Table 1. A brief history of phasing and imputation tools.
| Software | Published | Based on | Features | Complexity | |
|---|---|---|---|---|---|
| Phasing | PHASE v 1.0 [12] | 2001 | Coalescent approximation | Improved error rates are reduced by >50% relative to its nearest competitor | quadratic O(n2) |
| HAPI-UR [13] | 2012 | Li & Stephens HMM | Used windows of sites instead of specific markers; led to higher accuracy | linear O(nm) | |
| Eagle 2 [14] | 2016 | Li & Stephens HMM | pBWT on a large reference panel condensed into a set of compact tree structures that losslessly model haplotype structure | linear O(nm) | |
| Phasing & Imputation | fastPHASE [8] | 2006 | Li & Stephens HMM | Faster but less accurate than Phase | linear O(nm) |
| Beagle v. 1.0 [15] | 2007 | Li & Stephens HMM | Uses bifurcating tree structure (aka haplotype-cluster model) | quadratic O(n2) | |
| Beagle v. 2.0, 3.0 [16, 17] | 2009 | Li & Stephens HMM | Uses bifurcating tree structure (aka haplotype-cluster model) | quadratic O(n2) | |
| Beagle v. 4.0 [18] | 2018 | Li & Stephens HMM | Abandoned bifurcating model to adopt a flexible choice of haplotypes for reference similar to IMPUTE 2 | quadratic O(n2) | |
| Beagle v. 5.2 [19] | 2021 | Li & Stephens HMM | Introduction of progressive phasing algorithm to handle hundreds of millions of markers | linear O(nm) | |
| IMPUTE 2 [20] | 2009 | Li & Stephens HMM | Flexible choice of haplotypes for reference panel; quadratic computational complexity meant inefficient | linear O(nm) | |
| IMPUTE 4 [21] | 2018 | Li & Stephens HMM | Speed up haplotype imputation step | quadratic O(n2) | |
| IMPUTE 5 [22] | 2019 | Li & Stephens HMM | Uses positional BWT to choose haplotypes for each window | linear O(nm) | |
| MACH [23] | 2010 | Li & Stephens HMM | An iteratively updated phase of each study sample | linear O(m+n) | |
| SHAPEIT 1 [24] | 2011 | Li & Stephens HMM | Flexible choice of the panel but computationally efficient | linear O(n+m) | |
| SHAPEIT 2 [25] | 2013 | Li & Stephens HMM | Combined best aspects of SHAPEIT 1 and IMPUTE 2 to increase accuracy and efficiency | quadratic O(mn2) | |
| SHAPEIT 3 [26] | 2016 | Li & Stephens HMM | Increased scalability from SHAPEIT 2 | quadratic O(n2) | |
| SHAPEIT 4 [27] | 2018 | Li & Stephens HMM | pBWT to choose haplotypes for local window | linear O(nm) | |
| Imputation | Minimac [28] | 2012 | Li & Stephens HMM | Pre-phased imputation | linear O(nm) |
| Minimac 2 [29] | 2014 | Li & Stephens HMM | Improved version and bug fixing | linear O(nm) | |
| Minimac 3 [30] | 2015 | Li & Stephens HMM | State-space reduction to reduce computational complexity and cost | linear O(nm) | |
| Minimac4 [31] | 2018 | Li & Stephens HMM | Improved version and bug fixing | linear O(nm) |
A timeline and brief description of landmark and popular phasing and imputation algorithms and their computational complexities