Counting $N$ Queens
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914866996969472 |
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| author | Polson, Nick Sokolov, Vadim |
| author_facet | Polson, Nick Sokolov, Vadim |
| contents | Gauss proposed the problem of how to enumerate the number of solutions for placing $N$ queens on an $N\times N$ chess board, so no two queens attack each other. The N-queen problem is a classic problem in combinatorics. We describe a variety of Monte Carlo (MC) methods for counting the number of solutions. In particular, we propose a quantile re-ordering based on the Lorenz curve of a sum that is related to counting the number of solutions. We show his approach leads to an efficient polynomial-time solution. Other MC methods include vertical likelihood Monte Carlo, importance sampling, slice sampling, simulated annealing, energy-level sampling, and nested-sampling. Sampling binary matrices that identify the locations of the queens on the board can be done with a Swendsen-Wang style algorithm. Our Monte Carlo approach counts the number of solutions in polynomial time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08830 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting $N$ Queens Polson, Nick Sokolov, Vadim Computation Gauss proposed the problem of how to enumerate the number of solutions for placing $N$ queens on an $N\times N$ chess board, so no two queens attack each other. The N-queen problem is a classic problem in combinatorics. We describe a variety of Monte Carlo (MC) methods for counting the number of solutions. In particular, we propose a quantile re-ordering based on the Lorenz curve of a sum that is related to counting the number of solutions. We show his approach leads to an efficient polynomial-time solution. Other MC methods include vertical likelihood Monte Carlo, importance sampling, slice sampling, simulated annealing, energy-level sampling, and nested-sampling. Sampling binary matrices that identify the locations of the queens on the board can be done with a Swendsen-Wang style algorithm. Our Monte Carlo approach counts the number of solutions in polynomial time. |
| title | Counting $N$ Queens |
| topic | Computation |
| url | https://arxiv.org/abs/2407.08830 |