Leap generators for composition schemes

TL;DR

This paper extends leap generators to supercritical composition schemes, providing asymptotically uniform random generation methods with linear time complexity for complex combinatorial structures.

Contribution

It generalizes leap generators to composition schemes, analyzing their asymptotic uniformity and applicability to various combinatorial classes.

Findings

Generators have linear time complexity under certain conditions.

The leap distribution is asymptotically uniform with a specific rate.

Applicable to classes like Pólya trees and planar maps.

Abstract

Leap generators have been introduced in [Duchon et al.'04] for exact-size random generation of structures in a class of the form $\mathcal{C}=\mathrm{Seq}(\mathcal{B})$ (sequence construction), in the supercritical case. We extend these generators to supercritical composition schemes $\mathcal{C}=\mathcal{A}\circ\mathcal{B}$. Compared to the sequence construction, the obtained exact-size random generator for $\mathcal{C}$ still has linear time complexity (under conditions on the sampling complexity in $\mathcal{A}$ and $\mathcal{B}$), but perfect uniformity of the distribution is lost in general. However the distribution on $\mathcal{C}_n$, called leap distribution, is asymptotically uniform, the total variation distance from the uniform distribution being $(c+o(1))n^{-1/2}$ for an explicit constant $c$. These generators are simple to implement and can be applied to several classes of walks and trees, in particular P\'olya trees. Leap generators can also be given for certain critical composition schemes, those relating planar map families, where this time the total variation distance to the uniform distribution is $\sim c\,n^{-1/3}$ for an explicit constant $c$.