Two notes on genome rearrangement.

A central problem in genome rearrangement is finding a most parsimonious rearrangement scenario using certain rearrangement operations. An important problem of this type is sorting a signed genome by reversals and translocations (SBRT). Hannenhalli and Pevzner presented a duality theorem for SBRT which leads to a polynomial time algorithm for sorting a multi-chromosomal genome using a minimum number of reversals and translocations. However, there is one case for which their theorem and algorithm fail. We describe that case and suggest a correction to the theorem and the polynomial algorithm. The solution of SBRT uses a reduction to the problem of sorting a signed permutation by reversals (SBR). The best extant algorithms for SBR require quadratic time. The common approach to solve SBR is by finding a safe reversal using the overlap graph or the interleaving graph of a permutation. We describe a family of signed permutations which proves a quadratic lower bound on the number of affected vertices in the overlap/interleaving graph during any optimal sorting scenario. This implies, in particular, an Omega(n3) lower bound for Bergeron's algorithm.