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. 2019 Apr 2;14:11. doi: 10.1186/s13015-019-0144-9

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© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.

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Fig. 2 — Work done to calculate the sum of haplotype probabilities at a site for the conventional and our sublinear forward algorithm. Using the example that at site i, $ϕ_{i} (o_{i}) = {h_{3}}$ , we illustrate the number of arithmetic operations used in (i) the conventional $O (n k)$ Li and Stephens HMM recurrence relations. ii Our procedure specified in Eq. (12). Black lines correspond to arithmetic operations; operations which cannot be parallelized over j are colored yellow