COMPLEXITY OF PARALLEL COMPUTATION ON MATROIDS.

The complexity of the S-search problem for matroids is studied. The main result is a lower bound on any probabilistic algorithm for the S-search problem that acquires information about the input system by interrogating an independence oracle. It is proved that the expected time of any such probabilistic algorithm that uses a subexponential number of processors is OMEGA greater than n**1**/**3- epsilon ). This is one of the first nontrivial, superalgorithmic lower bounds on a randomized parallel computation. It implies that in the authors' model of computation Random-NC is strictly contained in P. Another consequence of the lower bounds is that the O greater than ROOT n) time probabilistic upper bound for arbitrary independence systems is close to optimal and cannot be significantly improved, even for matroids. However, this O( ROOT n) upper bound can be improved in a different sense for matroids - it can be made deterministic, still with polynomially many processors. Finally, it is shown that the lower bound can be beaten for the special case of graphic matroids.