For a partition π of s and a fixed i, 1 ≤ i ≤ |se|, let V(π, c, i) be the value of the self-alignment such that s1 is mapped to spaces with score  each, sj is optimally aligned with sj+1 for j = 1, 2, ..., k -2, pre(se, |se| - i) is scored according to the above two cases, and the last i letters in se are mapped to spaces with score  each. We have