A uniform analysis of trie structures that store prefixing-keys with application to doubly-chained prefixing-tries

Abstract

Tries are data structures for storing sets where each element is represented by a key that can be viewed as a string of characters over a finite alphabet. These structures have been extensively studied and analyzed under several probability models. All of these models, however, preclude the occurrence of sets in which the key of one element is a prefix of that of another — such a key is called a prefixing-key.

This paper presents an average case analysis of several trie varieties, which we generically called doubly-chained prefixingtries, for representing sets with possible prefixing-keys. The underlying probability model, which we call the prefix model, P _h,n,m assumes as equally likely all n-element sets whose keys are composed of at most h characters from a fixed alphabet of size m. For each of the trie varieties analyzed, we derive exact formulas for the expected space required to store such a set, and the average time required to retrieve an element given its key, as functions of h, n, and m.

Our approach to the analysis is of interest in its own right. It provides a unifying framework for computing the expectations of a wide class of random variables with respect to the prefix model. This class includes the cost functions of the trie varieties analyzed here.

The research of this author was supported in part by the National Science Foundation under Grant CCR-9010445, and CCR-9410592.

The research of this author was supported in part by the National Science Foundation under Grant IRI-9117153.