← 研踪 PaperTrailarXiv:2605.21921 · v3
On weighted partial triangulations of convex polygons
首次提交 2026-05-21 10:46
版本更新 2026-11-05 02:45
雷达观测 2026-11-05 12:12
摘要
We study the problem of sampling weighted partial triangulations of a convex polygon with $n+2$ sides. We consider the distribution $π_{n,λ}$ under which each partial triangulation $σ$ is assigned probability proportional to $λ^{|σ|}$, where $λ>0$ is a model parameter and $|σ| \in \{0,\dots,n-1\}$ 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. Our main result is a simple exact sampling algorithm for $π_{n,λ}$ with expected running time $O\big((\min\{n,n\sqrtλ\}+1)\log n\big)$, which is optimal up to the logarithmic factor.
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v3 · 2026-11-05 02:45 · 最新观测版本
On weighted partial triangulations of convex polygons
We study the problem of sampling weighted partial triangulations of a convex polygon with $n+2$ sides. We consider the distribution $π_{n,λ}$ under which each partial triangulation $σ$ is assigned probability proportional to $λ^{|σ|}$, where $λ>0$ is a model parameter and $|σ| \in \{0,\dots,n-1\}$ 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. Our main result is a simple exact sampling algorithm for $π_{n,λ}$ with expected running time $O\big((\min\{n,n\sqrtλ\}+1)\log n\big)$, which is optimal up to the logarithmic factor.
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