Maximal Motif Discovery in a Sliding Window

Abstract

Motifs are relatively short sequences that are biologically significant, and their discovery in molecular sequences is a well-researched subject. A don’t care is a special letter that matches every letter in the alphabet. Formally, a motif is a sequence of letters of the alphabet and don’t care letters. A motif $\tilde{m}_{d,k}$ that occurs at least k times in a sequence is maximal if it cannot be extended (to the left or right) nor can it be specialised (that is, its $d' \le d$ don’t cares cannot be replaced with letters from the alphabet) without reducing its number of occurrences. Here we present a new dynamic data structure, and the first on-line algorithm, to discover all maximal motifs in a sliding window of length $\ell$ on a sequence x of length n in $\mathcal {O}(nd\ell + d\lceil \frac{\ell }{w}\rceil \cdot \sum _{i = \ell }^{n-1} |{\textsc {diff}}_{i-1}^{i}|)$ time, where w is the size of the machine word and ${\textsc {diff}}_{i-1}^{i}$ is the symmetric difference of the sets of occurrences of maximal motifs at $x[i-\ell \mathinner {.\,.}i-1]$ and at $x[i-\ell +1 \mathinner {.\,.}i]$.