Figure 1

Example of gap extension penalty calculation. This figure shows an example of columns a₈ and b₁₂ being aligned. '*', '·', and '-' denote a residue, a static null, and a dynamic null, respectively. We assume that piecewise linear gap cost g(x) is max_{k = 1,2}{-(u _k x + v _k )} and critical gap length x _c (= ⌊(v₂ - v₁)/(u₁ - u₂)⌋) is 4. Gap extension penalty is u₁ if x ≤ 4, otherwise u₂. Running gap profile vectors ${\widehat{S}}_{a_8}$ and ${\widehat{E}}_{a_8}^{+}$ are {(1, $w_{A_1}$)} and {(1, $w_{A_2}$ + $w_{A_3}$ + $w_{A_4}$), (3, $w_{A_2}$ + $w_{A_3}$), (11, $w_{A_2}$)}, respectively. Dynamic gap information $D_{7,11}^0$(A) is {(0, 1), (2, 2), (7, 1)}. Segment profile $F_{a_8}$ is {(1, $w_{A_2}$), (3, $w_{A_2}$ + $w_{A_3}$), (5, $w_{A_2}$ + $w_{A_3}$ + $w_{A_4}$)}. Similarly, the profile vectors of B are defined: ${\widehat{S}}_{b_{12}}$ = {(7, $w_{B_1}$)}, ${\widehat{E}}_{b_{12}}^{+}$ = {(0, $w_{B_2}$)}, $D_{7,11}^0$(B) is empty, and $F_{b_{12}}$ = {(1, 0), (9, $w_{B_2}$)}. In what follows, we consider the non-trivial calculation of the gap extension penalty with respect to the gap of B₁, the target gap. By using ${\widehat{S}}_{b_{11}}$ and $D_{7,11}^0$(A), we find that the two dynamic gaps specified by (2, 2) and (7, 1) in $D_{7,11}^0$(A) are partially and completely aligned with the target gap, respectively. Consequently, the total number of nulls aligned with null columns of dynamic gaps to be removed is 2. Therefore, the number of columns of B₁ is 5(= 7 - 2). By subtracting 5 from 8 (the end position of the segment), the starting position of A, 3, is obtained. Then, the gap extension penalty with respect to the gap of B₁ is $w_{B_1}$ (F₁·u₁ + F₂·u₂) where F₁ = $w_{A_2}$ + $w_{A_3}$ and F₂ = $w_{A_4}$. Note that A₁ is not involved in the gap extension penalty because a_1,8 is a null.