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.

翻译：我们研究了对凸多边形加权部分三角剖分进行抽样的问题。考虑这样一种分布：每个部分三角剖分 $σ$ 的选择概率与 $λ^{|σ|}$ 成正比，其中 $λ>0$ 是一个模型参数，$|σ|$ 表示 $σ$ 中对角线的数量。该模型属于一类广泛的加权几何划分问题，包括网格三角剖分和二进制细分，并与若干经典组合结构密切相关，例如凸多边形的完全三角剖分及其相关的 Catalan 结构。尽管先前的研究主要集中于马尔可夫链方法，通常仅能提供次优的混合时间界限，但我们在本文中提出了一种直接且高效的精确抽样方法。我们的主要成果是一个随机化算法，该算法能在期望时间 $O\big((n\sqrtλ+1)\log n\big)$ 内（对所有足够大的 $n$）从目标分布中输出精确样本。这为加权部分三角剖分提供了一种近乎最优的抽样算法，成为基于马尔可夫链技术的一种有力替代方案。