A linear time algorithm for the inversion median problem in circular bacterial genomes

In the median problem, we are given a distance or dissimilarity measure d  , three genomes G1,G2G1,G2, and G3G3, and we want to find a genome G   (a median) such that the sum ∑i=13d(G,Gi) is minimized. The median problem is a special case of the multiple genome rearrangement problem, where one wants to find a phylogenetic tree describing the most “plausible” rearrangement scenario for multiple species. The median problem is NP-hard for both the breakpoint and the reversal distance. To the best of our knowledge, there is no approach yet that takes biological constraints on genome rearrangements into account. In this paper, we make use of the fact that in circular bacterial genomes the predominant mechanism of rearrangement are inversions that are centered around the origin or the terminus of replication and single gene inversions. These constraints simplify the median problem significantly. More precisely, we show that the median problem for the reversal distance can be solved in linear time for circular bacterial genomes.