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

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.