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

Algorithm 3

The 𝑂(𝑛𝑙𝑡) time and linear-space algorithm for haplotype blocking with constraints on diversity and the number of tag SNPs.

HBPTS (𝑖,𝑗,𝑇)    ▷ Lis: List blocks covered by 𝑇 tag SNPs in [𝑖,𝑗] with maximized total length.

Input: Interval [𝑖,𝑗] and number of tag SNPs 𝑇.

Output: The boundaries of blocks covered by 𝑇 tag SNPs.

Global variable: 𝐿,𝑅,𝑆,𝐴,𝐵.    ▷  𝐿 and 𝑅 are used to store the good partner pointers 𝐿_𝑖 and

       𝑅_𝑖 which have been preprocessed dependent on diversity constraints 𝐷,

       𝑆 is used to store the tag SNPs required for each feasible blocks,

      two dimensional array 𝐴 and 𝐵 are global temporary working storages.

(1) if  𝑇≤𝑡₀  then

(2)  for  𝑡←0  to  T  do

(3)     for  𝑥←𝑖  to  𝑗  do

(4)     Directly compute 𝐴[𝑡,𝑥]=𝑓(𝑖,𝑥,𝑡) according to recursion relation 3.

(5)  Trace back on 𝐴 array to output the boundaries of blocks covered by 𝑇 tag SNPs.

(6)  return

(7) for  𝑡←0  to  𝑇2+𝑡₀2  do

   ▷ Compute 𝐴[𝑡mod(𝑡₀𝑇+1),𝑥]=𝑓(𝑖,𝑥,𝑡),∀𝑥∈[𝑖⋯𝑗−1],𝑡∈2−𝑡₀2…𝑇2+𝑡₀2.

(8)    for  𝑥←𝑖  to  𝑗−1  do

(9)    Compute 𝐴[𝑡mod(𝑡₀+1),𝑥]=𝑓(𝑖,𝑥,𝑡) by formula 3.

(10) for  𝑡←0  to  𝑇2+𝑡₀2  do

  ▷ Compute 𝐵[𝑡mod(𝑡₀+1),𝑥]=𝑓(𝑥+1,𝑗,𝑡),∀𝑥∈[𝑖⋯𝑗−1], 𝑇𝑡∈2−𝑡₀2…𝑇2+𝑡₀2.

(11)    for  𝑥←𝑗  down to  𝑖+1  do

(12)    Compute 𝐵[𝑡mod(𝑡₀+1),𝑥]=𝑓(𝑥,𝑗,𝑡) by formula 3 with opposite direction.

(13) Find (𝑥^∗,𝑡^∗) = arg max_{𝑖≤𝑥≤𝑗,⌊𝑇/2⌋−⌊𝑡0/2⌋≤𝑡≤⌊𝑇/2⌋+⌈𝑡0/2⌉}{𝐴[𝑡,𝑥]+𝐵[𝑇−𝑡,𝑥]}.

  ▷ Pointer back tracking to find the 𝑥^∗, and 𝑡^∗

(14) HBPTS (𝑖,𝑥^∗,𝑡^∗)     ▷ recursive call to report blocks in interval [i, x*] with t* tags.

(15) HBPTS (𝑥^∗+1,𝑗,𝑇−𝑡^∗)    ▷ recursive call to report blocks in interval [𝑥^∗,𝑗] with 𝑇−𝑡^∗ tags.