Superstrings with multiplicities

A superstring of a set of words P = s1, · · · , sp is a string that contains each word of P as substring. Given P, the well known Shortest Linear Superstring problem (SLS), asks for a shortest superstring of P. In a variant of SLS, called Multi-SLS, each word si comes with an integer m(i), its multiplicity, that sets a constraint on its number of occurrences, and the goal is to find a shortest superstring that contains at least m(i) occurrences of si. Multi-SLS generalizes SLS and is obviously as hard to solve, but it has been studied only in special cases (with words of length 2 or with a fixed number of words). The approximability of Multi-SLS in the general case remains open. Here, we study the approximability of Multi-SLS and that of the companion problem Multi-SCCS, which asks for a shortest cyclic cover instead of shortest superstring. First, we investigate the approximation of a greedy algorithm for maximizing the compression offered by a superstring or by a cyclic cover: the approximation ratio is 1/2 for Multi-SLS and 1 for Multi-SCCS. Then, we exhibit a linear time approximation algorithm, Concat-Greedy, and show it achieves a ratio of 4 regarding the superstring length. This demonstrates that for both measures Multi-SLS belongs to the class of APX problems. © 2018 Yoshifumi Sakai; licensed under Creative Commons License CC-BY.Peer reviewe

Linking BWT and XBW via Aho-Corasick automaton : Applications to run-length encoding

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Convergence of the Number of Period Sets in Strings

Consider words of length n. The set of all periods of a word of length n is a subset of {0, 1, 2, . . ., n−1}. However, any subset of {0, 1, 2, . . ., n−1} is not necessarily a valid set of periods. In a seminal paper in 1981, Guibas and Odlyzko proposed to encode the set of periods of a word into an n long binary string, called an autocorrelation, where a one at position i denotes the period i. They considered the question of recognizing a valid period set, and also studied the number of valid period sets for strings of length n, denoted κn. They conjectured that ln(κn) asymptotically converges to a constant times ln2(n). Although improved lower bounds for ln(κn)/ln2(n) were proposed in 2001, the question of a tight upper bound has remained open since Guibas and Odlyzko’s paper. Here, we exhibit an upper bound for this fraction, which implies its convergence and closes this longstanding conjecture. Moreover, we extend our result to find similar bounds for the number of correlations: a generalization of autocorrelations which encodes the overlaps between two strings

Practical lower and upper bounds for the Shortest Linear Superstring

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DNA Slippage Occurs at Microsatellite Loci without Minimal Threshold Length in Humans: A Comparative Genomic Approach

The dynamics of microsatellite, or short tandem repeats (STRs), is well documented for long, polymorphic loci, but much less is known for shorter ones. For example, the issue of a minimum threshold length for DNA slippage remains contentious. Model-fitting methods have generally concluded that slippage only occurs over a threshold length of about eight nucleotides, in contradiction with some direct observations of tandem duplications at shorter repeated sites. Using a comparative analysis of the human and chimpanzee genomes, we examined the mutation patterns at microsatellite loci with lengths as short as one period plus one nucleotide. We found that the rates of tandem insertions and deletions at microsatellite loci strongly deviated from background rates in other parts of the human genome and followed an exponential increase with STR size. More importantly, we detected no lower threshold length for slippage. The rate of tandem duplications at unrepeated sites was higher than expected from random insertions, providing evidence for genome-wide action of indel slippage (an alternative mechanism generating tandem repeats). The rate of point mutations adjacent to STRs did not differ from that estimated elsewhere in the genome, except around dinucleotide loci. Our results suggest that the emergence of STR depends on DNA slippage, indel slippage, and point mutations. We also found that the dynamics of tandem insertions and deletions differed in both rates and size at which these mutations take place. We discuss these results in both evolutionary and mechanistic terms