Divide and conquer Algorithms Issue#817 (#938)

* add ons in string directory - Bayer_Moore_Search

* created divide_and_conquer folder and added max_sub_array_sum.py under it (issue #817)

divide_and_conquer/max_sub_array_sum.py

@@ -0,0 +1,72 @@
+""" 
+Given a array of length n, max_sub_array_sum() finds the maximum of sum of contiguous sub-array using divide and conquer method.
+
+Time complexity : O(n log n)
+
+Ref : INTRODUCTION TO ALGORITHMS THIRD EDITION (section : 4, sub-section : 4.1, page : 70)
+
+"""
+
+
+def max_sum_from_start(array):
+    """ This function finds the maximum contiguous sum of array from 0 index
+
+    Parameters : 
+    array (list[int]) : given array
+    
+    Returns : 
+    max_sum (int) : maximum contiguous sum of array from 0 index
+
+    """
+    array_sum = 0
+    max_sum = float("-inf")
+    for num in array:
+        array_sum += num
+        if array_sum > max_sum:
+            max_sum = array_sum
+    return max_sum
+
+
+def max_cross_array_sum(array, left, mid, right):
+    """ This function finds the maximum contiguous sum of left and right arrays
+
+    Parameters : 
+    array, left, mid, right (list[int], int, int, int) 
+    
+    Returns : 
+    (int) :  maximum of sum of contiguous sum of left and right arrays
+
+    """
+
+    max_sum_of_left = max_sum_from_start(array[left:mid+1][::-1])
+    max_sum_of_right = max_sum_from_start(array[mid+1: right+1])
+    return max_sum_of_left + max_sum_of_right
+
+
+def max_sub_array_sum(array, left, right):
+    """ This function finds the maximum of sum of contiguous sub-array using divide and conquer method
+
+    Parameters : 
+    array, left, right (list[int], int, int) : given array, current left index and current right index
+    
+    Returns : 
+    int :  maximum of sum of contiguous sub-array
+
+    """
+
+    # base case: array has only one element
+    if left == right:
+        return array[right]
+    
+    # Recursion
+    mid = (left + right) // 2
+    left_half_sum = max_sub_array_sum(array, left, mid)
+    right_half_sum = max_sub_array_sum(array, mid + 1, right)
+    cross_sum = max_cross_array_sum(array, left, mid, right)
+    return max(left_half_sum, right_half_sum, cross_sum)
+
+
+array = [-2, -5, 6, -2, -3, 1, 5, -6]
+array_length = len(array)
+print("Maximum sum of contiguous subarray:", max_sub_array_sum(array, 0, array_length - 1))
+