The Average Profile of Suffix Trees

Abstract

The internal profile of a tree structure denotes the number of internal nodes found at a specific level of the tree. Similarly, the external profile denotes the number of leaves on a level. The profile is of great interest because of its intimate connection to many other parameters of trees. For instance, the depth, fill-up level, height, path length, shortest path, and size of trees can each be interpreted in terms of the profile. The current study is motivated by the work of Park et al. [22], which was a comprehensive study of the profile of tries constructed from independent strings (also, each string generated by a memoryless source). In the present paper, however, we consider suffix trees, which are constructed from suffixes of a common string. The dependency between suffixes demands a careful, intricate treatment of overlaps in words. We precisely analyze the average internal and external profiles of suffix trees generated by a memoryless source. We utilize combinatorics on words (in particular, autocorrelation, i.e., the degree to which a word overlaps with itself) generating functions, singularity analysis, and the Mellin transform. We make comparisons of the average profile of suffix trees to the average profile of tries constructed from independent strings. We emphasize that our methods are extensible to higher moments. The present report describes the first moment of both the internal and external profiles of suffix trees.