The next high-priority phase of human genomics will involve the development of a full Haplotype Map of the human genome [12]. It will be used in large-scale screens of populations to associate specific haplotypes with specific complex genetic-influenced diseases. A prototype Haplotype Mapping strategy is presently being finalized by an NIH workinggroup. The biological key to that strategy is the surprising fact that genomic DNA can be partitioned into long blocks where genetic recombination has been rare, leading to strikingly fewer distinct haplotypes in the population than previously expected [12, 6, 21, 7]. In this paper

Haplotyping, also known as haplotype phase prediction, is the problem of predicting likely haplotypes based on genotype data. This problem, which has strong practical applications, can be approached using both statistical as well as combinatorial methods. While the most direct combinatorial approach, maximum parsimony, leads to NP-complete problems, the perfect phylogeny model proposed by Gusfield yields a problem, called PPH, that can be solved in polynomial (even linear) time. Even this may not be fast enough when the whole genome is studied, leading to the question of whether parallel algorithms can be used to solve the PPH problem. In the present paper we answer this question affirmatively, but we also give lower complexity bounds on its complexity. In detail, we show that the problem lies in Mod2L, a subclass of the circuit complexity class NC 2, and is hard for logarithmic space and thus presumably not in NC 1. We also investigate variants of the PPH problem that have been studied in the literature, like the perfect path phylogeny haplotyping problem and the combined problem where a perfect phylogeny of maximal parsimony is sought, and show that some of these variants are TC 0-complete or lie in AC 0. 1

Haplotyping, also known as haplotype phase prediction, is the problem of predicting likely haplotypes based on genotype data. One fast haplotyping method is based on an evolutionary model where a perfect phylogenetic tree is sought that explains the observed data. Unfortunately, when data entries are missing, as is often the case in real laboratory data, the resulting formal problem IPPH, which stands for incomplete perfect phylogeny haplotyping, is NP-complete and no theoretical results are known concerning its approximability, fixed-parameter tractability or exact algorithms for it. Even radically simplified versions, such as the restriction to phylogenetic trees consisting of just two directed paths from a given root, are still NP-complete, but here, at least, a fixed-parameter algorithm is known. We generalize this algorithm to arbitrary tree topologies and present the first theoretical analysis of an algorithm that works on arbitrary instances of the original IPPH problem. At the same time we also show that restricting the tree topology does not always make finding phylogenies easier: while the incomplete directed perfect phylogeny problem is well-known to be solvable in polynomial time, we show that the same problem restricted to path topologies is NP-complete. 1