BackgroundAn absent word of a word y of length n is a word that does not occur in y. It is a minimal absent word if all its proper factors occur in y. Minimal absent words have been computed in genomes of organisms from all domains of life; their computation also provides a fast alternative for measuring approximation in sequence comparison. There exists an -time and -space algorithm for computing all minimal absent words on a fixed-sized alphabet based on the construction of suffix automata (Crochemore et al., 1998). No implementation of this algorithm is publicly available. There also exists an -time and -space algorithm for the same problem based on the construction of suffix arrays (Pinho et al., 2009). An implementation of this algorithm was also provided by the authors and is currently the fastest available.ResultsOur contribution in this article is twofold: first, we bridge this unpleasant gap by presenting an -time and -space algorithm for computing all minimal absent words based on the construction of suffix arrays; and second, we provide the respective implementation of this algorithm. Experimental results, using real and synthetic data, show that this implementation outperforms the one by Pinho et al. The open-source code of our implementation is freely available at http://github.com/solonas13/maw.ConclusionsClassical notions for sequence comparison are increasingly being replaced by other similarity measures that refer to the composition of sequences in terms of their constituent patterns. One such measure is the minimal absent words. In this article, we present a new linear-time and linear-space algorithm for the computation of minimal absent words based on the suffix array.
We consider the Combinatorial RNA Design problem, a minimal instance of RNA design where one must produce an RNA sequence that adopts a given secondary structure as its minimal free-energy structure. We consider two free-energy models where the contributions of base pairs are additive and independent: the purely combinatorial Watson-Crick model, which only allows equallycontributing A − U and C − G base pairs, and the real-valued Nussinov-Jacobson model, which associates arbitrary energies to A − U, C − G and G − U base pairs.We first provide a complete characterization of designable structures using restricted alphabets and, in the four-letter alphabet, provide a complete characterization for designable structures without unpaired bases. When unpaired bases are allowed, we characterize extensive classes of (non-)designable structures, and prove the closure of the set of designable structures under the stutter operation. Membership of a given structure to any of the classes can be tested in Θ(n) time, including the generation of a solution sequence for positive instances.Finally, we consider a structure-approximating relaxation of the design, and provide a Θ(n) algorithm which, given a structure S that avoids two trivially non-designable motifs, transforms S into a designable structure constructively by adding at most one base-pair to each of its stems.
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