Phase changes in random m-ary search trees and generalized quicksort

We propose a uniform approach to describing the phase change of the limiting distribution of space measures in random m-ary search trees: the space requirement, when properly normalized, is asymptotically normally distributed for m ≤ 26 and does not have a fixed limit distribution for m > 26. The tools are based on the method of moments and asymptotic solutions of differential equations, and are applicable to secondary cost measures of quicksort with median-of-(2t+ 1) for which the same phase change occurs at t = 58. Both problems are essentially special cases of the generalized quicksort of Hennequin in which a sample of m(t+ 1)− 1 elements are used to select m− 1 equi-spaced ranks that are used in turn to partition the input into m subfiles. A complete description of the numbers at which the phase change occurs is given. For example, when m is fixed and t varies, the phase change occurs at (m, t) = {(2, 58), (3, 19), (4, 10), (5, 6), (6, 4), · · · }. We also indicate some applications of our approach to other problems including bucket recursive trees. A general framework on “asymptotic transfers” of the underlying recurrence is also given.