Banded Smith-Waterman (BSW) — DP Table Inspector
Objective function (affine gap):
H(i,j) = max{ H(i-1,j-1) + S(X(i),Y(j)), E(i,j), F(i,j) }
E(i,j) = max{ H(i-1,j) - g_o, E(i-1,j) - g_e, 0 }   — deletion (gap in Y)
F(i,j) = max{ H(i,j-1) - g_o, F(i,j-1) - g_e, 0 }   — insertion (gap in X)
Banded constraint: |i - j| ≤ w — only cells within the band are computed.
Affine gap: opening a new gap costs g_o; extending an existing gap costs g_e.
Out of band Boundary (i=0 or j=0) Match Mismatch Gap (E/F > 0) Max H

H(i,j) main score matrix

S(X(i), Y(j)) match / mismatch scores

E(i,j) deletion (gap in Y)

F(i,j) insertion (gap in X)