#### Smith Waterman

p
```"""
https://en.wikipedia.org/wiki/Smith%E2%80%93Waterman_algorithm
The Smith-Waterman algorithm is a dynamic programming algorithm used for sequence
alignment. It is particularly useful for finding similarities between two sequences,
such as DNA or protein sequences. In this implementation, gaps are penalized
linearly, meaning that the score is reduced by a fixed amount for each gap introduced
in the alignment. However, it's important to note that the Smith-Waterman algorithm
supports other gap penalty methods as well.
"""

def score_function(
source_char: str,
target_char: str,
match: int = 1,
mismatch: int = -1,
gap: int = -2,
) -> int:
"""
Calculate the score for a character pair based on whether they match or mismatch.
Returns 1 if the characters match, -1 if they mismatch, and -2 if either of the
characters is a gap.
>>> score_function('A', 'A')
1
>>> score_function('A', 'C')
-1
>>> score_function('-', 'A')
-2
>>> score_function('A', '-')
-2
>>> score_function('-', '-')
-2
"""
if "-" in (source_char, target_char):
return gap
return match if source_char == target_char else mismatch

def smith_waterman(
query: str,
subject: str,
match: int = 1,
mismatch: int = -1,
gap: int = -2,
) -> list[list[int]]:
"""
Perform the Smith-Waterman local sequence alignment algorithm.
Returns a 2D list representing the score matrix. Each value in the matrix
corresponds to the score of the best local alignment ending at that point.
>>> smith_waterman('ACAC', 'CA')
[[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 2], [0, 1, 0]]
>>> smith_waterman('acac', 'ca')
[[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 2], [0, 1, 0]]
>>> smith_waterman('ACAC', 'ca')
[[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 2], [0, 1, 0]]
>>> smith_waterman('acac', 'CA')
[[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 2], [0, 1, 0]]
>>> smith_waterman('ACAC', '')
[[0], [0], [0], [0], [0]]
>>> smith_waterman('', 'CA')
[[0, 0, 0]]
>>> smith_waterman('ACAC', 'CA')
[[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 2], [0, 1, 0]]

>>> smith_waterman('acac', 'ca')
[[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 2], [0, 1, 0]]

>>> smith_waterman('ACAC', 'ca')
[[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 2], [0, 1, 0]]

>>> smith_waterman('acac', 'CA')
[[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 2], [0, 1, 0]]

>>> smith_waterman('ACAC', '')
[[0], [0], [0], [0], [0]]

>>> smith_waterman('', 'CA')
[[0, 0, 0]]

>>> smith_waterman('AGT', 'AGT')
[[0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 2, 0], [0, 0, 0, 3]]

>>> smith_waterman('AGT', 'GTA')
[[0, 0, 0, 0], [0, 0, 0, 1], [0, 1, 0, 0], [0, 0, 2, 0]]

>>> smith_waterman('AGT', 'GTC')
[[0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 2, 0]]

>>> smith_waterman('AGT', 'G')
[[0, 0], [0, 0], [0, 1], [0, 0]]

>>> smith_waterman('G', 'AGT')
[[0, 0, 0, 0], [0, 0, 1, 0]]

>>> smith_waterman('AGT', 'AGTCT')
[[0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0], [0, 0, 0, 3, 1, 1]]

>>> smith_waterman('AGTCT', 'AGT')
[[0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 2, 0], [0, 0, 0, 3], [0, 0, 0, 1], [0, 0, 0, 1]]

>>> smith_waterman('AGTCT', 'GTC')
[[0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 2, 0], [0, 0, 0, 3], [0, 0, 1, 1]]
"""
# make both query and subject uppercase
query = query.upper()
subject = subject.upper()

# Initialize score matrix
m = len(query)
n = len(subject)
score = [[0] * (n + 1) for _ in range(m + 1)]
kwargs = {"match": match, "mismatch": mismatch, "gap": gap}

for i in range(1, m + 1):
for j in range(1, n + 1):
# Calculate scores for each cell
match = score[i - 1][j - 1] + score_function(
query[i - 1], subject[j - 1], **kwargs
)
delete = score[i - 1][j] + gap
insert = score[i][j - 1] + gap

# Take maximum score
score[i][j] = max(0, match, delete, insert)

return score

def traceback(score: list[list[int]], query: str, subject: str) -> str:
r"""
Perform traceback to find the optimal local alignment.
Starts from the highest scoring cell in the matrix and traces back recursively
until a 0 score is found. Returns the alignment strings.
>>> traceback([[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 2], [0, 1, 0]], 'ACAC', 'CA')
'CA\nCA'
>>> traceback([[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 2], [0, 1, 0]], 'acac', 'ca')
'CA\nCA'
>>> traceback([[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 2], [0, 1, 0]], 'ACAC', 'ca')
'CA\nCA'
>>> traceback([[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 2], [0, 1, 0]], 'acac', 'CA')
'CA\nCA'
>>> traceback([[0, 0, 0]], 'ACAC', '')
''
"""
# make both query and subject uppercase
query = query.upper()
subject = subject.upper()
# find the indices of the maximum value in the score matrix
max_value = float("-inf")
i_max = j_max = 0
for i, row in enumerate(score):
for j, value in enumerate(row):
if value > max_value:
max_value = value
i_max, j_max = i, j
# Traceback logic to find optimal alignment
i = i_max
j = j_max
align1 = ""
align2 = ""
gap = score_function("-", "-")
# guard against empty query or subject
if i == 0 or j == 0:
return ""
while i > 0 and j > 0:
if score[i][j] == score[i - 1][j - 1] + score_function(
query[i - 1], subject[j - 1]
):
# optimal path is a diagonal take both letters
align1 = query[i - 1] + align1
align2 = subject[j - 1] + align2
i -= 1
j -= 1
elif score[i][j] == score[i - 1][j] + gap:
# optimal path is a vertical
align1 = query[i - 1] + align1
align2 = f"-{align2}"
i -= 1
else:
# optimal path is a horizontal
align1 = f"-{align1}"
align2 = subject[j - 1] + align2
j -= 1

return f"{align1}\n{align2}"

if __name__ == "__main__":
query = "HEAGAWGHEE"
subject = "PAWHEAE"

score = smith_waterman(query, subject, match=1, mismatch=-1, gap=-2)
print(traceback(score, query, subject))
```