feat(dynamic_programming): add Needleman-Wunsch global sequence align…

dynamic_programming/needleman_wunsch.py

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+"""Needleman-Wunsch algorithm for global sequence alignment.
+
+Reference:
+    https://en.wikipedia.org/wiki/Needleman%E2%80%93Wunsch_algorithm
+
+The Needleman-Wunsch algorithm (1970) is a dynamic programming algorithm
+used in bioinformatics and computational biology to find the optimal global
+alignment between two sequences (such as DNA, RNA, or protein sequences).
+
+Unlike local alignment algorithms (e.g., Smith-Waterman), which find the
+highest-scoring local sub-regions, Needleman-Wunsch aligns both sequences across
+their entire lengths from start to finish.
+
+Algorithm:
+1. Initialization:
+   - Construct a matrix of size (m + 1) x (n + 1) where m and n are sequence lengths.
+   - Initialize boundary conditions:
+     score_matrix[i][0] = i * gap_score
+     score_matrix[0][j] = j * gap_score
+
+2. Matrix Filling (Recurrence Relation):
+   For each cell (i, j):
+     diagonal = score_matrix[i - 1][j - 1] + (match_score if seq1[i-1] == seq2[j-1]
+                                              else mismatch_score)
+     deletion = score_matrix[i - 1][j] + gap_score
+     insertion = score_matrix[i][j - 1] + gap_score
+     score_matrix[i][j] = max(diagonal, deletion, insertion)
+
+3. Traceback:
+   - Start from the bottom-right cell (m, n) and trace back to (0, 0).
+   - At each step, determine which direction (diagonal, up, or left) produced the
+     maximum score, assembling the aligned sequences in reverse order.
+
+Complexity:
+    Time Complexity:  O(m * n) where m and n are the lengths of the sequences.
+    Space Complexity: O(m * n) to store the score matrix for traceback.
+"""
+
+from __future__ import annotations
+
+
+def needleman_wunsch(
+    sequence1: str,
+    sequence2: str,
+    match_score: int = 1,
+    mismatch_score: int = -1,
+    gap_score: int = -1,
+) -> tuple[str, str, int]:
+    """Compute the optimal global sequence alignment using Needleman-Wunsch.
+
+    Parameters:
+        sequence1: The first input sequence to align.
+        sequence2: The second input sequence to align.
+        match_score: Score awarded when two characters match (default: 1).
+        mismatch_score: Penalty score when characters do not match (default: -1).
+        gap_score: Penalty score for introducing a gap '-' (default: -1).
+
+    Returns:
+        A tuple containing:
+        - aligned_sequence1: The first aligned sequence with inserted gaps.
+        - aligned_sequence2: The second aligned sequence with inserted gaps.
+        - alignment_score: The total optimal alignment score.
+
+    Raises:
+        ValueError: If gap_score is positive (gap must be neutral or a penalty).
+
+    Examples:
+        >>> # Wikipedia classic example
+        >>> needleman_wunsch(
+        ...     "GCATGCG", "GATTACA", match_score=1, mismatch_score=-1, gap_score=-1
+        ... )
+        ('GCA-TGCG', 'G-ATTACA', 0)
+
+        >>> # Identical sequences
+        >>> needleman_wunsch(
+        ...     "ACGT", "ACGT", match_score=2, mismatch_score=-1, gap_score=-2
+        ... )
+        ('ACGT', 'ACGT', 8)
+
+        >>> # Completely mismatched sequences
+        >>> needleman_wunsch(
+        ...     "AAAA", "TTTT", match_score=1, mismatch_score=-1, gap_score=-2
+        ... )
+        ('AAAA', 'TTTT', -4)
+
+        >>> # One sequence is empty
+        >>> needleman_wunsch("AGTC", "", match_score=1, mismatch_score=-1, gap_score=-1)
+        ('AGTC', '----', -4)
+
+        >>> # Both sequences are empty
+        >>> needleman_wunsch("", "")
+        ('', '', 0)
+
+        >>> # Protein sequence example
+        >>> needleman_wunsch(
+        ...     "HEAGAWGHEE", "PAWHEAE", match_score=2, mismatch_score=-1, gap_score=-2
+        ... )
+        ('HEAGAWGHE-E', '---PAW-HEAE', -1)
+
+        >>> # Invalid gap score
+        >>> needleman_wunsch("A", "C", gap_score=5)
+        Traceback (most recent call last):
+            ...
+        ValueError: gap_score must be non-positive (<= 0)
+    """
+    if gap_score > 0:
+        msg = "gap_score must be non-positive (<= 0)"
+        raise ValueError(msg)
+
+    first_sequence_length = len(sequence1)
+    second_sequence_length = len(sequence2)
+
+    # Initialize the (m + 1) x (n + 1) dynamic programming score matrix
+    score_matrix = [
+        [0] * (second_sequence_length + 1) for _ in range(first_sequence_length + 1)
+    ]
+
+    # Fill base-case boundary penalties
+    for row_index in range(first_sequence_length + 1):
+        score_matrix[row_index][0] = row_index * gap_score
+    for col_index in range(second_sequence_length + 1):
+        score_matrix[0][col_index] = col_index * gap_score
+
+    # Populate the score matrix using dynamic programming
+    for row_index in range(1, first_sequence_length + 1):
+        for col_index in range(1, second_sequence_length + 1):
+            char1 = sequence1[row_index - 1]
+            char2 = sequence2[col_index - 1]
+            substitution = match_score if char1 == char2 else mismatch_score
+
+            diagonal_score = score_matrix[row_index - 1][col_index - 1] + substitution
+            deletion_score = score_matrix[row_index - 1][col_index] + gap_score
+            insertion_score = score_matrix[row_index][col_index - 1] + gap_score
+
+            score_matrix[row_index][col_index] = max(
+                diagonal_score, deletion_score, insertion_score
+            )
+
+    # Traceback from bottom-right (m, n) to top-left (0, 0)
+    aligned_chars_first: list[str] = []
+    aligned_chars_second: list[str] = []
+    curr_row = first_sequence_length
+    curr_col = second_sequence_length
+
+    while curr_row > 0 or curr_col > 0:
+        if curr_row > 0 and curr_col > 0:
+            char1 = sequence1[curr_row - 1]
+            char2 = sequence2[curr_col - 1]
+            substitution = match_score if char1 == char2 else mismatch_score
+
+            # Check if diagonal step was optimal
+            if (
+                score_matrix[curr_row][curr_col]
+                == score_matrix[curr_row - 1][curr_col - 1] + substitution
+            ):
+                aligned_chars_first.append(char1)
+                aligned_chars_second.append(char2)
+                curr_row -= 1
+                curr_col -= 1
+                continue
+
+        # Check if vertical step (gap in second sequence) was optimal
+        if (
+            curr_row > 0
+            and score_matrix[curr_row][curr_col]
+            == score_matrix[curr_row - 1][curr_col] + gap_score
+        ):
+            aligned_chars_first.append(sequence1[curr_row - 1])
+            aligned_chars_second.append("-")
+            curr_row -= 1
+        else:
+            # Horizontal step (gap in first sequence)
+            aligned_chars_first.append("-")
+            aligned_chars_second.append(sequence2[curr_col - 1])
+            curr_col -= 1
+
+    aligned_sequence1 = "".join(reversed(aligned_chars_first))
+    aligned_sequence2 = "".join(reversed(aligned_chars_second))
+    final_score = score_matrix[first_sequence_length][second_sequence_length]
+
+    return aligned_sequence1, aligned_sequence2, final_score
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()