Patricia tries again revisited

The Patricia trie is a simple modification of a regular trie. By eliminating unary branching nodes, the Patricia achieves better performance than regular tries. However, the question is: how much on the average is the Patricia better? This paper offers a thorough answer to this question by considering some statistics of the number of nodes examined in a successful search and an unsuccessful search in the Patricia tries. It is shown that for the Patricia containing n records the average of the successful search length S n asymptotically becomes 1/ h 1 · ln n + O (1), and the variance of S n is either var S n = c · ln n + 0 (1) for an asymmetric Patricia or var S n = 0 (1) for a symmetric Patricia, where h 1 is the entropy of the alphabet over which the Patricia is built and c is an explicit constant. Higher moments of S n are also assessed. The number of nodes examined in an unsuccessful search U n is studied only for binary symmetric Patricia tries. We prove that the m th moment of the unsuccessful search length EU m n satisfies lim n →∞ EU m n /log m 2 n = 1, and the variance of U n is var U n = 0.87907. These results suggest that Patricia tries are very well balanced trees in the sense that a random shape of Patriciatries resembles the shape of complete trees that are ultimately balanced trees.