The computation of genomic distances has been a very active field of computational comparative genomics over the past 25 years. Substantial results include the polynomial-time computability of the inversion distance by Hannenhalli and Pevzner in 1995 and the introduction of the double cut and join distance by Yancopoulos et al. in 2005. Both results, however, rely on the assumption that the genomes under comparison contain the same set of unique markers (syntenic genomic regions, sometimes also referred to as genes). In 2015, Shao et al. relax this condition by allowing for duplicate markers in the analysis. This generalized version of the genomic distance problem is NP-hard, and they give an integer linear programming (ILP) solution that is efficient enough to be applied to real-world datasets. A restriction of their approach is that it can be applied only to balanced genomes that have equal numbers of duplicates of any marker. Therefore, it still needs a delicate preprocessing of the input data in which excessive copies of unbalanced markers have to be removed. In this article, we present an algorithm solving the genomic distance problem for natural genomes, in which any marker may occur an arbitrary number of times. Our method is based on a new graph data structure, the multi-relational diagram, that allows an elegant extension of the ILP by Shao et al. to count runs of markers that are under- or over-represented in one genome with respect to the other and need to be inserted or deleted, respectively. With this extension, previous restrictions on the genome configurations are lifted, for the first time enabling an uncompromising rearrangement analysis. Any marker sequence can directly be used for the distance calculation. The evaluation of our approach shows that it can be used to analyze genomes with up to a few 10,000 markers, which we demonstrate on simulated and real data.

The computation of genomic distances has been a very active field of computational comparative genomics over the last 25 years. Substantial results include the polynomial-time computability of the inversion distance by Hannenhalli and Pevzner in 1995 and the introduction of the double-cut and join (DCJ) distance by Yancopoulos, Attie and Friedberg in 2005. Both results, however, rely on the assumption that the genomes under comparison contain the same set of unique markers (syntenic genomic regions, sometimes also referred to as genes). In 2015, Shao, Lin and Moret relax this condition by allowing for duplicate markers in the analysis. This generalized version of the genomic distance problem is NP-hard, and they give an ILP solution that is efficient enough to be applied to real-world datasets. A restriction of their approach is that it can be applied only to balanced genomes, that have equal numbers of duplicates of any marker. Therefore it still needs a delicate preprocessing of the input data in which excessive copies of unbalanced markers have to be removed. In this paper we present an algorithm solving the genomic distance problem for natural genomes, in which any marker may occur an arbitrary number of times. Our method is based on a new graph data structure, the multi-relational diagram, that allows an elegant extension of the ILP by Shao, Lin and Moret to count runs of markers that are under-or over-represented in one genome with respect to the other and need to be inserted or deleted, respectively. With this extension, previous restrictions on the genome configurations are lifted, for the first time enabling an uncompromising rearrangement analysis. Any marker sequence can directly be used for the distance calculation.The evaluation of our approach shows that it can be used to analyze genomes with up to a few ten thousand markers, which we demonstrate on simulated and real data. Source code and test data are available from https://gitlab.ub.uni-bielefeld.de/gi/ding. ······ ······ ······ ······ ······ ······ ······ ······ J J J J ······ ······ ········· ········· ········· ········· ········· ········· ········· ········· ········· ········· ········· J J J J ········· ········· ········· ········· J J J J r r r r r r r r r r r r r r r r

Genomic regions under positive selection harbour variation linked for example to adaptation. Most tools for detecting positively selected variants have computational resource requirements rendering them impractical on population genomic datasets with hundreds of thousands of individuals or more. We have developed and implemented an efficient haplotype-based approach able to scan large datasets and accurately detect positive selection. We achieve this by combining a pattern matching approach based on the positional Burrows-Wheeler transform with model-based inference which only requires the evaluation of closed-form expressions. We evaluate our approach with simulations, and find it to be both sensitive and specific. The computational resource requirements quantified using UK Biobank data indicate that our implementation is scalable to population genomic datasets with millions of individuals. Our approach may serve as an algorithmic blueprint for the era of ‘big data’ genomics: a combinatorial core coupled with statistical inference in closed form.

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