Minimal Absent Words in a Sliding Window and Applications to On-Line Pattern Matching

摘要

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. in Inf Process Lett 67:111-117, 1998; Belazzougui et al. in ESA 8125:133-144, 2013; Barton et al. in BMC Bioinform 15:388, 2014). Minimal absent words are used for data compression (Crochemore et al. in Proc IEEE 88:1756-1768, 2000, Ota and Morita in Theoret Comput Sci 526:108-119, 2014) and for alignment-free sequence comparison by utilizing a metric based on minimal absent words (Chairungsee and Crochemore in Theoret Comput Sci 450:109-116, 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. in Bioinformatics 31:2421-2425, 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.