On weighted partial triangulations of convex polygons

TL;DR

This paper introduces an efficient randomized algorithm for exact sampling of weighted partial triangulations of convex polygons, improving over traditional Markov chain methods with better expected runtime.

Contribution

It presents a novel direct sampling algorithm with provably optimal expected time complexity for weighted partial triangulations, expanding the toolkit for geometric partition problems.

Findings

Expected sampling time is O((n√λ+1) log n) for large n.

Provides a nearly optimal sampling method for weighted partial triangulations.

Offers an alternative to Markov chain approaches with improved efficiency.

Abstract

We study the problem of sampling weighted partial triangulations of a convex polygon. We consider the distribution where each partial triangulation $\sigma$ is chosen with probability proportional to $\lambda^{|\sigma|}$, where $\lambda>0$ is a model parameter and $|\sigma|$ denotes the number of diagonals in $\sigma$. 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 $O\big((n\sqrt{\lambda}+1)\log n\big)$ 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.