This paper describes a new compiler algorithm to reduce the number of barrier synchronisations in parallelised programs. A preliminary technique to rapidly determine critical data dependences is developed. This forms the basis of the Fast First Sink (FFS) algorithm which places, provably, the minimal number of barriers in polynomial time for codes with a regular structure. This algorithm is implemented in a prototype compiler and applied to three well known benchmarks. Preliminary results show that it outperforms an existing state-of the-art commercial compiler.