A Simpler 1.5-Approximation Algorithm for Sorting By Transpositions

An important problem in genome rearrangements is sorting permutations by transpositions. Its complexity is still open, and two rather complicated 1.5-approximation algorithms for sorting linear permutations are known (Bafna and Pevzner, 96 and Christie, 98). In this paper, we observe that the problem of sorting circular permutations by transpositions is equivalent to the problem of sorting linear permutations by transpositions. Hence, all algorithms for sorting linear permutations by transpositions can be used to sort circular permutations. Our main result is a new 1.5-approximation algorithm, which is considerably simpler than the previous ones, and achieves running time which is equal to the best known. Moreover, the analysis of the algorithm is significantly less involved, and provides a good starting point for studying related open problems

A 1.375-Approximation Algorithm for Sorting by Transpositions

Sorting permutations by transpositions is an important problem in genome rearrangements. A transposition is a rearrangement operation in which a segment is cut out of the permutation and pasted in a di#erent location. The complexity of this problem is still open and it has been a ten-year-old open problem to improve the best known 1.5-approximation algorithm. In this paper we provide a 1.375-approximation algorithm for sorting by transpositions. The algorithm is based on a new upper bound on the diameter of 3-permutations. In addition, we present some new results regarding the transposition diameter: We improve the lower bound for the transposition diameter of the symmetric group, and determine the exact transposition diameter of 2-permutations and simple permutations

'Andean-centred' genera in the short-branch clade of Annonaceae : testing biogeographical hypotheses using phylogeny reconstruction and molecular dating

Aim: We test biogeographical hypotheses regarding the origin of Andean-centred plant groups by reconstructing phylogeny in the short-branch clade (SBC) of Annonaceae, and estimating the timing of diversifications in four apparently Andean-centred genera: Cremastosperma R.E.Fr., Klarobelia Chatrou, Malmea R.E.Fr. and Mosannona Chatrou. The SBC includes species distributed in both the Old and New World tropics. A number of the Neotropical genera display 'Andean-centred' distribution patterns, with high species richness on both sides of the Andes mountain range. In particular, we test whether these groups could have originated on the South American continent during the time frame of the Andean orogeny [from c. 23 Ma (Miocene) to the present]. Methods: Chloroplast DNA sequences were used to reconstruct phylogeny in related Annonaceae taxa plus outgroups, under maximum parsimony and Bayesian inference. The markers rbcL, trnL-trnF and psbA-trnH were sampled for 96 accessions to test the monophyly of each of the genera, and thus whether they might be para- or polyphyletic with respect to related groups distributed across Amazonia. To determine the sister groups of the four genera, the additional markers matK, ndhF, trnT-trnL, trnS-trnG and atpB-rbcL were sampled for 23 of the 96 accessions. Molecular dating techniques (nonparametric rate-smoothing; penalized likelihood; Bayesian inference) were then applied to estimate the age of the crown group of each genus and the age of their sister groups. Results: Monophyly was confirmed in Cremastosperma, Malmea and Mosannona. The monotypic genus Pseudephedranthus Aristeg. was found to be nested within Klarobelia, the species of which otherwise formed a monophyletic group, and a South American-centred (SAC) clade was identified. The SAC clade comprises all the SBC genera distributed in South America and generally to a limited extent into Central America, but not those endemic to Africa, Asia and Central America. Age estimations for clades within the SBC were no older than around 60 Myr; those for the crown groups of Cremastosperma, Klarobelia, Malmea and Mosannona fell largely within the last 10-20 Myr. Main conclusions: The distribution patterns of Cremastosperma, Klarobelia, Malmea and Mosannona are not the arbitrary result of the definition of para- or polyphyletic groups. We infer the presence of a common ancestor of the four genera in South America, but not by vicariance of an ancestral population on Gondwana. The age estimations, instead, may suggest that the SAC clade originated in South America by dispersal across the Boreotropics. Although the strength of this test was limited by imprecision in the molecular dating results, the ages of crown groups of the four genera suggest that diversifications occurred within the time frame of the orogeny of the Northern Andes