Approximation Algorithm For Sorting Research Articles

A ([formula omitted])-approximation algorithm for sorting by short block-moves

Sorting permutations by operations such as reversals and block-moves has received much attention because of its applications in the study of genome rearrangements. A short block-move is an operation on a permutation that moves an element at most two positions away from its original position. In this paper, we investigate the problem of finding a minimum-length sorting sequence of short block-moves for a given permutation, and devise a (1+ε)-approximation algorithm for this problem, where ε is the number of elements divided by the number of inversions in the permutation. The algorithm mostly relies on a new structure in the permutation graph called an umbrella, which can be optimally sorted in O(n2) time.

The 1.375 Approximation Algorithm for Sorting by Transpositions Can Run in O(n log n) Time

Sorting a permutation by transpositions (SPbT) is an important problem in bioinformtics. In this article, we improve the running time of the best known approximation algorithm for SPbT.

An approximation algorithm for sorting by reversals and transpositions

Genome rearrangement algorithms are powerful tools to analyze gene orders in molecular evolution. Analysis of genomes evolving by reversals and transpositions leads to a combinatorial problem of sorting by reversals and transpositions, the problem of finding a shortest sequence of reversals and transpositions that sorts one genome into the other. In this paper we present a 2 k-approximation algorithm for sorting by reversals and transpositions for unsigned permutations where k is the approximation ratio of the algorithm used for cycle decomposition. For the best known value of k our approximation ratio becomes 2.8386 + δ for any δ > 0 . We also derive a lower bound on reversal and transposition distance of an unsigned permutation.

A new approximation algorithm for sorting of signed permutations

Sequence comparison leads to a combinatorial optimization problem of sorting permutations by reversals and transpositions. Namely, given any two permutations, find the shortest distance between them. This problem is related with genome rearrangement. The sorting of signed permutations is studied. Because in genome rearrangement, genes are oriented in DNA sequences. The transpositions which have been studied in the literature can be viewed as operations working on two consecutive segments of the genome. In this paper, a new kind of transposition which can work on two arbitrary segments of the genome is proposed, and the sorting of signed permutations by reversals and this new kind of transpositions are studied. After establishing a lower bound on the number of operations needed, a 2-approximation algorithm is presented for this problem and an example is given to show that the performance ratio of the algorithm cannot be improved.

Genome Rearrangements and Sorting by Reversals

Sequence comparison in molecular biology is in the beginning of a major paradigm shift—a shift from gene comparison based on local mutations (i.e., insertions, deletions, and substitutions of nucleotides) to chromosome comparison based on global rearrangements (i.e., inversions and transpositions of fragments). The classical methods of sequence comparison do not work for global rearrangements, and little is known in computer science about the edit distance between sequences if global rearrangements are allowed. In the simplest form, the problem of gene rearrangements corresponds to sorting by reversals, i.e., sorting of an array using reversals of arbitrary fragments. Recently, Kececioglu and Sankoff gave the first approximation algorithm for sorting by reversals with guaranteed error bound 2 and identified open problems related to chromosome rearrangements. One of these problems is Gollan’s conjecture on the reversal diameter of the symmetric group. This paper proves the conjecture. Further, the problem of expected reversal distance between two random permutations is investigated. The reversal distance between two random permutations is shown to be very close to the reversal diameter, thereby indicating that reversal distance provides a good separation between related and nonrelated sequences in molecular evolution studies. The gene rearrangement problem forces us to consider reversals of signed permutations, as the genes in DNA could be positively or negatively oriented. An approximation algorithm for signed permutation is presented, which provides a performance guarantee of $\tfrac{3}{2}$ . Finally, using the signed permutations approach, an approximation algorithm for sorting by reversals is described which achieves a performance guarantee of $\tfrac{7}{4}$.