On weighted partial triangulations of convex polygons

2026-05-20

AI Summary

This work addresses the problem of exact sampling from the distribution over partial triangulations of a convex polygon, where each triangulation is weighted proportionally to an exponential function of its number of diagonals. The authors propose the first direct randomized sampling algorithm that does not rely on Markov chains, leveraging combinatorial structure analysis and geometric probability modeling to achieve exact sampling from the target distribution. This approach overcomes the efficiency limitations inherent in traditional Markov chain methods, achieving an expected time complexity of $O((n\sqrt{\lambda} + 1)\log n)$, which significantly outperforms existing techniques and demonstrates near-optimal performance on large-scale instances.

Abstract

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.

Problem

• weighted partial triangulations
• convex polygons
• exact sampling
• geometric partition
• Catalan structures

Innovation

• exact sampling
• weighted partial triangulations
• randomized algorithm
• convex polygon
• efficient sampling