Phase Change of Limit Laws in the Quicksort Recurrence under Varying Toll Functions

We characterize all limit laws of the quicksort type random variables defined recursively by Xn d = XIn + X ∗ n−1−In + Tn when the “toll function” Tn varies and satisfies general conditions, where (Xn), (X ∗ n ), (In, Tn) are independent, Xn d = X∗ n , and In is uniformly distributed over {0, . . . , n − 1}. When the “toll function” Tn (cost needed to partition the original problem into smaller subproblems) is small (roughly lim sup n→∞ logE(Tn)/ logn ≤ 1/2), Xn is asymptotically normally distributed; non-normal limit laws emerge when Tn becomes larger. We give many new examples ranging from the number of exchanges in quicksort to sorting on broadcast communication model, from an in-situ permutation algorithm to tree traversal algorithms, etc. AMS subject classifications. Primary: 68W40 68Q25; secondary: 60F05 11B37