Consider the sequence alignment problem over a four-letter alphabet {z₁,z₂,z₃,z₄}, with a given gap cost and given mismatch costs. Assume that each of these parameters is a positive integer.

Suppose you are given two strings A = a₁a₂...a_m and B = b₁b₂...b_n and a proposed alignment between them. Give an O(mn) algorithm to decide whether this alignment is the unique minimum-cost alignment between A and B. Please give explanation.

This is problem 6.18 from Algorithm Design by John Kleinberg, Eva Tardos.