Genome rearrangement algorithms

With the increasing amount of sequenced genomes, a comparison of species based on these data becomes more and more interesting. In contrast to the classical approach, where only point mutations were considered, genome rearrangement problems ignore small mutations and only consider large-scale mutations that change the gene order on the chromosomes. This makes these problems a powerful tool when studying organisms that have diverged millions of years ago, or very fast evolving genomes, like those of cancerous cells. A large variety of problems arises from genome rearrangements, like the calculation of evolutionary distances, the reconstruction of evolutionary events and the gene order of hypothetical ancestors, or even the reconstruction of whole phylogenetic trees. Due to the high complexity of most of these problems, the algorithms are based on highly simplified models of the reality. For example, most algorithms consider only a single type or a small set of evolutionary events. Therefore, one biological problem results in several algorithmic problems, depending on the chosen model. Genome rearrangement problems are often very challenging. For many problems, it is not known whether they can be solved exactly in polynomial time w.r.t. the size of the genomes (like sorting by transpositions), others are known to be NP-hard even for the most simple models, like the median problem or the multiple genome rearrangement problem. For some problems, their complexity depends on the chosen model. Many of the problems can be solved in polynomial time if every gene occurs in each genome exactly once, but become NP-hard as soon as duplications are allowed. From a computer scientist's point of view, this raises both theoretical and algorithmic tasks. For each problem, it is desirable either to find an exact algorithm or to prove that the problem is NP-hard. In some cases, the examination of a simplified version of the problem and the connection between this simplified problem and the actual problem can be helpful (actually, this approach helped Hannenhalli and Pevzner to obtain their famous result about sorting by reversals). If a problem has been shown to iii Summary be NP-hard, still many instances of the problem can be solved exactly and efficiently by clever algorithms. Where even this strategy fails, we seek for efficient heuristic algorithms. The major results contained in this thesis are as follows: • We provide a linear time transformation from an arbitrary permutation into its equivalent simple permutation, and an O(n log …