Global patterns of human DNA sequence variation (haplo-types) defined by common single nucleotide polymorphisms (SNPs) have important implications for identifying disease associations and human traits. Recent genetics research reveals that SNPs within certain haplotype blocks induce only a few distinct common haplotypes in the majority of the population. The existence of haplotype block structure has serious implications for association-based methods for the mapping of disease genes. Our ultimate goal is to select haplotype block designations that best capture the structure within the data. Here in this paper we propose several efficient combinatorial algorithms related to selecting interesting haplotype blocks under different diversity functions that generalizes many previous results in the literatures. In particular, given an m × n haplotype matrix A, we show linear time algorithms for finding all interval diversities, farthest sites, and the longest block within A. For selecting the multiple long blocks with diversity constraint, we show that selecting k blocks with longest total length can be be found in O(nk) time. We also propose linear time algorithms in calculating the all intra-longest-blocks and all intra-k-longest-blocks.
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