On overabundant words and their application to biological sequence analysis

摘要

The observed frequency of the longest proper prefix, the longest proper suffix, and the longest infix of a word w in a given sequence x can be used for classifying w as avoided or overabundant. The definitions used for the expectation and deviation of w in this statistical model were described and biologically justified by Brendel et al. (1986) [1]. We have very recently introduced a time-optimal algorithm for computing all avoided words of a given sequence over an integer alphabet (2017) [2]. In this article, we extend this study by presenting an O(n)-time and O(n)-space algorithm for computing all overabundant words in a sequence x of length n over an integer alphabet. Our main result is based on a new non-trivial combinatorial property of the suffix tree tau of x: the number of distinct factors of x whose longest infix is the label of an explicit node of tau is no more than 3n - 4. We further show that the presented algorithm is time-optimal by proving that O(n) is a tight upper bound for the number of overabundant words. Finally, we present experimental results, using both synthetic and real data, which justify the effectiveness and efficiency of our approach in practical terms. (C) 2018 Elsevier B.V. All rights reserved.