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

Attribute Value Reordering For Efficient Hybrid OLAP

The normalization of a data cube is the ordering of the attribute values. For large multidimensional arrays where dense and sparse chunks are stored differently, proper normalization can lead to improved storage efficiency. We show that it is NP-hard to compute an optimal normalization even for 1x3 chunks, although we find an exact algorithm for 1x2 chunks. When dimensions are nearly statistically independent, we show that dimension-wise attribute frequency sorting is an optimal normalization and takes time O(d n log(n)) for data cubes of size n^d. When dimensions are not independent, we propose and evaluate several heuristics. The hybrid OLAP (HOLAP) storage mechanism is already 19%-30% more efficient than ROLAP, but normalization can improve it further by 9%-13% for a total gain of 29%-44% over ROLAP

Genome Rearrangement Problems

Various global rearrangements of permutations, such as reversals and transpositions, have recently become of interest because of their applications in computational molecular biology. A reversal is an operation that reverses the order of a substring of a permutation. A transposition is an operation that swaps two adjacent substrings of a permutation. The problem of determining the smallest number of reversals required to transform a given permutation into the identity permutation is called sorting by reversals. Similar problems can be defined for transpositions and other global rearrangements. Related to sorting by reversals is the problem of establishing the reversal diameter. The reversal diameter of Sn (the symmetric group on n elements) is the maximum number of reversals required to sort a permutation of length n. Of course, diameter problems can be posed for other global rearrangements. These various problems are of interest because the permutations can be used to represent sequences of genes in chromosomes, and the global rearrangements then represent evolutionary events. As a result, we call these problems genome rearrangement problems. Genome rearrangement problems seem to be unlike previously studied algorithmic problems on sequences, so new methods have had to be developed to deal with them. These methods predominantly employ graphs to model permutation structure. However, even using these methods, often a genome rearrangement problem has no obvious polynomial-time algorithm, and in some cases can be shown to be NP-hard. For example, the problem of sorting by reversals is NP-hard, whereas the computational complexity of sorting by transpositions is open. For problems like these, it is natural to seek polynomial-time approximation algorithms that achieve an approximation guarantee. In this thesis, we study several genome rearrangement problems as interesting and challenging algorithmic problems in their own right, including some problems for which the global rearrangement has no immediate biological equivalent. For example, we define a block-interchange to be a rearrangement that swaps any two substrings of the permutation. We examine, in particular, how the graph theoretic models relate to the genome rearrangement problems that we study. The major new results contained in this thesis are as follows: We present a 3/2-approximation algorithm for sorting by reversals. This is the best known approximation algorithm for the problem, and improves upon the 7/4 approximation bound of the previous best algorithm. We give a polynomial-time algorithm for a significant special case of sorting by reversals, thereby disproving a conjecture of Kececioglu and Sankoff, who had suggested that this special case was likely to be NP-hard. We analyse the structure of the so-called cpcle graph of a permutation in the context of sorting by transpositions, and thereby gain a deeper insight into this problem. Among the consequences are; a tighter lower bound for the problem, a simpler 3/2-aproximation algorithm than had previously been described, and algorithms that, in empirical tests, almost always find the exact transposition distance of random permutations. We introduce a natural generalisation of sorting by transpositions called sorting by block-interchanges, and present a polynomial-time algorithm for this problem. We initiate the study of analogous problems on strings over a fixed length alphabet. We establish upper and lower bounds and diameter results for the problems over a binary alphabet. We also prove that the problems analogous to sorting by reversals and sorting by block-interchanges are NP-hard. (Abstract shortened by ProQuest.)

Ordering Metro Lines by Block Crossings

A problem that arises in drawings of transportation networks is to minimize the number of crossings between different transportation lines. While this can be done efficiently under specific constraints, not all solutions are visually equivalent. We suggest merging crossings into block crossings, that is, crossings of two neighboring groups of consecutive lines. Unfortunately, minimizing the total number of block crossings is NP-hard even for very simple graphs. We give approximation algorithms for special classes of graphs and an asymptotically worst-case optimal algorithm for block crossings on general graphs. That is, we bound the number of block crossings that our algorithm needs and construct worst-case instances on which the number of block crossings that is necessary in any solution is asymptotically the same as our bound