Added code for memory function implementation

dynamic_programming/knapsack.py

@@ -1,6 +1,21 @@
 """
 Given weights and values of n items, put these items in a knapsack of capacity W to get the maximum total value in the knapsack.
 """
+def MF_knapsack(i,wt,val,j):
+    '''
+    This code involves the concept of memory functions. Here we solve the subproblems which are needed
+    unlike the below example
+    F is a 2D array with -1s filled up
+    '''
+    global F  # a global dp table for knapsack
+    if F[i][j] < 0:
+        if j < wt[i - 1]:
+            val = MF_knapsack(i - 1,wt,val,j)
+        else:
+            val = max(MF_knapsack(i - 1,wt,val,j),MF_knapsack(i - 1,wt,val,j - wt[i - 1]) + val[i - 1])
+        F[i][j] = val
+    return F[i][j]
+
 def knapsack(W, wt, val, n):
     dp = [[0 for i in range(W+1)]for j in range(n+1)]
 
@@ -12,13 +27,16 @@ def knapsack(W, wt, val, n):
                 dp[i][w] = dp[i-1][w]
 
     return dp[n][w]
-if __name__ == "__main__":
+
+if __name__ == '__main__':
+    '''
+    Adding test case for knapsack
+    '''
     val = [3,2,4,4]
     wt = [4,3,2,3]
-    W = 6
     n = 4
-    '''
-    Should give 8
-    '''
-    print(knapsack(W,wt,val,n))
-
+    w = 6
+    F = [[0]*(w + 1)] + [[0] + [-1 for i in range(w + 1)] for j in range(n + 1)]
+    print(knapsack(w,wt,val,n))
+    print(MF_knapsack(n,wt,val,w))  # switched the n and w 
+