BACKGROUND Hidden Markov models of haplotype inheritance such as the Li and Stephens model allow for computationally tractable probability calculations using the forward algorithm 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. RESULTS To make the forward algorithm for the haploid Li and Stephens model computationally tractable for these datasets, we have created a numerically exact version of the algorithm with observed average case sublinear runtime with respect to reference panel size k when tested against the 1000 Genomes dataset. CONCLUSIONS We show a forward algorithm which avoids any tradeoff between runtime and model complexity. Our algorithm makes use of two general strategies which might be applicable to improving the time complexity of other future sequence analysis algorithms: sparse dynamic programming matrices and lazy evaluation.

中文翻译：

---

单倍体 Li 和 Stephens 模型的平均情况亚线性前向算法。

背景技术单倍型遗传的隐马尔可夫模型（例如Li和Stephens模型）允许使用前向算法进行计算上易于处理的概率计算，只要模型中使用的代表性参考组足够小。具体而言，除非使用启发式近似，否则单倍体 Li 和 Stephens 模型及其变体在参考面板大小中是线性的。然而，数千至数十万个体的测序项目正在进行中，预计其他项目的数量将达到数百万。结果 为了使单倍体 Li 和 Stephens 模型的前向算法在计算上易于处理这些数据集，我们创建了算法的数值精确版本，在针对 1000 个基因组数据集进行测试时，观察到相对于参考面板大小 k 的平均情况次线性运行时间。结论我们展示了一种前向算法，它避免了运行时间和模型复杂性之间的任何权衡。我们的算法利用了两种可能适用于提高其他未来序列分析算法的时间复杂度的通用策略：稀疏动态规划矩阵和惰性求值。