Alignment-free sequence comparison using absent words

Abstract

Sequence comparison is a prerequisite to virtually all comparative genomic analyses. It is often realised by sequence alignment techniques, which are computationally expensive. This has led to increased research into alignment-free techniques, which are based on measures referring to the composition of sequences in terms of their constituent patterns. These measures, such as q-gram distance, are usually computed in time linear with respect to the length of the sequences. In this paper, we focus on the complementary idea: how two sequences can be efficiently compared based on information that does not occur in the sequences. A word is an absent word of some sequence if it does not occur in the sequence. An absent word is minimal if all its proper factors occur in the sequence. Here we present the first linear-time and linear-space algorithm to compare two sequences by considering all their minimal absent words. In the process, we present results of combinatorial interest, and also extend the proposed techniques to compare circular sequences. We also present an algorithm that, given a word x of length n, computes the largest integer for which all factors of x of that length occur in some minimal absent word of x in time and space O(n). Finally, we show that the known asymptotic upper bound on the number of minimal absent words of a word is tight.