On weighted partial triangulations of convex polygons

Abstract

We study the problem of sampling weighted partial triangulations of a convex polygon with $n+2$ sides. We consider the distribution $\pi_{n,\lambda}$ under which each partial triangulation $\sigma$ is assigned probability proportional to $\lambda^{|\sigma|}$, where $\lambda>0$ is a model parameter and $|\sigma| \in \{0,\dots,n-1\}$ 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. Our main result is a simple exact sampling algorithm for $\pi_{n,\lambda}$ with expected running time $O\big((\min\{n,n\sqrt{\lambda}\}+1)\log n\big)$, which is optimal up to the logarithmic factor.