Minimum Star Partitions of Simple Polygons in Polynomial Time
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914381522010112 |
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| author | Abrahamsen, Mikkel Blikstad, Joakim Nusser, André Zhang, Hanwen |
| author_facet | Abrahamsen, Mikkel Blikstad, Joakim Nusser, André Zhang, Hanwen |
| contents | We devise a polynomial-time algorithm for partitioning a simple polygon $P$ into a minimum number of star-shaped polygons. The question of whether such an algorithm exists has been open for more than four decades [Avis and Toussaint, Pattern Recognit., 1981] and it has been repeated frequently, for example in O'Rourke's famous book [Art Gallery Theorems and Algorithms, 1987]. In addition to its strong theoretical motivation, the problem is also motivated by practical domains such as CNC pocket milling, motion planning, and shape parameterization.
The only previously known algorithm for a non-trivial special case is for $P$ being both monotone and rectilinear [Liu and Ntafos, Algorithmica, 1991]. For general polygons, an algorithm was only known for the restricted version in which Steiner points are disallowed [Keil, SIAM J. Comput., 1985], meaning that each corner of a piece in the partition must also be a corner of $P$. Interestingly, the solution size for the restricted version may be linear for instances where the unrestricted solution has constant size. The covering variant in which the pieces are star-shaped but allowed to overlap--known as the Art Gallery Problem--was recently shown to be $\exists\mathbb R$-complete and is thus likely not in NP [Abrahamsen, Adamaszek and Miltzow, STOC 2018 & J. ACM 2022]; this is in stark contrast to our result. Arguably the most related work to ours is the polynomial-time algorithm to partition a simple polygon into a minimum number of convex pieces by Chazelle and Dobkin [STOC, 1979 & Comp. Geom., 1985]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_10631 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Minimum Star Partitions of Simple Polygons in Polynomial Time Abrahamsen, Mikkel Blikstad, Joakim Nusser, André Zhang, Hanwen Computational Geometry We devise a polynomial-time algorithm for partitioning a simple polygon $P$ into a minimum number of star-shaped polygons. The question of whether such an algorithm exists has been open for more than four decades [Avis and Toussaint, Pattern Recognit., 1981] and it has been repeated frequently, for example in O'Rourke's famous book [Art Gallery Theorems and Algorithms, 1987]. In addition to its strong theoretical motivation, the problem is also motivated by practical domains such as CNC pocket milling, motion planning, and shape parameterization. The only previously known algorithm for a non-trivial special case is for $P$ being both monotone and rectilinear [Liu and Ntafos, Algorithmica, 1991]. For general polygons, an algorithm was only known for the restricted version in which Steiner points are disallowed [Keil, SIAM J. Comput., 1985], meaning that each corner of a piece in the partition must also be a corner of $P$. Interestingly, the solution size for the restricted version may be linear for instances where the unrestricted solution has constant size. The covering variant in which the pieces are star-shaped but allowed to overlap--known as the Art Gallery Problem--was recently shown to be $\exists\mathbb R$-complete and is thus likely not in NP [Abrahamsen, Adamaszek and Miltzow, STOC 2018 & J. ACM 2022]; this is in stark contrast to our result. Arguably the most related work to ours is the polynomial-time algorithm to partition a simple polygon into a minimum number of convex pieces by Chazelle and Dobkin [STOC, 1979 & Comp. Geom., 1985]. |
| title | Minimum Star Partitions of Simple Polygons in Polynomial Time |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2311.10631 |