Efficient Haplotype Block Partitioning and Tag SNP Selection Algorithms under Various Constraints

Algorithm 2

The 𝑂(𝑛𝑘) time and linear space algorithm for haplotype blocking.

LIS(𝑘,𝑖,𝑗)    ▷ Lis: List 𝑘 blocks in [𝑖,𝑗] with maximized total length.

Input: Interval [𝑖,𝑗] and number of blocks 𝑘.

Output: 𝑘's blocks and their boundaries.

Global variable: 𝐿,𝑅,𝑌,𝐴,𝐵,𝐶.    ▷  𝐿 and 𝑅 are used to store the good partner pointers 𝐿[𝑥]^′𝑠 and 𝑅[𝑥]^′𝑠

      which have been preprocessed dependent on diversity constraints 𝐷,𝑌 is used to store the

      results of LIS(𝑘,𝑖,𝑗), 𝐴,𝐵,𝐶 are global temporary working storages.

(1) if  𝑘≤1  then

(2)  Append the boundaries of the longest block in [𝑖,𝑗] to 𝑌

(3)   return

(4) for  𝑥←𝑖  to  𝑗  do    ▷ Initiate the boundary condition of 𝑓(𝑘,𝑖,𝑗).

(5)     𝐶[𝑥]←0

(6) for  𝑦←1  to  𝑘2  do    ▷ Compute 𝑘𝐴[𝑥]=𝑓2,𝑖,𝑥, ∀𝑥∈[𝑖…𝑗−1].

(7)    for  𝑥←𝑖  to  𝑗−1  do

(8)     if  𝑥=𝑖  then

(9)     𝑡𝑒𝑚𝑝1←1

(10)   else

(11)    𝑡𝑒𝑚𝑝1←𝐴[𝑥−1]

(12)    if  𝐿[𝑥]≤𝑖  then     ▷  𝐿[𝑥]∉[𝑖,𝑗], exceeding the boundary region.

(13)    𝑡𝑒𝑚𝑝2←𝑥−𝑖+1

(14)    else

(15)    𝑡𝑒𝑚𝑝2←𝐶[𝐿[𝑥]−1]+𝑥−𝐿[𝑥]+1

(16)    𝐴[𝑥]←max{𝑡𝑒𝑚𝑝1,𝑡𝑒𝑚𝑝2}

(17)   copy 𝐴[𝑥] to 𝐶[𝑥] for next iteration of 𝑥   ▷  𝐶[𝑥] is used to store the temporary results of 𝑓(𝑘,𝑖,𝑗).

(18) for  𝑥←𝑖  to  𝑗  do   ▷ Initiate the boundary condition of 𝑓(𝑘,𝑖,𝑗).

(19)   𝐶[𝑥]←0

(20) for  𝑦←1  to  𝑘2  do    ▷ Compute 𝑘𝐵[𝑥]=𝑓2,𝑥+1,𝑗,∀𝑥∈[𝑖…𝑗−1].

(21)   for  𝑥←𝑗  downto  𝑖+1  do

(22)     if  𝑥=𝑗  then

(23)     𝑡𝑒𝑚𝑝1←1

(24)     else

(25)     𝑡𝑒𝑚𝑝1←𝐴[𝑥+1]

(26)     if  𝑅[𝑥]≥𝑗  then    ▷  𝑅[𝑥]∉[𝑖,𝑗], exceeding the boundary region.

(27)     𝑡𝑒𝑚𝑝2←𝑗−𝑥+1

(28)     else

(29)     𝑡𝑒𝑚𝑝2←𝐶[𝑅[𝑥]+1]+𝑅[𝑥]−𝑥+1

(30)     𝐵[𝑥]←max{𝑡𝑒𝑚𝑝1,𝑡𝑒𝑚𝑝2}

(31)   copy 𝐵[𝑥] to 𝐶[𝑥] for next iteration of 𝑥   ▷  𝐶[𝑥] is used to store the temporary results of 𝑓(𝑘,𝑖,𝑗).

(32) 𝑡𝑒𝑚𝑝←−∞

(33) for  𝑥←𝑖  to  𝑗−1  do    ▷ Find 𝑥^∗=argmax_{𝑖≤𝑥≤𝑗}{𝐴[𝑥]+𝐵[𝑥]}.

(34)    if  (𝐴[𝑥]+𝐵[𝑥+1])>𝑡𝑒𝑚𝑝  then

(35)    𝑥^∗←𝑥

(36)    𝑡𝑒𝑚𝑝←(𝐴[𝑥]+𝐵[𝑥+1])

(37) LIS𝑘2,𝑖,𝑥^∗      ▷ recursive call.

(38) LIS𝑘2,𝑥^∗+1,𝑗     ▷ recursive call.