The average complexity of deterministic and randomized parallel comparison sorting algorithms

In practice, the average time of (deterministic or randomized) sorting algorithms seems to be more relevant than the worst case time of deterministic algorithms. Still, the many known complexity bounds for parallel comparison sorting include no nontrivial lower bounds for the average time required to sort by comparisons n elements with p processors (via deterministic or randomized algorithms). We show that for p ≥ n this time is Θ (log n/log(1 + p/n)), (it is easy to show that for p ≤ n the time is Θ (n log n/p) = Θ (log n/(p/n)). Therefore even the average case behaviour of randomized algorithms is not more efficient than the worst case behaviour of deterministic ones.