On Sorting by Flanked Transpositions.

Transposition is a well-known genome rearrangement event that switches two consecutive segments on a genome. The problem of sorting permutations by transpositions has attracted a great amount of interest since it was introduced by Bafna and Pevzner in 1995. However, empirical evidence has reported that, in many genomes, the participation of repeat segments is inevitable during genome evolution and the breakpoints where a transposition occurs are most likely accompanied by a triple of repeated segments. For example, a transposition will transform r x r y z r into r y z r x r , where r is a relative short repeat appearing three times and x and y are long segments involved in the transposition. For this transposition event, the neighbors of segments x and y remain the same before and after the transposition. This type of transposition is called flanked transposition. In this paper, we investigate the problem of sorting by flanked transpositions, which requires a series of flanked transpositions to transform one genome into another. First, we present an O ( n ) expected running time algorithm to determine if a genome can be transformed into the other genome by a series of flanked transposition for a special case, where each adjacency (roughly two neighbors of two element in the genome) appears once in both input genomes. We then extend the decision algorithm to work for the general case with the same expected running time O ( n ). Finally, we show that the new version, sorting by minimum number of flanked transpositions is also NP-hard.