Sequence Comparison.pdf

3.6 Space-Saving Strategies 53
score-only pass over the dynamic-programming matrix to locate the first and last
nodes of an optimal local alignment, then delivering a global alignment between
these two nodes by applying Hirschberg’s approach.
Algorithm LINEAR ALIGN(A = a 1 a 2 ...a m , B = b 1 b 2 ...b n , i 1 , j 1 , i 2 , j 2 )
begin
if i 1 + 1 ≤ i 2 or j 1 + 1 ≤ j 2 then
Output the aligned pairs for the maximum-score path from (i 1 , j 1 ) to (i 2 , j 2 )
else
i mid ←⌊(i 1 + i 2 )/2⌋
// Find the maximum scores from (i 1 , j 1 )
S − [ j 1 ] ← 0
for j ← j 1 + 1 to j 2 do S − [ j] ← S − [ j − 1] − β
for i ← i 1 + 1 to i mid do
s ← S − [ j 1 ]
c ← S − [ j 1 ] − β
S − [ j 1 ] ← c
for j ← j 1 + ⎧1 to j 2 do
⎨ S − [ j] − β
c ← max c − β
⎩
s + σ(a i ,b j )
s ← S − [ j]
S − [ j] ← c
// Find the maximum scores to (i 2 , j 2 )
S + [ j 2 ] ← 0
for j ← j 2 − 1 down to j 1 do S + [ j] ← S + [ j + 1] − β
for i ← i 2 − 1 down to i mid do
s ← S + [ j 2 ]
c ← S + [ j 2 ] − β
S + [ j 2 ] ← c
for j ← j 2 −⎧
1 down to j 1 do
⎨ S + [ j] − β
c ← max c − β
⎩
s + σ(a i+1 ,b j+1 )
s ← S + [ j]
S + [ j] ← c
// Find where maximum-score path crosses row i mid
j mid ← value j ∈ [ j 1 , j 2 ] that maximizes S − [ j]+S + [ j]
LINEAR ALIGN(A,B, i 1 , j 1 , i mid , j mid )
LINEAR ALIGN(A,B, i mid , j mid , i 2 , j 2 )
end
Fig. 3.18 Computation of an optimal global alignment in linear space.