We present a new parallel algorithm for the two processors scheduling problem. The algorithm uses only O(n 3 ) processores and takes time O(log 2 n) time on a PRAM. In order to prove the above bounds we show how to compute in NC the lexicographically first matching for a special kind of convex bipartite graphs. This work was done during the visit of the first and second authors to Patras University and it is supported by the Ministry of Industry, Energy and Technology of Greece, by a bilateral research agreement between Greece and Germany and by the ESPRIT II Basic Research Actions Program of the EC under contract No. 3075 (project ALCOM) 1 1 Introduction A classical problem in scheduling theory is to find an optimal nonpreemtive schedule for a collection of unit length tasks subject to precedence constraints. We are given n tasks to be executed on m processors. Each task requires exactly one unit of execution and can run on any processor. A directed acyclic graph specifies the..