On weighted partial triangulations of convex polygons
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913151300141056 |
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| author | Blanca, Antonio Stauffer, Alexandre Stuhl, Izabella |
| author_facet | Blanca, Antonio Stauffer, Alexandre Stuhl, Izabella |
| contents | We study the problem of sampling weighted partial triangulations of a convex polygon. We consider the distribution where each partial triangulation $σ$ is chosen with probability proportional to $λ^{|σ|}$, where $λ>0$ is a model parameter and $|σ|$ denotes the number of diagonals in $σ$. This model belongs to a broad class of weighted geometric partition problems that include lattice triangulations and dyadic tilings, and is closely related to several classical combinatorial structures, including the full triangulations of a convex polygon and the associated Catalan structures. While prior work has largely focused on Markov chain approaches, often only providing suboptimal mixing time bounds, we provide a direct efficient method for exact sampling. Our main result is a randomized algorithm that outputs an exact sample from the target distribution in expected time $O\big((n\sqrtλ+1)\log n\big)$ for all sufficiently large $n$. This provides a nearly optimal sampling algorithm for weighted partial triangulations, offering a compelling alternative to Markov chain-based techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21921 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On weighted partial triangulations of convex polygons Blanca, Antonio Stauffer, Alexandre Stuhl, Izabella Discrete Mathematics Combinatorics We study the problem of sampling weighted partial triangulations of a convex polygon. We consider the distribution where each partial triangulation $σ$ is chosen with probability proportional to $λ^{|σ|}$, where $λ>0$ is a model parameter and $|σ|$ denotes the number of diagonals in $σ$. This model belongs to a broad class of weighted geometric partition problems that include lattice triangulations and dyadic tilings, and is closely related to several classical combinatorial structures, including the full triangulations of a convex polygon and the associated Catalan structures. While prior work has largely focused on Markov chain approaches, often only providing suboptimal mixing time bounds, we provide a direct efficient method for exact sampling. Our main result is a randomized algorithm that outputs an exact sample from the target distribution in expected time $O\big((n\sqrtλ+1)\log n\big)$ for all sufficiently large $n$. This provides a nearly optimal sampling algorithm for weighted partial triangulations, offering a compelling alternative to Markov chain-based techniques. |
| title | On weighted partial triangulations of convex polygons |
| topic | Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2605.21921 |