Upper Bounds to Genome Rearrangement Problem using Prefix Transpositions

A Genome rearrangement problem studies large-scale mutations on a set of DNAs in living organisms. Various rearrangements like reversals, transpositions, translocations, fissions, fusions, and combinations and different variations have been studied extensively by computational biologists and computer scientists over the past four decades. From a mathematical point of view, a genome is represented by a permutation. The genome rearrangement problem is interpreted as a problem that transforms one permutation into another in a minimum number of moves under certain constraints depending on the chosen rearrangements. Finding the minimum number of moves is equivalent to sorting the permutation with the given rearrangement. A transposition is an operation on a permutation that moves a sublist of a permutation to a different position in the same permutation. A \emph{Prefix Transposition}, as the name suggests, is a transposition that moves a sublist which is a prefix of the permutation. In this thesis, we study prefix transpositions on permutations and present a better upper bound for sorting permutations with prefix transpositions. A greedy algorithm called the \emph{generalised sequence length algorithm} is defined as an extension of the sequence length algorithm where suitable alternate moves are also considered. This algorithm is used to sequentially improve the upper bound to $n-\log_{3.3} n$ and $n-\log_3 n$. In the latter part of the thesis, we defined the concept of a \emph{block}. We used it along with the greedy moves of the generalised sequence length algorithm to get an upper bound of $n-\log_2 n$ to sort permutations by prefix transpositions.