Haplotype Threading Using the Positional Burrows-Wheeler Transform

In the classic model of population genetics, one haplotype (query) is considered as a mosaic copy of segments from a number of haplotypes in a panel, or threading the haplotype through the panel. The Li and Stephens model parameterized this problem using a hidden Markov model (HMM). However, HMM algorithms are linear to the sample size, and can be very expensive for biobank-scale panels. Here, we formulate the haplotype threading problem as the Minimal Positional Substring Cover problem, where a query is represented by a mosaic of a minimal number of substring matches from the panel. We show that this problem can be solved by a sequential set of greedy set maximal matches. Moreover, the solution space can be bounded by the left-most and the right-most solutions by the greedy approach. Based on these results, we formulate and solve several variations of this problem. Although our results are yet to be generalized to the cases with mismatches, they offer a theoretical framework for designing methods for genotype imputation and haplotype phasing.