Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909363220774912 |
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| author | Eppstein, David |
| author_facet | Eppstein, David |
| contents | We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in the number of surrounding cycles. As a consequence, we can list the non-crossing Hamiltonian paths or the polygonalizations, in time polynomial in the output size, by filtering the output of simple backtracking algorithms for non-crossing paths or surrounding cycles respectively. To prove these results we relate the numbers of non-crossing structures to two easily-computed parameters of the point set: the minimum number of points whose removal results in a collinear set, and the number of points interior to the convex hull. These relations also lead to polynomial-time approximation algorithms for the numbers of structures of all four types, accurate to within a constant factor of the logarithm of these numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_00147 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time Eppstein, David Computational Geometry Data Structures and Algorithms We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in the number of surrounding cycles. As a consequence, we can list the non-crossing Hamiltonian paths or the polygonalizations, in time polynomial in the output size, by filtering the output of simple backtracking algorithms for non-crossing paths or surrounding cycles respectively. To prove these results we relate the numbers of non-crossing structures to two easily-computed parameters of the point set: the minimum number of points whose removal results in a collinear set, and the number of points interior to the convex hull. These relations also lead to polynomial-time approximation algorithms for the numbers of structures of all four types, accurate to within a constant factor of the logarithm of these numbers. |
| title | Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time |
| topic | Computational Geometry Data Structures and Algorithms |
| url | https://arxiv.org/abs/2303.00147 |