Canonical labelling of Latin squares in average-case polynomial time
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911869307977728 |
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| author | Gill, Michael J. Mammoliti, Adam Wanless, Ian M. |
| author_facet | Gill, Michael J. Mammoliti, Adam Wanless, Ian M. |
| contents | A Latin square of order $n$ is an $n\times n$ matrix in which each row and column contains each of $n$ symbols exactly once. For $ε>0$, we show that with high probability a uniformly random Latin square of order $n$ has no proper subsquare of order larger than $n^{1/2}\log^{1/2+ε}n$. Using this fact we present a canonical labelling algorithm for Latin squares of order $n$ that runs in average time bounded by a polynomial in $n$.
The algorithm can be used to solve isomorphism problems for many combinatorial objects that can be encoded using Latin squares, including quasigroups, Steiner triple systems, Mendelsohn triple systems, $1$-factorisations, nets, affine planes and projective planes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06205 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Canonical labelling of Latin squares in average-case polynomial time Gill, Michael J. Mammoliti, Adam Wanless, Ian M. Combinatorics 05B15 A Latin square of order $n$ is an $n\times n$ matrix in which each row and column contains each of $n$ symbols exactly once. For $ε>0$, we show that with high probability a uniformly random Latin square of order $n$ has no proper subsquare of order larger than $n^{1/2}\log^{1/2+ε}n$. Using this fact we present a canonical labelling algorithm for Latin squares of order $n$ that runs in average time bounded by a polynomial in $n$. The algorithm can be used to solve isomorphism problems for many combinatorial objects that can be encoded using Latin squares, including quasigroups, Steiner triple systems, Mendelsohn triple systems, $1$-factorisations, nets, affine planes and projective planes. |
| title | Canonical labelling of Latin squares in average-case polynomial time |
| topic | Combinatorics 05B15 |
| url | https://arxiv.org/abs/2402.06205 |