Consider a set S which contains M bi-allelic SNP markers a₁,a₂...,a_M in K populations

and S_i contains M_i SNP markers s_i1,s_i2,...,s_iMi in population i. First, we estimated pairwise LD measure r² for each SNP pair within each population. Two markers s_im and s_in are said to be in strong LD if the r²(s_im,s_in) is greater than or equal to a pre-specified threshold value r₀. Both are considered tag SNP for each other, in that s_im can be used as a surrogate for s_in, or vice versa.

Our aim is to find a tag SNP set, denoted by T, such that for ∀s_im ∈S_i, i=1,...,K, ∃α_j ∈T that satisfies r²(α_j,S_im) ≥ r₀. In our presentation, we introduce intermediate SNP sets, P and Q_i, i = 1,...,K.

where, P_i is called the candidate set which contains all the SNPs in population i that are eligible to be chosen as a tag SNP, Q_i contains SNPs in population i that are already tagged by at least one of tag SNPs in T, i.e. ∀s_im ∈Q_i, _i = 1,...,K, ∃α_j ∈T that satisfies r²(α_j,S_im) ≥ r₀. We implemented several algorithms in TAGster to select tag SNP set T.

Algorithm 1: A greedy algorithm for single or multiple populations

1. Set T = ∅, P₁ = S₁ and Q₁ = ∅, for any 1 =1,...,K;

2. For each SNP α_j in P, calculate

If α_j ∈P₁
If α_j ∉P₁

3. Find the SNP α_max that has the highest

, and add α_max to T. If α_max ∈P_i, add any SNP s_im in P_i with r²(α_max, s_im) ≥ r₀ to Q₁ and then exclude α_max from P_i;

4. Repeat Steps 2-3 until Q_i=S₁ for any 1=1,...,K;

Algorithm 2: An optimal solution for single population tag SNP

An exhaustive Search is performed within each population to find minimal number of population specific tag SNPs T_i for i= 1,...,K.

1. Set T_i = ∅ and P_i=S_i, for i=1,...,K;

2. Within population i, partition SNPs in P_i into disjoint precinct P_ij, j= 1,...,n, so that r²(s_im,s_in)<r₀ for any two SNPs s_im and s_in that belong to different precincts.

3. Within a precinct P_ij,

• For any two SNPs s_im and s_in in precinct P_ij, if
  
  
  
  ,we exclude one with smaller
  
  
  
  from precinct P_ij
• Conduct an exhaustive search to find a set of minimum number of tag SNPs for SNPs in precinct P_ij and add these tag SNPs into T_i;

4. Repeat step (3) for each precinct

Algorithm 3: Two-stage solution for multi-populations

1. Conduct Algorithm 2 within each population to select a set of population specific tag SNPs T_i for i = 1,...,K;

2. Set T = ∅, P_i = S_i for i = 1,...,K;

3. For each SNP t_ij in T_i, find and SNP s_im (s_im ∈P_i and s_im ∉T_i) that satify r²(t_ij,S_im) ≥ r₀ and then add them as well as t_ij into LD bin B_ij and exclude them for P_i

4. With each LD bin B_ij, set T_ij = ∅. Find any SNP s_im in B_ij that satify r²(s_im,S_in) ≥ r₀ for any SNP s_in in B_ij, and then add s_im to T_ij;

5. Set

. For each SNP τ in P, l= 1,...,|P|, construct a one dimensional array A_l with K elements, where

6. Cluster SNPs in P so that any two SNPs τ_m and τ_n in a cluster satisfy

7. Set Ψ = ∅. Find one SNP τ_l in each cluster with maximum

and add it to Ψ.

8. Cluster SNPs in Ψ so that any two SNPs τ_m and τ_n in a cluster satisfy

9. For each cluster, set LD bin set B= ∅, record the LD bins in each population that can be tagged by any SNP in the cluster to B, and then conduct an exhaustive search to find a minimum set of tag SNPs in the cluster that can tag all LD bins in B. Add this set of SNPs to T.