Profiles of PATRICIA Tries

Abstract

A PATRICIA trie is a trie in which non-branching paths are compressed. The external profile $$B_{n,k}$$ , defined to be the number of leaves at level k of a PATRICIA trie on n nodes, is an important “summarizing” parameter, in terms of which several other parameters of interest can be formulated. Here we derive precise asymptotics for the expected value and variance of $$B_{n,k}$$ , as well as a central limit theorem with error bound on the characteristic function, for PATRICIA tries on n infinite binary strings generated by a memoryless source with bias $$p > 1/2$$ for $$k\sim \alpha \log n$$ with $$\alpha \in (1/\log (1/q) + \varepsilon , 1/\log (1/p) - \varepsilon )$$ for any fixed $$\varepsilon > 0$$ . In this range, $$ {\mathbb {E}}[B_{n,k}] = \varTheta ( {\mathrm {Var}}[B_{n,k}])$$ , and both are of the form $$\varTheta (n^{\beta (\alpha )}/\sqrt{\log n})$$ , where the $$\varTheta $$ hides bounded, periodic functions in $$\log n$$ whose Fourier series we explicitly determine. The compression property leads to extra terms in the Poisson functional equations for the profile which are not seen in tries or digital search trees, resulting in Mellin transforms which are only implicitly given in terms of the moments of $$B_{m,j}$$ for various m and j. Thus, the proofs require information about the profile outside the main range of interest. Our derivations rely on analytic techniques, including Mellin transforms, analytic de-Poissonization, the saddle point method, and careful bounding of complex functions.