G-lattices for an Unrooted Perfect Phylogeny

We look at the Pure Parsimony problem and the Perfect Phylogeny Haplotyping problem. From the Pure Parsimony problem we consider structures of genotypes called g-lattices. These structures either provide solutions or give bounds to the pure parsimony problem. In particular, we investigate which of these structures supports an unrooted perfect phylogeny, a condition that adds biological interpretation. By understanding which g-lattices support an unrooted perfect phylogeny, we connect two of the standard biological inference rules used to recreate how genetic diversity propagates across generations

Pure Parsimony Xor Haplotyping

The haplotype resolution from xor-genotype data has been recently formulated as a new model for genetic studies. The xor-genotype data is a cheaply obtainable type of data distinguishing heterozygous from homozygous sites without identifying the homozygous alleles. In this paper we propose a formulation based on a well-known model used in haplotype inference: pure parsimony. We exhibit exact solutions of the problem by providing polynomial time algorithms for some restricted cases and a fixed-parameter algorithm for the general case. These results are based on some interesting combinatorial properties of a graph representation of the solutions. Furthermore, we show that the problem has a polynomial time k-approximation, where k is the maximum number of xor-genotypes containing a given SNP. Finally, we propose a heuristic and produce an experimental analysis showing that it scales to real-world large instances taken from the HapMap project

Parsimony-based genetic algorithm for haplotype resolution and block partitioning

This dissertation proposes a new algorithm for performing simultaneous haplotype resolution and block partitioning. The algorithm is based on genetic algorithm approach and the parsimonious principle. The multiloculs LD measure (Normalized Entropy Difference) is used as a block identification criterion. The proposed algorithm incorporates missing data is a part of the model and allows blocks of arbitrary length. In addition, the algorithm provides scores for the block boundaries which represent measures of strength of the boundaries at specific positions. The performance of the proposed algorithm was validated by running it on several publicly available data sets including the HapMap data and comparing results to those of the existing state-of-the-art algorithms. The results show that the proposed genetic algorithm provides the accuracy of haplotype decomposition within the range of the same indicators shown by the other algorithms. The block structure output by our algorithm in general agrees with the block structure for the same data provided by the other algorithms. Thus, the proposed algorithm can be successfully used for block partitioning and haplotype phasing while providing some new valuable features like scores for block boundaries and fully incorporated treatment of missing data. In addition, the proposed algorithm for haplotyping and block partitioning is used in development of the new clustering algorithm for two-population mixed genotype samples. The proposed clustering algorithm extracts from the given genotype sample two clusters with substantially different block structures and finds haplotype resolution and block partitioning for each cluster

Graph algorithms for the haplotyping problem

Evidence from investigations of genetic differences among human beings shows that genetic diseases are often the results of genetic mutations. The most common form of these mutations is single nucleotide polymorphism (SNP). A complete map of all SNPs in the human genome will be extremely valuable for studying the relationships between specific haplotypes and specific genetic diseases. Some recent discoveries show that the DNA sequence of human beings can be partitioned into long blocks where genetic recombination has been rare. Then, inferring both haplotypes from chromosome sequences is a biologically meaningful research topic, which has compounded mathematical and computational problems.;We are interested in the algorithmic implications to infer haplotypes from long blocks of DNA that have not undergone recombination in populations. The assumption justifies a model of haplotype evolution---haplotypes in a population evolves along a coalescent, based on the standard population-genetic assumption of infinite sites, which as a rooted tree is a perfect phylogeny. The Perfect Phylogeny Haplotyping (PPH) Problem was introduced by Daniel Gusfield in 2002. A nearly linear-time solution to the PPH problem (O( nmalpha(nm)), where alpha is the extremely slowly growing inverse Ackerman function) is provided. However, it is very complex and difficult to implement. So far, even the best practical solution to the PPH problem has the worst-case running time of O( nm2). D. Gusfield conjectured that a linear-time ( O(nm)) solution to the PPH problem should be possible.;We solve the conjecture of Gusfield by introducing a linear-time algorithm for the PPH problem. Different kinds of posets for haplotype matrices and genotype matrices are designed and the relationships between them are studied. Since redundant calculations can be avoided by the transitivity of partial ordering in posets, we design a linear-time (O(nm )) algorithm for the PPH problem that provides all the possible solutions from an input. The algorithm is fully implemented and the simulation shows that it is much faster than previous methods