  We show that the bipartite perfect matching problem is in quasi-NC$^2$. That
is, it has uniform circuits of quasi-polynomial size $n^{O(\log n)}$, and
$O(log^2 n)$ depth. Previously, only an exponential upper bound was known on
the size of such circuits with poly-logarithmic depth.
  We obtain our result by an almost complete derandomization of the famous
Isolation Lemma when applied to yield an efficient randomized parallel
algorithm for the bipartite perfect matching problem.
