Sorting by weighted reversals, transpositions, and inverted transpositions.

During evolution, genomes are subject to genome rearrangements that alter the ordering and orientation of genes on the chromosomes. If a genome consists of a single chromosome (like mitochondrial, chloroplast, or bacterial genomes), the biologically relevant genome rearrangements are (1) inversions--also called reversals--where a section of the genome is excised, reversed in orientation, and reinserted and (2) transpositions, where a section of the genome is excised and reinserted at a new position in the genome; if this also involves an inversion, one speaks of an inverted transposition. To reconstruct ancient events in the evolutionary history of organisms, one is interested in finding an optimal sequence of genome rearrangements that transforms a given genome into another genome. It is well known that this problem is equivalent to the problem of "sorting" a signed permutation into the identity permutation. In this paper, we provide a 1.5-approximation algorithm for sorting by weighted reversals, transpositions and inverted transpositions for biologically realistic weights.