An algorithm for collapsing sign alternating sequences of real numbers

AbstractA table with two rows and n columns may be thought of as two vectors with n components. The distance between the two rows then corresponds to the norm of the difference between the rows. We examine the problem of how to collapse the adjacent columns of the table while keeping the norm of the difference as large as possible. First a stepwise algorithm is given which achieves this end with respect to the norm of the vector of differences. After proving the optimality of the stepwise solution we extend the result to the norm which arises from minimizing the number of persons misclassified. The same algorithm suffices

Representing and decomposing genomic structural variants as balanced integer flows on sequence graphs

The study of genomic variation has provided key insights into the functional role of mutations. Predominantly, studies have focused on single nucleotide variants (SNV), which are relatively easy to detect and can be described with rich mathematical models. However, it has been observed that genomes are highly plastic, and that whole regions can be moved, removed or duplicated in bulk. These structural variants (SV) have been shown to have significant impact on the phenotype, but their study has been held back by the combinatorial complexity of the underlying models. We describe here a general model of structural variation that encompasses both balanced rearrangements and arbitrary copy-numbers variants (CNV). In this model, we show that the space of possible evolutionary histories that explain the structural differences between any two genomes can be sampled ergodically