On avoided words, absent words, and their application to biological sequence analysis

BACKGROUND: The deviation of the observed frequency of a word w from its expected frequency in a given sequence x is used to determine whether or not the word is avoided. This concept is particularly useful in DNA linguistic analysis. The value of the deviation of w, denoted by [Formula: see text] , effectively characterises the extent of a word by its edge contrast in the context in which it occurs. A word w of length [Formula: see text] is a [Formula: see text] -avoided word in x if [Formula: see text] , for a given threshold [Formula: see text] . Notice that such a word may be completely absent from x. Hence, computing all such words naïvely can be a very time-consuming procedure, in particular for large k. RESULTS: In this article, we propose an [Formula: see text] -time and [Formula: see text] -space algorithm to compute all [Formula: see text] -avoided words of length k in a given sequence of length n over a fixed-sized alphabet. We also present a time-optimal [Formula: see text] -time algorithm to compute all [Formula: see text] -avoided words (of any length) in a sequence of length n over an integer alphabet of size [Formula: see text] . In addition, we provide a tight asymptotic upper bound for the number of [Formula: see text] -avoided words over an integer alphabet and the expected length of the longest one. We make available an implementation of our algorithm. Experimental results, using both real and synthetic data, show the efficiency and applicability of our implementation in biological sequence analysis. CONCLUSIONS: The systematic search for avoided words is particularly useful for biological sequence analysis. We present a linear-time and linear-space algorithm for the computation of avoided words of length k in a given sequence x. We suggest a modification to this algorithm so that it computes all avoided words of x, irrespective of their length, within the same time complexity. We also present combinatorial results with regards to avoided words and absent words.