Python Program for Cutting a Rod | DP-13

Given a rod of length n inches and an array of prices that contains prices of all pieces of size smaller than n. Determine the maximum value obtainable by cutting up the rod and selling the pieces. For example, if length of the rod is 8 and the values of different pieces are given as following, then the maximum obtainable value is 22 (by cutting in two pieces of lengths 2 and 6)


length   | 1   2   3   4   5   6   7   8  
--------------------------------------------
price    | 1   5   8   9  10  17  17  20

And if the prices are as following, then the maximum obtainable value is 24 (by cutting in eight pieces of length 1)

length   | 1   2   3   4   5   6   7   8  
--------------------------------------------
price    | 3   5   8   9  10  17  17  20

Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

Following is simple recursive implementation of the Rod Cutting problem. The implementation simply follows the recursive structure mentioned above.

Python

# A Naive recursive solution  
# for Rod cutting problem 
import sys 
  
# A utility function to get the 
# maximum of two integers 
def max(a, b): 
    return a if (a > b) else b 
      
# Returns the best obtainable price for a rod of length n  
# and price[] as prices of different pieces 
def cutRod(price, n): 
    if(n <= 0): 
        return 0
    max_val = -sys.maxsize-1
      
    # Recursively cut the rod in different pieces   
    # and compare different configurations 
    for i in range(0, n): 
        max_val = max(max_val, price[i] + 
                      cutRod(price, n - i - 1)) 
    return max_val 
  
# Driver code 
arr = [1, 5, 8, 9, 10, 17, 17, 20] 
size = len(arr) 
print("Maximum Obtainable Value is", cutRod(arr, size)) 
  
# This code is contributed by 'Smitha Dinesh Semwal' 

Output:

Maximum Obtainable Value is 22

Considering the above implementation, following is recursion tree for a Rod of length 4.

cR() ---> cutRod() 

                             cR(4)
                  /        /           
                 /        /              
             cR(3)       cR(2)     cR(1)   cR(0)
            /  |         /         |
           /   |        /          |  
      cR(2) cR(1) cR(0) cR(1) cR(0) cR(0)
     /        |          |
    /         |          |   
  cR(1) cR(0) cR(0)      cR(0)
   /
 /
CR(0)

In the above partial recursion tree, cR(2) is being solved twice. We can see that there are many subproblems which are solved again and again. Since same suproblems are called again, this problem has Overlapping Subprolems property. So the Rod Cutting problem has both properties (see this and this) of a dynamic programming problem. Like other typical Dynamic Programming(DP) problems, recomputations of same subproblems can be avoided by constructing a temporary array val[] in bottom up manner.

Python

# A Dynamic Programming solution for Rod cutting problem 
INT_MIN = -32767
  
# Returns the best obtainable price for a rod of length n and 
# price[] as prices of different pieces 
def cutRod(price, n): 
    val = [0 for x in range(n + 1)] 
    val[0] = 0
  
    # Build the table val[] in bottom up manner and return 
    # the last entry from the table 
    for i in range(1, n + 1): 
        max_val = INT_MIN 
        for j in range(i): 
             max_val = max(max_val, price[j] + val[i-j-1]) 
        val[i] = max_val 
  
    return val[n] 
  
# Driver program to test above functions 
arr = [1, 5, 8, 9, 10, 17, 17, 20] 
size = len(arr) 
print("Maximum Obtainable Value is " + str(cutRod(arr, size))) 
  
# This code is contributed by Bhavya Jain 

Output:

Maximum Obtainable Value is 22

Please refer complete article on Cutting a Rod | DP-13 for more details!

My Personal Notes

Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

Python

Python

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