Solving Simple Stochastic Games with few Random Nodes faster using Bland's Rule

The best algorithm so far for solving Simple Stochastic Games is Ludwig's randomized algorithm which works in expected $2^{O(\sqrt{n})}$ time. We first give a simpler iterative variant of this algorithm, using Bland's rule from the simplex algorithm, which uses exponentially less random bits than Ludwig's version. Then, we show how to adapt this method to the algorithm of Gimbert and Horn whose worst case complexity is $O(k!)$, where $k$ is the number of random nodes. Our algorithm has an expected running time of $2^{O(k)}$, and works for general random nodes with arbitrary outdegree and probability distribution on outgoing arcs.