The Expected Profile of Digital Search Trees ∗ December 10 , 2009

A digital search tree (DST) is a fundamental data structure on words that finds myriad of applications from the popular Lempel-Ziv’78 data compression scheme to distributed hash tables. It is a digital tree in which strings (keys, words) are stored directly in (internal) nodes. The profile of a DST measures the number of nodes at the same distance from the root; it is a function of the number of stored strings and the distance from the root. Most parameters of DST (e.g., depth, height, fill-up) can be expressed in terms of the profile. We make here the first step towards deciphering an asymptotic behavior of the DST profile, a long standing open problem in the analysis of algorithms and combinatorics. Throughout we assume that strings stored in DST are generated by a memoryless source. We present a precise analysis of the average profile described by a sophisticated recurrence equation that we solve by analytic methods. This analysis is surprisingly demanding but once it is carried out it reveals unusually intriguing and interesting behavior. The average profile undergoes several phase transitions when moving from the root to the longest path: at first it resembles a full tree until it abruptly starts growing polynomially and oscillating in this range. These results are derived by methods of analytic combinatorics such as generating functions, Mellin transform, Poissonization and de-Poissonization, the saddle-point method, singularity analysis and uniform asymptotic analysis.