Expected External Profile of PATRICIA Tries

Abstract

We consider PATRICIA tries on n random binary strings generated by a memoryless source with parameter p ≥ 1/2. For both the symmetric (p = 1/2) and asymmetric cases, we analyze asymptotics of the expected value of the external profile at level k = k(n), defined to be the number of leaves at level k. We study three natural ranges of k with respect to n. For k bounded, the mean profile decays exponentially with respect to n. For k growing logarithmically with n, the parameter exhibits polynomial growth in n, with some periodic fluctuations. Finally, for k = Θ(n), we see super-exponential decay, again with periodic fluctuations. Our derivations rely on analytic techniques, including Mellin transforms, analytic depoissonization, and the saddle point method. To cover wider ranges of k and n and provide more intuitive insights, we also use methods of applied mathematics, including asymptotic matching and linearization.