Asymptotic Behavior of the Height in a Digital Search Tree and the Longest Phrase of the Lempel-Ziv Scheme

Wojciech Szpankowskit Department of Computer Science Purdue University W. Lafayette, IN 47907 U.S.A. spa~cs.purdue.edu We study the height of a digital search tree (DST in short) built from n random strings generated by an unbiased memoryless source (i.e., all symbols are equally likely). We shall argue that the height of such a tree is equivalent to the length of the longest phrase in the Lempel-Ziv parsing scheme that partitions a random sequence into n phrases. We also analyze the longest phrase in the Lempel-Ziv scheme in which a string of fixed length m is parsed into a random number of phrases. In the course of our analysis, we shall identify four natural regions of the height distribution and characterize them asymptotically for large n. In particular, for the region where most of the probability mass is concentrated, the asymptotic distribution of the height exhibits an exponen~ tial of a Gaussian distribution (with an oscillating term) around the most probable value k1 = llog2 n + J2log2n log2(J210g2n) + lo~2 ~J + 1. More precisely, we shall prove that the asymptotic distribution of a digital search tree is either concentrated on the one point k1 or the two points k1 -1 and kb which actually proves (slightly modified) Kesten's conjecture quoted in [2J. FinallYl we compare our findings for DST with the asymptotic distributions of the height (recently obtained by us) for other digital trees such as tries and PATRlCIA tries. We derive these results by a combination of analytic methods such as generating functions, Laplace transform, the saddle point method and ideas of applied mathematics such as linearization, asymptotic matching and the WKB method. We also present detailed numerical verification of our results. Key Worns: Digital search trees, Lempel-Ziv algorithm, height distribution, longest phrase distribution, Laplace transform, saddle point method, matched asymptotics, linearization, WKB method, elliptic theta function. -This work was supported by DOE Grant DE-FG02-!J6ER2516B. 'The work of this author was supported by NSF Grants NCR-9415491 and CCR-!J804760.