Transforming cabbage into turnip: Polynomial algorithm for sorting signed permutations by reversals

Genomes frequently evolve by reversals rho(i,j) that transform a gene order pi(1) ..., pi(i)pi(i+1) ... pi(j-1)pi(j) ... pi(n) into pi(1) ... pi(i)pi( j-1) ... pi(i+1)pi(j) ... pi(n). Reversal distance between permutations pi and sigma is the minimum number of reversals to transform pi into sigma Ana lysis of genome rearrangements in molecular biology started in the late 193 0's, when Dobzhansky and Sturtevant published a milestone paper presenting a rearrangement scenario with 17 inversions between the species of Drosophi la. Analysis of genomes evolving by inversions leads to a combinatorial pro blem of sorting by reversals studied in detail recently. We study sorting o f signed permutations by reversals, a problem that adequately models rearra ngements in small genomes like chloroplast or mitochondrial DNA. The previo usly suggested approximation algorithms for sorting signed permutations by reversals compute the reversal distance between permutations with an astoni shing accuracy for both simulated and biological data. We prove a duality t heorem explaining this intriguing performance and show that there exists a "hidden" parameter that allows one to compute the reversal distance between signed permutations in polynomial time.