Transfer theorems and asymptotic distributional results for m-ary search trees

• Authors: James Allen Fill;Nevin Kapur
• Venue: Random Structures & Algorithms
• Year: 2005

We derive asymptotics of moments and identify limiting distributions, under the random permutation model on m-ary search trees, for functionals that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the so-called shape functional fall under this framework. The approach is based on establishing transfer theorems that link the order of growth of the input into a particular (deterministic) recurrence to the order of growth of the output. The transfer theorems are used in conjunction with the method of moments to establish limit laws. It is shown that: (i) for small toll sequences (tn) [roughly, tn = O(n1/2)] we have asymptotic normality if m ≤ 26 and typically periodic behavior if m ≥ 27; (ii) for moderate toll sequences [roughly, tn = ω(n1/2) but tn = o(n)] we have convergence to nonnormal distributions if m ≤ m0 (where m0 ≥ 26) and typically periodic behavior if m ≥ m0 + 1; and (iii) for large toll sequences [roughly, tn = ω(n)] we have convergence to nonnormal distributions for all values of m. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005Research for both authors supported by NSF Grants DMS-9803780 and DMS-0104167, and by The Johns Hopkins University's Acheson J. Duncan Fund for the Advancement of Research in Statistics.Research supported by NSF Grant 0049092 and carried out primarily while this author was affiliated with what is now the Department of Applied Mathematics and Statistics at The Johns Hopkins University.