# Generating a Random Sink-free Orientation in Quadratic Time

### Abstract

A *sink-free orientation* of a finite undirected graph is a choice of orientation for each edge such that every vertex has out-degree at least 1. Bubley and Dyer (1997) use Markov Chain Monte Carlo to sample approximately from the uniform distribution on sink-free orientations in time $O(m^3 \log (1 / \varepsilon))$, where $m$ is the number of edges and $\varepsilon$ the degree of approximation. Huber (1998) uses coupling from the past to obtain an exact sample in time $O(m^4)$. We present a simple randomized algorithm inspired by Wilson's cycle popping method which obtains an exact sample in mean time at most $O(nm)$, where $n$ is the number of vertices.