Add Damerau-Levenshtein distance algorithm (#10159)

* Add Damerau-Levenshtein distance algorithm

* fix: precommit check

* fix: doc correction

* [pre-commit.ci] auto fixes from pre-commit.com hooks

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* refactor: use variable for length and doc correction

* Update damerau_levenshtein_distance.py

* Update damerau_levenshtein_distance.py

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>

strings/damerau_levenshtein_distance.py

@@ -0,0 +1,71 @@
+"""
+This script is a implementation of the Damerau-Levenshtein distance algorithm.
+
+It's an algorithm that measures the edit distance between two string sequences
+
+More information about this algorithm can be found in this wikipedia article:
+https://en.wikipedia.org/wiki/Damerau%E2%80%93Levenshtein_distance
+"""
+
+
+def damerau_levenshtein_distance(first_string: str, second_string: str) -> int:
+    """
+    Implements the Damerau-Levenshtein distance algorithm that measures
+    the edit distance between two strings.
+
+    Parameters:
+        first_string: The first string to compare
+        second_string: The second string to compare
+
+    Returns:
+        distance: The edit distance between the first and second strings
+
+    >>> damerau_levenshtein_distance("cat", "cut")
+    1
+    >>> damerau_levenshtein_distance("kitten", "sitting")
+    3
+    >>> damerau_levenshtein_distance("hello", "world")
+    4
+    >>> damerau_levenshtein_distance("book", "back")
+    2
+    >>> damerau_levenshtein_distance("container", "containment")
+    3
+    >>> damerau_levenshtein_distance("container", "containment")
+    3
+    """
+    # Create a dynamic programming matrix to store the distances
+    dp_matrix = [[0] * (len(second_string) + 1) for _ in range(len(first_string) + 1)]
+
+    # Initialize the matrix
+    for i in range(len(first_string) + 1):
+        dp_matrix[i][0] = i
+    for j in range(len(second_string) + 1):
+        dp_matrix[0][j] = j
+
+    # Fill the matrix
+    for i, first_char in enumerate(first_string, start=1):
+        for j, second_char in enumerate(second_string, start=1):
+            cost = int(first_char != second_char)
+
+            dp_matrix[i][j] = min(
+                dp_matrix[i - 1][j] + 1,  # Deletion
+                dp_matrix[i][j - 1] + 1,  # Insertion
+                dp_matrix[i - 1][j - 1] + cost,  # Substitution
+            )
+
+            if (
+                i > 1
+                and j > 1
+                and first_string[i - 1] == second_string[j - 2]
+                and first_string[i - 2] == second_string[j - 1]
+            ):
+                # Transposition
+                dp_matrix[i][j] = min(dp_matrix[i][j], dp_matrix[i - 2][j - 2] + cost)
+
+    return dp_matrix[-1][-1]
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()