An average-case sublinear exact Li and Stephens forward algorithm

Hidden Markov models of haplotype inheritance such as the Li and Stephens model allow for computationally tractable probability calculations using the forward algorithms as long as the representative reference panel used in the model is sufficiently small. Specifically, the monoploid Li and Stephens model and its variants are linear in reference panel size unless heuristic approximations are used. However, sequencing projects numbering in the thousands to hundreds of thousands of individuals are underway, and others numbering in the millions are anticipated.To make the Li and Stephens forward algorithm for these datasets computationally tractable, we have created a numerically exact version of the algorithm with observed average case 𝒪(nk0.35) runtime, avoiding any tradeoff between runtime and model complexity. We demonstrate that our approach also provides a succinct data structure for general purpose haplotype data storage. We discuss generalizations of our algorithmic techniques to other hidden Markov models.2012 ACM Subject ClassificationTheory of computation ⟶ Streaming, sublinear and near linear time algorithms; Applied computing ⟶ BioinformaticsSupplement Materialhttps://github.com/yoheirosen/sublinear-Li-Stephens.FundingThis work was supported by the National Human Genome Research Institute of the National Institutes of Health under Award Number 5U54HG007990, the National Heart, Lung, and Blood Institute of the National Institutes of Health under Award Number 1U01HL137183-01, and grants from the W.M. Keck foundation and the Simons Foundation.AcknowledgementsWe would like to thank Jordan Eizenga for his helpful discussions throughout the development of this work.