An asymptotic theory for Cauchy-Euler differential equations with applications to the analysis of algorithms

Cauchy–Euler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We study in this paper a very general framework for Cauchy–Euler equations and propose an asymptotic theory that covers almost all applications where Cauchy–Euler equations appear. Our approach is very general and requires almost no background on differential equations. Indeed the whole theory can be stated in terms of recurrences instead of functions. Old and new applications of the theory are given. New phase changes of limit laws of new variations of quicksort are systematically derived. We apply our theory to about a dozen of diverse examples in quicksort, binary search trees, urn models, increasing trees, etc.  2002 Elsevier Science (USA). All rights reserved. * Corresponding author. E-mail address: [email protected] (H.-K. Hwang). 1 Partially supported by National Science Council of ROC under the Grant NSC-90-2115-M-133001. 2 Partially supported by National Science Council of ROC under the Grants NSC-90-2118-M-001034 and NSC-90-2118-M-001-028. 0196-6774/02/$ – see front matter  2002 Elsevier Science (USA). All rights reserved. PII: S0196-6774(02)0 02 08 -0 178 H.-H. Chern et al. / Journal of Algorithms 44 (2002) 177–225