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 supn log E(Tn)/ log n 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 Key words. Quicksort, binary search trees, analysis of algorithms, limit distribution, method of moments, contraction method Abbreviated title: Limit laws of qu...

• Type: Journal
• Year: 2002
• Where: SIAMCOMP
• Authors: Hsien-Kuei Hwang, Ralph Neininger
• Keywords: Algorithms, Limit Laws, Quicksort, SIAMCOMP 2002