Analysis of the average depth in a suffix tree under a Markov model

Abstract

International audience In this report, we prove that under a Markovian model of order one, the average depth of suffix trees of index n is asymptotically similar to the average depth of tries (a.k.a. digital trees) built on n independent strings. This leads to an asymptotic behavior of $(\log{n})/h + C$ for the average of the depth of the suffix tree, where $h$ is the entropy of the Markov model and $C$ is constant. Our proof compares the generating functions for the average depth in tries and in suffix trees; the difference between these generating functions is shown to be asymptotically small. We conclude by using the asymptotic behavior of the average depth in a trie under the Markov model found by Jacquet and Szpankowski ([JaSz91]).

Average Size of a Suffix Tree for Markov Sources

“…Thus we need to estimate d n (w) and q n (w) for all w ∈ A * . We denote B k the set of words of length k that do not overlap with themselves over more than k/2 symbols (see [9,5,3] for more precise definition). To be precise w ∈ A k − B k if there exist j > k/2 and v ∈ A j and (u 1 , u 2 ) ∈ A k−j such that w = u 1 v = vu 2 .…”

“…To be precise w ∈ A k − B k if there exist j > k/2 and v ∈ A j and (u 1 , u 2 ) ∈ A k−j such that w = u 1 v = vu 2 . This set plays a fundamental role in the analysis and it is already proven in [3] that…”

“…They find myriad of applications in computer science and telecommunications, most notably in algorithms on strings, data compressions (Lempel-Ziv'77 scheme), and codes. Despite this, little is still known about their typical behaviors for general probabilistic models (see [5,1,3]).…”

“…An analysis of the depth in a Markovian trie has been presented earlier in [12]. A rigorous analysis of the depth of suffix tree was first presented in [5] for memoryless sources, and then extended in [3] to Markov sources. We should point out that depth grows like O(log n) which makes the analysis manageable.…”

“…The proof of the convergence of the average size of the suffix tree to the average size of the trie borrows many fundamental elements of the depth analysis in [3], for example the term q n (w) (see next section), but the extension of the depth analysis to the size analysis require the introduction of a new term d n (w) which has non trivial properties. The analysis of average size of the trie in a Markovian model has been made by several author before but surprisingly we could not find a clear statement about the periodic case.…”