Sequence alignment. When a new gene is discovered, a standard approach to understanding it...

Sequence alignment. When a new gene is discovered, a standard approach to understanding its function is to look through a database of known genes and find close matches. The closeness of two genes is measured by the extent to which they are aligned. To formalize this, think of a gene as being a long string over an alphabet ∑ = {A, C, G, T}. Consider two genes (strings) x = ATGCC and y = TACGC A. An alignment of x and y is a way of matching up these two strings by writing them in columns, for instance:

— A T — G C C

T A — C G C A

Here the “-” indicates a “gap.” The characters of each string must appear in order, and each column must contain a character from at least one of the strings. The score of an alignment is specified by a scoring matrix δ of size (|∑| + 1) x (|∑| + 1), where the extra row and column are to accommodate gaps. For instance the preceding alignment has the following score:

Give a dynamic programming algorithm that takes as input two strings x[1...n] and y[1...m] and a scoring matrix δ, and returns the highest-scoring alignment. The running time should be O (mn).