  The stable roommates problem with $n$ agents has worst case complexity
$O(n^2)$ in time and space. Random instances can be solved faster and with less
memory, however. We introduce an algorithm that has average time and space
complexity $O(n^\frac{3}{2})$ for random instances. We use this algorithm to
simulate large instances of the stable roommates problem and to measure the
probabilty $p_n$ that a random instance of size $n$ admits a stable matching.
Our data supports the conjecture that $p_n = \Theta(n^{-1/4})$.
