Average complexity of deterministic and randomized parallel comparison-sorting algorithms

In practice, the average time of (deterministic of 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 comparison 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 behavior of randomized algorithms is not more efficient than the worst-case behavior of deterministic ones.