Uniform random sampling of planar graphs in linear time

TL;DR

This paper presents an efficient linear-time algorithm for uniform random sampling of labelled planar graphs, significantly improving previous methods with a combination of Boltzmann samplers, bijections, and analytic combinatorics.

Contribution

It introduces a novel algorithm leveraging Boltzmann samplers and combinatorial bijections for linear-time uniform sampling of planar graphs.

Findings

Expected linear time for approximate-size sampling

Quadratic expected time for exact-size sampling

Significant reduction from previous $O(n^7)$ complexity

Abstract

This article introduces new algorithms for the uniform random generation of labelled planar graphs. Its principles rely on Boltzmann samplers, as recently developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the Boltzmann framework, a suitable use of rejection, a new combinatorial bijection found by Fusy, Poulalhon and Schaeffer, as well as a precise analytic description of the generating functions counting planar graphs, which was recently obtained by Gim\'enez and Noy. This gives rise to an extremely efficient algorithm for the random generation of planar graphs. There is a preprocessing step of some fixed small cost. Then, the expected time complexity of generation is quadratic for exact-size uniform sampling and linear for approximate-size sampling. This greatly improves on the best previously known time complexity for exact-size uniform sampling of planar graphs with $n$ vertices, which was a little over $O(n^7)$.