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 <italic>successful search</italic> and an <italic>unsuccessful search</italic> in the Patricia tries. It is shown that for the Patricia containing <italic>n</italic> records the average of the successful search length <italic>S<subscrpt>n</subscrpt></italic> asymptotically becomes 1/<italic>h</italic><subscrpt>1</subscrpt> · ln <italic>n</italic> + <italic>O</italic>(1), and the variance of <italic>S<subscrpt>n</subscrpt></italic> is either var <italic>S<subscrpt>n</subscrpt></italic> = <italic>c</italic> · ln <italic>n</italic> + <italic>0</italic>(1) for an asymmetric Patricia or var <italic>S<subscrpt>n</subscrpt></italic> = <italic>0</italic>(1) for a symmetric Patricia, where <italic>h</italic><subscrpt>1</subscrpt> is the entropy of the alphabet over which the Patricia is built and <italic>c</italic> is an explicit constant. Higher moments of <italic>S<subscrpt>n</subscrpt></italic> are also assessed. The number of nodes examined in an unsuccessful search <italic>U<subscrpt>n</subscrpt></italic> is studied only for binary symmetric Patricia tries. We prove that the <italic>m</italic>th moment of the unsuccessful search length <italic>EU<supscrpt>m</supscrpt><subscrpt>n</subscrpt></italic> satisfies lim<subscrpt><italic>n</italic>→∞</subscrpt> <italic>EU<supscrpt>m</supscrpt><subscrpt>n</subscrpt></italic>/log<supscrpt><italic>m</italic></supscrpt><subscrpt>2</subscrpt><italic>n</italic> = 1, and the variance of <italic>U<subscrpt>n</subscrpt></italic> is var <italic>U<subscrpt>n</subscrpt></italic> = 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.