Python Program for Cutting a Rod | DP-13

17 Jan, 2022
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
Following is simple recursive implementation of the Rod Cutting problem. The implementation simply follows the recursive structure mentioned above.
Python3
# A Naive recursive solution# for Rod cutting problemimport sys # A utility function to get the# maximum of two integersdef 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 piecesdef 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 codearr = [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'
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 subproblems are called again, this problem has Overlapping Subproblems 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.
Python3
# A Dynamic Programming solution for Rod cutting problemINT_MIN = -32767 # Returns the best obtainable price for a rod of length n and# price[] as prices of different piecesdef 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 functionsarr = [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
Maximum Obtainable Value is 22
 
Please refer complete article on Cutting a Rod | DP-13 for more details! 
anikakapoor
amartyaghoshgfg
Python Programs
Writing code in comment?
Please use ide.geeksforgeeks.org,
generate link and share the link here.
Python Program to check Armstrong Number
Python program to interchange first and last elements in a list
Differences and Applications of List, Tuple, Set and Dictionary in Python
Appending to list in Python dictionary
Python Program for simple interest
Python Program for Merge Sort
Python Program for n-th Fibonacci number
Python | Find most frequent element in a list
Python | Remove spaces from a string
Python Program to find largest element in an array