Question

I need to solve the follow problem withing O(nm).
n = |T|
m = |P|
where T,P two strings
f is a scoring function.

the algorithm should return a substring T’ of T such that score(P,T’) value is the maximum.

score(A,B) is the max val for alignment A and B according f.

I know I can get it from DIST matrix which is a Monge matrix if f is discrete (meaning the diagonals of the matrix has weights not larger than C which is a constant, and the horizontal and vertical edges is 0 or some other constant), but in this case the f is a general function from (sigma * {-})x(sigma * {-}) to R (where ‘-‘ is a gap).

any ideas?

Answer 1

You’ve noticed that there are several algorithms that compute a shortest path in a graph whose arcs are (i, j) → (i + 1, j), (i + 1, j + 1), (i, j + 1). The most general form of this algorithm would allow every arc length to be specified separately, with the following meanings.

• (i, j) → (i + 1, j): cost of aligning the (i+1)th letter of P with a gap in T
• (i, j) → (i + 1, j + 1): cost of aligning the (i+1)th letter of P with the (j+1)th letter of T
• (i, j) → (i, j + 1): cost of aligning a gap in P with the (j+1)th letter of T

Costs can be negative. To solve your substring problem, make the costs of all of the (i, j) → (i, j + 1) arcs zero so that we can delete from T without penalty.