 Local sequence alignment without mismatches

Choose one contiguous substring from each string and align them under the same matching and gap rules as global sequence alignment without mismatches. Let $L_{i,j}$ maximize the score over aligned suffixes of the two prefixes. Zero permits a fresh empty alignment; the other cases extend one ending at adjacent prefix indices. The answer is the largest table entry, which can occur before either input ends. Starting all boundaries at zero yields $O(mn)$ time for positive lengths. Alignment traceback stops on reaching zero.