Some results on tries with adaptive branching

Abstract

We study a modification of digital trees (or tries) with adaptive multi-digit branching. Such tries can dynamically adjust degrees of their nodes by choosing the number of digits to be processed per each lookup. While we do not specify any particular method for selecting the degrees of nodes, we assume that such selection can be accomplished by examining the number of strings remaining in each sub-tree, and/or estimating parameters of the input distribution. We call this class of digital trees adaptive multi-digit tries (or AMD-tries) and provide a preliminary analysis of their expected behavior in a memoryless model. We establish the following results: 1) there exist AMD-tries attaining a constant (O(1)) expected time of a successful search; 2) there exist AMD-tries consuming a linear (O(n), n is the number of strings inserted) amount of space; 3) both constant search time and linear space usage can be attained if the (memoryless) source is symmetric. We accompany our analysis with a brief survey of several known types of adaptive trie structures, and show how our analysis extends (and/or complements) the previous results.