max_sub_array.py: Add Maximum Subarray

Divide and Conquer method to solve maximum subarray problem

closes NITSkmOS#215

knapsack_problem/Python/knapsack.py

@@ -0,0 +1,63 @@
+"""
+Here Knapsack problem has been implemented using Greedy Approach
+where the weight and and corresponding profit of some items has been
+given. And also the maximum capacity of a sack(bag) is given. we have
+to take items so that capacity does not exceed and we get maximum profit.
+For more information vist- <https://en.wikipedia.org/wiki/Knapsack_problem>
+"""
+
+
+def knapsack(profit, weight, capacity):
+    """
+    :param profit: array of profit of the items
+    :param weight: array of weight of the items
+    :param capacity: capacity of the sack
+    :return: maximum profit and the fraction of items
+    """
+
+    # array of profit/weight ratio
+    ratio = [v / w for v, w in zip(profit, weight)]
+
+    # a list of (0, 1, ..., n-1)
+    index = list(range(len(profit)))
+
+    # index is sorted according to ratio in descending order
+    index.sort(key=lambda i: ratio[i], reverse=True)
+
+    # max_profit is the maximum profit gained
+    max_profit = 0
+
+    # fraction is the fraction in which items should be taken
+    fraction = [0] * len(profit)
+
+    for i in index:
+        if weight[i] <= capacity:
+            fraction[i] = 1
+            max_profit += profit[i]
+            capacity -= weight[i]
+        else:
+            fraction[i] = capacity / weight[i]
+            max_profit += profit[i] * fraction[i]
+            break
+
+    return max_profit, fraction
+
+
+def main():
+    # profit is array of profit of the items
+    # weight is array of weight of the items
+    # capacity is capacity of the sack
+
+    profit = [50, 60, 80]
+    weight = [10, 30, 20]
+    capacity = 50
+
+    # max_profit is the maximum profit gained
+    # fraction is the fraction in which items should be taken
+    max_profit, fraction = knapsack(profit, weight, capacity)
+    print('Maximum profit:', max_profit)
+    print('Items should be taken in fraction of:', fraction)
+
+
+if __name__ == '__main__':
+    main()