Sorting by Block Moves

The research in this thesis is focused on the problem of Block Sorting, which has applications in Computational Biology and in Optical Character Recognition (OCR). A block in a permutation is a maximal sequence of consecutive elements that are also consecutive in the identity permutation. BLOCK SORTING is the process of transforming an arbitrary permutation to the identity permutation through a sequence of block moves. Given an arbitrary permutation π and an integer m, the Block Sorting Problem, or the problem of deciding whether the transformation can be accomplished in at most m block moves has been shown to be NP-hard. After being known to be 3-approximable for over a decade, block sorting has been researched extensively and now there are several 2-approximation algorithms for its solution. This work introduces new structures on a permutation, which are called runs and ordered pairs, and are used to develop two new approximation algorithms. Both the new algorithms are 2-approximation algorithms, yielding the approximation ratio equal to the current best. This work also includes an analysis of both the new algorithms showing they are 2-approximation algorithms

On Sorting under Special Transpositions

In this paper, we study a genome rearrangement primitive called block moves. This primitive as a special case of another well studied primitive transposition. We revisit the problem of BLOCK SORTING, which is a sorting problem under the primitive block moves in this work. BLOCK SORTING has been shown to be NP-Complete, and a couple of results have designed factor 2 approximation algorithms for the problem - the best known till date. However whether the problem is APX-Hard, or an improvement over the factor 2 approximation algorithms have been interesting open problems. We design a new factor 2 approximation algorithm for BLOCK SORTING. Our algorithm is equal to the best known in terms of approximation ratio, however, our approach is much simpler and is linear time (O (n)) as compared to the cubic (O (n3)) and quadratic (O (n2)) run-times of the existing algorithms for the problem