Affine gaps penalty

To align sequences S, T, consider as usual the prefixes 43#43 of S and 42#42 of T. Any alignment of these two prefixes is one of the following three types:

1.

S ------i
T ------j
alignment of 43#43 and 42#42 where characters S(i) and T(j) are aligned opposite each other. This includes both the case that S_i = T_j and that 73#73.

2.

S ------i_ _ _ _ _ _ _
T -------------j
alignment of 43#43 and 42#42 where character S_i is aligned to a character strictly to the left of character T_j. Therefore, the alignment ends with a gap in S.

3.

S --------------i
T ------j_ _ _ _ _ _ _ _
alignment of 43#43 and 42#42 where character S_i is aligned to a character strictly to the right of character T_j. Therefore, the alignment ends with a gap in T.

 
74#74
Hence the base conditions are:

75#75

and the recursive computation of V(i, j) will be:

76#76

while

77#77

The optimal value alignment is the maximum value in the nth row or m column.
Complexity

• Time complexity. As before O(nm), as we only compute four matrices instead of one.
• Space complexity. There's a need to save four matrices (for E, F, G, and V respectively) during the computation. Hence, O(nm) space is needed, for the trivial implementation.