Limit Laws for Heights in Generalized Tries and PATRICIA Tries

Wojciech Szpankowskit Department of Computer Science Purdue University W. LafayeUe, IN 47907 UB.A. spa@cs.purdue.edu We consider digital trees such as (generalized) tries and 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. It turns out that this height also represents the number of random questions required to recognize n distinct objects. We shall identiry three natural regions of the height distribution~. For tries, in the region where most of the probability mas~ is concentrated, the asymptotic distribution is of extreme value type (Le., double exponential distribution). Surprisinglyenough, the height of the PATIUCIA trie behaves quite differently in this region; It exhibits an exponential of a Gaussian distribution (with an oscillating tcnn) around the most probable value k l = llog2 n + /21og2 n ~ ~J+l. In fact, the asymptotic distribution of PATIUCIA height concentrates on one or two points. For most n all the ma."iS is concentrated at kI , however, there exist subsequences of n such that the mass is on the two points hI 1 and hI, or kI and hi + 1. We derive these 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 method. We present also some numerical verification of our results.