add: median of medians algorithm in maths

maths/median_of_medians.py

@@ -0,0 +1,69 @@
+"""
+Median of Medians Algorithm
+Guarantees O(n) worst-case time complexity for finding the k-th smallest element.
+Reference: https://en.wikipedia.org/wiki/Median_of_medians
+"""
+
+
+def partition(arr: list, pivot: int) -> tuple[list, list, list]:
+    """
+    Partition array into elements less than, equal to, and greater than pivot.
+
+    >>> partition([3, 1, 4, 1, 5], 3)
+    ([1, 1], [3], [4, 5])
+    >>> partition([7, 7, 7], 7)
+    ([], [7, 7, 7], [])
+    """
+    low = [x for x in arr if x < pivot]
+    equal = [x for x in arr if x == pivot]
+    high = [x for x in arr if x > pivot]
+    return low, equal, high
+
+
+def median_of_medians(arr: list, rank: int) -> int:
+    """
+    Find the k-th smallest element in an unsorted list using Median of Medians.
+
+    Args:
+        arr: List of comparable elements
+        rank: 1-based index of the desired smallest element
+
+    Returns:
+        The k-th smallest element in arr
+
+    Raises:
+        ValueError: If rank is out of range
+
+    >>> median_of_medians([3, 1, 4, 1, 5, 9, 2, 6], 3)
+    2
+    >>> median_of_medians([7, 2, 10, 5], 1)
+    2
+    >>> median_of_medians([1, 2, 3, 4, 5], 5)
+    5
+    """
+    if not 1 <= rank <= len(arr):
+        msg = f"rank={rank} is out of range for array of length {len(arr)}"
+    raise ValueError(msg)
+
+    if len(arr) <= 5:
+        return sorted(arr)[rank - 1]
+
+    chunks = [arr[i : i + 5] for i in range(0, len(arr), 5)]
+    medians = [sorted(chunk)[len(chunk) // 2] for chunk in chunks]
+
+    pivot = median_of_medians(medians, len(medians) // 2 + 1)
+
+    low, equal, high = partition(arr, pivot)
+
+    if rank <= len(low):
+        return median_of_medians(low, rank)
+    elif rank <= len(low) + len(equal):
+        return pivot
+    else:
+        return median_of_medians(high, rank - len(low) - len(equal))
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()