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 ${\cal L}(X_n)= {\cal L}(X_{I_n}+X^*_{n-1-I_n}+T_n)$ when the "toll function" Tn varies and satisfies general conditions, where (Xn), (Xn*), (In, Tn) are independent, In is uniformly distributed over {0, . . .,n-1}, and ${\cal L}(X_n)={\cal L}(X_n^\ast)$. When the "toll function" Tn (cost needed to partition the original problem into smaller subproblems) is small (roughly $\limsup_{n\rightarrow\infty}\log E(T_n)/\log n\le 1/2$), Xn is asymptotically normally distributed; nonnormal limit laws emerge when Tn becomes larger. We give many new examples ranging from the number of exchanges in quicksort to sorting on a broadcast communication model, from an in-situ permutation algorithm to tree traversal algorithms, etc.