A Generalized Suffix Tree and its (Un)expected Asymptotic Behaviors

Suffix trees find several applications in computer science and telecommunications, most notably in algorithms on strings, data compressions, and codes. Despite this, very little is known about their typical behaviors. In a probabilistic framework, a family of suffix trees—further called b-suffix trees—built from the first n suffixes of a random word is considered. In this family a noncompact suffix tree (i.e., such that every edge is labeled by a single symbol) is represented by $b = 1$, and a compact suffix tree (i.e., without unary nodes) is asymptotically equivalent to $b \to \infty $ as $n \to \infty $. Several parameters of b-suffix trees are studied, namely, the depth of a given suffix, the depth of insertion, the height and the shortest feasible path. Some new results concerning typical (i.e., almost sure) behaviors of these parameters are established. These findings are used to obtain several insights into certain algorithms on words, molecular biology, and universal data compression schemes.