feat: add longest repeating subsequence algorithm

Add a dynamic programming implementation of the Longest Repeating Subsequence (LRS) algorithm. This finds the longest subsequence that occurs at least twice in a string, using a variation of LCS where the string is compared with itself but characters at the same index cannot be reused. Includes comprehensive doctests and type hints.

Author: JPisOP007

dynamic_programming/longest_repeating_subsequence.py

@@ -0,0 +1,88 @@
+"""
+Longest Repeating Subsequence (LRS)
+
+Given a string, find the length of the longest repeating subsequence, i.e., the
+longest subsequence that occurs at least twice. The two occurrences must use
+characters at different positions in the original string.
+
+This is a variation of the Longest Common Subsequence (LCS) problem where we find
+the LCS of the string with itself, with the constraint that characters at the same
+index position cannot both be used.
+
+Reference: https://en.wikipedia.org/wiki/Longest_common_subsequence_problem
+"""
+
+
+def longest_repeating_subsequence(string: str) -> int:
+    """
+    Find the length of the longest repeating subsequence in the given string.
+
+    A repeating subsequence is a subsequence that appears at least twice in the
+    string, where characters at the same index are not counted as part of both
+    subsequences simultaneously.
+
+    Uses dynamic programming with time complexity O(n^2) and space complexity O(n^2),
+    where n is the length of the input string.
+
+    Parameters
+    ----------
+    string : str
+        The input string to search for repeating subsequences.
+
+    Returns
+    -------
+    int
+        The length of the longest repeating subsequence.
+
+    Examples
+    --------
+    >>> longest_repeating_subsequence("aabb")
+    2
+    >>> longest_repeating_subsequence("aab")
+    1
+    >>> longest_repeating_subsequence("axxxy")
+    2
+    >>> longest_repeating_subsequence("abcabc")
+    3
+    >>> longest_repeating_subsequence("")
+    0
+    >>> longest_repeating_subsequence("a")
+    0
+    >>> longest_repeating_subsequence("abcdef")
+    0
+    >>> longest_repeating_subsequence("aaa")
+    2
+    >>> longest_repeating_subsequence("aaaa")
+    3
+    >>> longest_repeating_subsequence(12345)
+    Traceback (most recent call last):
+        ...
+    TypeError: Input must be a string, got int
+    """
+    if not isinstance(string, str):
+        msg = f"Input must be a string, got {type(string).__name__}"
+        raise TypeError(msg)
+
+    n = len(string)
+    if n == 0:
+        return 0
+
+    # dp[i][j] stores the length of the longest repeating subsequence
+    # considering string[0..i-1] and string[0..j-1]
+    dp = [[0] * (n + 1) for _ in range(n + 1)]
+
+    for i in range(1, n + 1):
+        for j in range(1, n + 1):
+            # Characters match and are at different positions
+            if string[i - 1] == string[j - 1] and i != j:
+                dp[i][j] = dp[i - 1][j - 1] + 1
+            else:
+                dp[i][j] = max(dp[i - 1][j], dp[i][j - 1])
+
+    return dp[n][n]
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()