Exact uniform sampling over catalan structures

TL;DR

This paper introduces a unified framework for exact uniform sampling of Catalan structures, achieving optimal coding and random bit efficiency through a recursive relation and a novel matrix and tree structure.

Contribution

The paper presents a new recursive framework and data structures for efficient, exact uniform sampling of Catalan structures, with optimal coding and randomness properties.

Findings

Linear-time sampling algorithms for various Catalan structures.

Optimal coding and minimal random bits required.

A unified recursive approach applicable to multiple structures.

Abstract

We present a new framework for creating elegant algorithms for exact uniform sampling of important Catalan structures, such as triangulations of convex polygons, Dyck words, monotonic lattice paths and mountain ranges. Along with sampling, we obtain optimal coding, and optimal number of random bits required for the algorithm. The framework is based on an original two-parameter recursive relation, where Ballot and Catalan numbers appear and which may be regarded as to demonstrate a generalized reduction argument. We then describe (a) a unique $n\times n$ matrix to be used for any of the problems -the common pre-processing step of our framework- and (b) a linear height tree, where leaves correspond one by one to all distinct solutions of each problem; sampling is essentially done by selecting a path from the root to a leaf - the main algorithm. Our main algorithm is linear for a number of the problems mentioned.