Added Median of Medians Algorithm (#9864)

* Added Median of Medians Algorithm

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* Update median_of_medians.py as per requested changes

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

searches/median_of_medians.py

@@ -0,0 +1,107 @@
+"""
+A Python implementation of the Median of Medians algorithm
+to select pivots for quick_select, which is efficient for
+calculating the value that would appear in the index of a
+list if it would be sorted, even if it is not already
+sorted. Search in time complexity O(n) at any rank
+deterministically
+https://en.wikipedia.org/wiki/Median_of_medians
+"""
+
+
+def median_of_five(arr: list) -> int:
+    """
+    Return the median of the input list
+    :param arr: Array to find median of
+    :return: median of arr
+
+    >>> median_of_five([2, 4, 5, 7, 899])
+    5
+    >>> median_of_five([5, 7, 899, 54, 32])
+    32
+    >>> median_of_five([5, 4, 3, 2])
+    4
+    >>> median_of_five([3, 5, 7, 10, 2])
+    5
+    """
+    arr = sorted(arr)
+    return arr[len(arr) // 2]
+
+
+def median_of_medians(arr: list) -> int:
+    """
+    Return a pivot to partition data on by calculating
+    Median of medians of input data
+    :param arr: The data to be checked (a list)
+    :return: median of medians of input array
+
+    >>> median_of_medians([2, 4, 5, 7, 899, 54, 32])
+    54
+    >>> median_of_medians([5, 7, 899, 54, 32])
+    32
+    >>> median_of_medians([5, 4, 3, 2])
+    4
+    >>> median_of_medians([3, 5, 7, 10, 2, 12])
+    12
+    """
+
+    if len(arr) <= 5:
+        return median_of_five(arr)
+    medians = []
+    i = 0
+    while i < len(arr):
+        if (i + 4) <= len(arr):
+            medians.append(median_of_five(arr[i:].copy()))
+        else:
+            medians.append(median_of_five(arr[i : i + 5].copy()))
+        i += 5
+    return median_of_medians(medians)
+
+
+def quick_select(arr: list, target: int) -> int:
+    """
+    Two way partition the data into smaller and greater lists,
+    in relationship to the pivot
+    :param arr: The data to be searched (a list)
+    :param target: The rank to be searched
+    :return: element at rank target
+
+    >>> quick_select([2, 4, 5, 7, 899, 54, 32], 5)
+    32
+    >>> quick_select([2, 4, 5, 7, 899, 54, 32], 1)
+    2
+    >>> quick_select([5, 4, 3, 2], 2)
+    3
+    >>> quick_select([3, 5, 7, 10, 2, 12], 3)
+    5
+    """
+
+    # Invalid Input
+    if target > len(arr):
+        return -1
+
+    # x is the estimated pivot by median of medians algorithm
+    x = median_of_medians(arr)
+    left = []
+    right = []
+    check = False
+    for i in range(len(arr)):
+        if arr[i] < x:
+            left.append(arr[i])
+        elif arr[i] > x:
+            right.append(arr[i])
+        elif arr[i] == x and not check:
+            check = True
+        else:
+            right.append(arr[i])
+    rank_x = len(left) + 1
+    if rank_x == target:
+        answer = x
+    elif rank_x > target:
+        answer = quick_select(left, target)
+    elif rank_x < target:
+        answer = quick_select(right, target - rank_x)
+    return answer
+
+
+print(median_of_five([5, 4, 3, 2]))