diff --git a/dynamic_programming/subsequence_algorithms.py b/dynamic_programming/subsequence_algorithms.py
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+"""
+Author  : Mehdi ALAOUI
+
+This is a pure Python implementation of Dynamic Programming solutions to:
+1. Longest Increasing Subsequence (LIS)
+2. Longest Common Subsequence (LCS)
+
+1. LIS Problem: Given an array, find the longest increasing sub-array and return it.
+Example: [10, 22, 9, 33, 21, 50, 41, 60, 80] -> [10, 22, 33, 41, 60, 80]
+
+2. LCS Problem: Given two sequences, find the length and content of the longest
+common subsequence that appears in both of them. A subsequence appears in the
+same relative order but not necessarily continuously.
+Example: "programming" and "gaming" -> "gaming"
+"""
+
+from __future__ import annotations
+
+
+# Longest Increasing Subsequence (LIS)
+def longest_subsequence(array: list[int]) -> list[int]:  # This function is recursive
+    """
+    Some examples
+    >>> longest_subsequence([10, 22, 9, 33, 21, 50, 41, 60, 80])
+    [10, 22, 33, 41, 60, 80]
+    >>> longest_subsequence([4, 8, 7, 5, 1, 12, 2, 3, 9])
+    [1, 2, 3, 9]
+    >>> longest_subsequence([9, 8, 7, 6, 5, 7])
+    [8]
+    >>> longest_subsequence([1, 1, 1])
+    [1, 1, 1]
+    >>> longest_subsequence([])
+    []
+    """
+    array_length = len(array)
+    # If the array contains only one element, we return it (it's the stop condition of
+    # recursion)
+    if array_length <= 1:
+        return array
+        # Else
+    pivot = array[0]
+    is_found = False
+    i = 1
+    longest_subseq: list[int] = []
+    while not is_found and i < array_length:
+        if array[i] < pivot:
+            is_found = True
+            temp_array = [element for element in array[i:] if element >= array[i]]
+            temp_array = longest_subsequence(temp_array)
+            if len(temp_array) > len(longest_subseq):
+                longest_subseq = temp_array
+        else:
+            i += 1
+
+    temp_array = [element for element in array[1:] if element >= pivot]
+    temp_array = [pivot, *longest_subsequence(temp_array)]
+    if len(temp_array) > len(longest_subseq):
+        return temp_array
+    else:
+        return longest_subseq
+
+
+# Longest Common Subsequence (LCS)
+def longest_common_subsequence(x: str, y: str):
+    """
+    Finds the longest common subsequence between two strings. Also returns the
+    The subsequence found
+
+    Parameters
+    ----------
+
+    x: str, one of the strings
+    y: str, the other string
+
+    Returns
+    -------
+    L[m][n]: int, the length of the longest subsequence. Also equal to len(seq)
+    Seq: str, the subsequence found
+
+    >>> longest_common_subsequence("programming", "gaming")
+    (6, 'gaming')
+    >>> longest_common_subsequence("physics", "smartphone")
+    (2, 'ph')
+    >>> longest_common_subsequence("computer", "food")
+    (1, 'o')
+    >>> longest_common_subsequence("", "abc")  # One string is empty
+    (0, '')
+    >>> longest_common_subsequence("abc", "")  # Other string is empty
+    (0, '')
+    >>> longest_common_subsequence("", "")  # Both strings are empty
+    (0, '')
+    >>> longest_common_subsequence("abc", "def")  # No common subsequence
+    (0, '')
+    >>> longest_common_subsequence("abc", "abc")  # Identical strings
+    (3, 'abc')
+    >>> longest_common_subsequence("a", "a")  # Single character match
+    (1, 'a')
+    >>> longest_common_subsequence("a", "b")  # Single character no match
+    (0, '')
+    >>> longest_common_subsequence("abcdef", "ace")  # Interleaved subsequence
+    (3, 'ace')
+    >>> longest_common_subsequence("ABCD", "ACBD")  # No repeated characters
+    (3, 'ABD')
+    """
+    # find the length of strings
+
+    assert x is not None
+    assert y is not None
+
+    m = len(x)
+    n = len(y)
+
+    # declaring the array for storing the dp values
+    dp = [[0] * (n + 1) for _ in range(m + 1)]
+
+    for i in range(1, m + 1):
+        for j in range(1, n + 1):
+            match = 1 if x[i - 1] == y[j - 1] else 0
+
+            dp[i][j] = max(dp[i - 1][j], dp[i][j - 1], dp[i - 1][j - 1] + match)
+
+    seq = ""
+    i, j = m, n
+    while i > 0 and j > 0:
+        match = 1 if x[i - 1] == y[j - 1] else 0
+
+        if dp[i][j] == dp[i - 1][j - 1] + match:
+            if match == 1:
+                seq = x[i - 1] + seq
+            i -= 1
+            j -= 1
+        elif dp[i][j] == dp[i - 1][j]:
+            i -= 1
+        else:
+            j -= 1
+
+    return dp[m][n], seq
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+
+    # Example usage for LIS
+    arr = [10, 22, 9, 33, 21, 50, 41, 60, 80]
+    lis = longest_subsequence(arr)
+    print(f"Longest Increasing Subsequence: {lis}")
+
+    # Example usage for LCS
+    str1 = "AGGTAB"
+    str2 = "GXTXAYB"
+    length, lcs = longest_common_subsequence(str1, str2)
+    print(f"Longest Common Subsequence: '{lcs}' with length {length}")