The Problem of Chromosome Reincorporation in DCJ Sorting and Halving

We study two problems in the double cut and join (DCJ) model: sorting – transforming one multilinear genome into another and halving – transforming a duplicated genome into a perfectly duplicated one. The DCJ model includes rearrangement operations such as reversals, translocations, fusions and fissions. We can also mimic transpositions or block interchanges by two operations: we extract an appropriate segment of a chromosome, creating a temporary circular chromosome, and in the next step we reinsert it in its proper place. Existing linear-time algorithms solving both problems ignore the constraint of reincorporating the temporary circular chromosomes immediately after their creation. For the restricted sorting problem only a quadratic algorithm was known, whereas the restricted halving problem was stated as open by Tannier, Zheng, and Sankoff. In this paper we address this constraint and show how to solve the problem of sorting in O(n logn) time and halving in O(n) time.