment: Improving Quadratic to Near Linear Time |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2403.13292 |