Minimal absent words in a sliding window & applications to on-line pattern matching

An absent (or forbidden) word of a word y is a word that does not occur in y. It is then called minimal if all its proper factors occur in y. There exist linear-time and linear-space algorithms for computing all minimal absent words of y (Crochemore et al., 1998, Belazzougui et al., 2013, Barton et al., 2014). Minimal absent words are used for data compression (Crochemore et al., 2000, Ota and Morita, 2014) and for alignment-free sequence comparison by utilizing a metric based on minimal absent words (Chairungsee and Crochemore, 2012). They are also used in molecular biology; for instance, three minimal absent words of the human genome were found to play a functional role in a coding region in Ebola virus genomes (Silva et al., 2015). In this article we introduce a new application of minimal absent words for on-line pattern matching. Specifically, we present an algorithm that, given a pattern x and a text y, computes the distance between x and every window of size |x| on y. The running time is O(σ|y|), where σ is the size of the alphabet. Along the way, we show an O(σ|y|)-time and O(σ|x|)-space algorithm to compute the minimal absent words of every window of size |x| on y, together with some new combinatorial insight on minimal absent words.