Limit laws for the height in PATRICIA tries

Abstract

We consider digital trees, namely, PATRICIA tries built from n random strings generated by an unbiased memoryless source (i.e., all symbols are equally likely). We study limit laws of the height, which is defined as the longest path in such trees. We shall identify three regions of the height distributions. In the region where most of the probability mass is concentrated, the asymptotic distribution of the PATRICIA height concentrates on one or two points around the most probable value k 1=⌊ log 2n+ 2 log 2n −3/2⌋+1 . For most n all the mass is concentrated at k 1, however, there exist subsequences of n such that the mass is on the two points k 1−1 and k 1, or k 1 and k 1+1. This should be contrasted with the limiting distribution of height for standard tries that is of extreme value type (i.e., double exponential distribution). We derive our results by a combination of analytic methods such as generating functions, Mellin transform, the saddle point method and ideas of applied mathematics such as linearization, asymptotic matching and the WKB principle. We make certain assumptions about the forms of some asymptotic expansions and their asymptotic matching. We also present some numerical verification of our results.