Exact-size Sampling of Enriched Trees in Linear Time

TL;DR

This paper introduces a linear-time method for uniformly sampling enriched trees, which encode various combinatorial structures, by combining decorated Bienaymé--Galton--Watson trees with Boltzmann sampling techniques.

Contribution

It develops a novel linear-time sampling algorithm for enriched trees and related structures, unifying several combinatorial classes under a common framework.

Findings

Expected linear time sampling for enriched trees

Uniform generation of combinatorial classes like polygon dissections

Efficient sampling of critical Bienayme9--Galton--Watson trees with fixed outdegree sets

Abstract

Various combinatorial classes such as outerplanar graphs and maps, series-parallel graphs, substitution-closed classes of permutations and many more allow bijective encodings by so-called enriched trees, which are rooted trees with additional structure on the offspring of each node. Using this universal description we develop sampling procedures that uniformly generate objects from this classes with a given size $n$ in expected time $O(n)$.The key ingredient is a representation of enriched trees in terms of decorated Bienaym\'e--Galton--Watson trees, which allows us to develop a novel combination of Devroye's efficient sampler for trees (Devroye, 2012) with Boltzmann sampling techniques. Additionally, we construct expected linear time samplers for critical Bienaym\'e--Galton--Watson trees having exactly $n$ (out of $\ge n$ total) nodes with outdegree in some fixed set, enabling uniform generation for many combinatorial classes such as dissections of polygons.