Improving the Florentine algorithms: recovering algorithms for Motzkin and Schr\"oder paths

TL;DR

This paper introduces improved random sampling algorithms for Motzkin and Schr"oder paths that reduce rejection rates and enhance efficiency by building on Florentine algorithms with a novel recovery technique.

Contribution

It develops new sampling procedures that significantly lower rejection probability and improve time complexity for Motzkin and Schr"oder paths compared to previous Florentine algorithms.

Findings

Reduced rejection probability in sampling algorithms

Enhanced time complexity over previous methods

Effective recovery technique for path sampling

Abstract

We present random sampling procedures for Motzkin and Schr\"oder paths, following previous work on Dyck paths. Our algorithms follow the anticipated rejection method of the Florentine algorithms (Barcucci et al. 1994+), but introduce a recovery idea to greatly reduce the probability of rejection. They use an optimal amount of randomness and achieve a better time complexity than the Florentine algorithms.