Add a divide and conquer method in finding the maximum difference pair (#3692)

* A divide and conquer method in finding the maximum difference pair

* fix formatting issues

* fix formatting issues

* add doctest runner

divide_and_conquer/max_difference_pair.py

@@ -0,0 +1,47 @@
+from typing import List
+
+
+def max_difference(a: List[int]) -> (int, int):
+    """
+    We are given an array A[1..n] of integers, n >= 1. We want to
+    find a pair of indices (i, j) such that
+    1 <= i <= j <= n and A[j] - A[i] is as large as possible.
+
+    Explanation:
+    https://www.geeksforgeeks.org/maximum-difference-between-two-elements/
+
+    >>> max_difference([5, 11, 2, 1, 7, 9, 0, 7])
+    (1, 9)
+    """
+    # base case
+    if len(a) == 1:
+        return a[0], a[0]
+    else:
+        # split A into half.
+        first = a[: len(a) // 2]
+        second = a[len(a) // 2 :]
+
+        # 2 sub problems, 1/2 of original size.
+        small1, big1 = max_difference(first)
+        small2, big2 = max_difference(second)
+
+        # get min of first and max of second
+        # linear time
+        min_first = min(first)
+        max_second = max(second)
+
+        # 3 cases, either (small1, big1),
+        # (min_first, max_second), (small2, big2)
+        # constant comparisons
+        if big2 - small2 > max_second - min_first and big2 - small2 > big1 - small1:
+            return small2, big2
+        elif big1 - small1 > max_second - min_first:
+            return small1, big1
+        else:
+            return min_first, max_second
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()