Cutting a Rod | DP-13 - GeeksforGeeks

28 Mar, 2022
Given a rod of length n inches and an array of prices that includes 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 the length of the rod is 8 and the values of different pieces are given as the 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
A naive solution to this problem is to generate all configurations of different pieces and find the highest-priced configuration. This solution is exponential in terms of time complexity. Let us see how this problem possesses both important properties of a Dynamic Programming (DP) Problem and can efficiently be solved using Dynamic Programming.1) Optimal Substructure: We can get the best price by making a cut at different positions and comparing the values obtained after a cut. We can recursively call the same function for a piece obtained after a cut.Let cutRod(n) be the required (best possible price) value for a rod of length n. cutRod(n) can be written as follows.cutRod(n) = max(price[i] + cutRod(n-i-1)) for all i in {0, 1 .. n-1}2) Overlapping Subproblems The following is a simple recursive implementation of the Rod Cutting problem. 
The implementation simply follows the recursive structure mentioned above.  
Maximum Obtainable Value is 22
 Considering the above implementation, the following is the 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 solved twice. We can see that there are many subproblems that are solved again and again. Since the same subproblems are called again, this problem has the 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 the same subproblems can be avoided by constructing a temporary array val[] in a bottom-up manner. 
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// A Dynamic Programming solution for Rod cutting problem#include<iostream>#include <bits/stdc++.h>#include<math.h>using namespace std; // A utility function to get the maximum of two integersint max(int a, int b) { return (a > b)? a : b;} /* Returns the best obtainable price for a rod of length n and   price[] as prices of different pieces */int cutRod(int price[], int n){   int val[n+1];   val[0] = 0;   int i, j;    // Build the table val[] in bottom up manner and return the last entry   // from the table   for (i = 1; i<=n; i++)   {       int max_val = INT_MIN;       for (j = 0; j < i; j++)         max_val = max(max_val, price[j] + val[i-j-1]);       val[i] = max_val;   }    return val[n];} /* Driver program to test above functions */int main(){    int arr[] = {1, 5, 8, 9, 10, 17, 17, 20};    int size = sizeof(arr)/sizeof(arr[0]);    cout <<"Maximum Obtainable Value is "<<cutRod(arr, size);    getchar();    return 0;} // This code is contributed by shivanisinghss2110
// A Dynamic Programming solution for Rod cutting problem#include<stdio.h>#include<limits.h> // A utility function to get the maximum of two integersint max(int a, int b) { return (a > b)? a : b;} /* Returns the best obtainable price for a rod of length n and   price[] as prices of different pieces */int cutRod(int price[], int n){   int val[n+1];   val[0] = 0;   int i, j;    // Build the table val[] in bottom up manner and return the last entry   // from the table   for (i = 1; i<=n; i++)   {       int max_val = INT_MIN;       for (j = 0; j < i; j++)         max_val = max(max_val, price[j] + val[i-j-1]);       val[i] = max_val;   }    return val[n];} /* Driver program to test above functions */int main(){    int arr[] = {1, 5, 8, 9, 10, 17, 17, 20};    int size = sizeof(arr)/sizeof(arr[0]);    printf("Maximum Obtainable Value is %d", cutRod(arr, size));    getchar();    return 0;}
// A Dynamic Programming solution for Rod cutting problemclass RodCutting{    /* Returns the best obtainable price for a rod of       length n and price[] as prices of different pieces */    static int cutRod(int price[],int n)    {        int val[] = new int[n+1];        val[0] = 0;         // Build the table val[] in bottom up manner and return        // the last entry from the table        for (int i = 1; i<=n; i++)        {            int max_val = Integer.MIN_VALUE;            for (int j = 0; j < i; j++)                max_val = Math.max(max_val,                                   price[j] + val[i-j-1]);            val[i] = max_val;        }         return val[n];    }     /* Driver program to test above functions */    public static void main(String args[])    {        int arr[] = new int[] {1, 5, 8, 9, 10, 17, 17, 20};        int size = arr.length;        System.out.println("Maximum Obtainable Value is " +                            cutRod(arr, size));    }}/* This code is contributed by Rajat Mishra */
# 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
// A Dynamic Programming solution// for Rod cutting problemusing System;class GFG {     /* Returns the best obtainable       price for a rod of length n       and price[] as prices of       different pieces */    static int cutRod(int []price,int n)    {        int []val = new int[n + 1];        val[0] = 0;         // Build the table val[] in        // bottom up manner and return        // the last entry from the table        for (int i = 1; i<=n; i++)        {            int max_val = int.MinValue;            for (int j = 0; j < i; j++)                max_val = Math.Max(max_val,                          price[j] + val[i - j - 1]);            val[i] = max_val;        }         return val[n];    }         // Driver Code    public static void Main()    {        int []arr = new int[] {1, 5, 8, 9, 10, 17, 17, 20};        int size = arr.Length;        Console.WriteLine("Maximum Obtainable Value is " +                                        cutRod(arr, size));             }} // This code is contributed by Sam007
<?php// A Dynamic Programming solution for// Rod cutting problem /* Returns the best obtainable pricefor a rod of length n and price[] asprices of different pieces */function cutRod( $price, $n){    $val = array();    $val[0] = 0;    $i; $j;         // Build the table val[] in bottom    // up manner and return the last    // entry from the table    for ($i = 1; $i <= $n; $i++)    {        $max_val = PHP_INT_MIN;                 for ($j = 0; $j < $i; $j++)            $max_val = max($max_val,             $price[$j] + $val[$i-$j-1]);        $val[$i] = $max_val;    }         return $val[$n];} // Driver program to test above functions$arr = array(1, 5, 8, 9, 10, 17, 17, 20);$size = count($arr);echo "Maximum Obtainable Value is ",                      cutRod($arr, $size);     // This code is contributed by anuj_67.?>
<script>    // A Dynamic Programming solution    // for Rod cutting problem         /* Returns the best obtainable       price for a rod of length n       and price[] as prices of       different pieces */    function cutRod(price, n)    {        let val = new Array(n + 1);        val[0] = 0;          // Build the table val[] in        // bottom up manner and return        // the last entry from the table        for (let i = 1; i<=n; i++)        {            let max_val = Number.MIN_VALUE;            for (let j = 0; j < i; j++)                max_val = Math.max(max_val, price[j] + val[i - j - 1]);            val[i] = max_val;        }          return val[n];    }         let arr = [1, 5, 8, 9, 10, 17, 17, 20];    let size = arr.length;    document.write("Maximum Obtainable Value is " + cutRod(arr, size) + "n");</script>
Maximum Obtainable Value is 22
The Time Complexity of the above implementation is O(n^2), which is much better than the worst-case time complexity of Naive Recursive implementation.
3) Using the idea of Unbounded Knapsack.
This problem is very similar to the Unbounded Knapsack Problem, where there are multiple occurrences of the same item. Here the pieces of the rod.
Now I will create an analogy between Unbounded Knapsack and the Rod Cutting Problem. 
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C
Java
Python3
C#
Javascript
// CPP program for above approach#include <iostream>using namespace std; // Global Array for// the purpose of memoization.int t[9][9]; // A recursive program, using ,// memoization, to implement the// rod cutting problem(Top-Down).int un_kp(int price[], int length[],                    int Max_len, int n){     // The maximum price will be zero,    // when either the length of the rod    // is zero or price is zero.    if (n == 0 || Max_len == 0)    {        return 0;    }     // If the length of the rod is less    // than the maximum length, Max_lene will    // consider it.Now depending    // upon the profit,    // either Max_lene  we will take    // it or discard it.    if (length[n - 1] <= Max_len)    {        t[n][Max_len]            = max(price[n - 1]                      + un_kp(price, length,                           Max_len - length[n - 1], n),                  un_kp(price, length, Max_len, n - 1));    }     // If the length of the rod is    // greater than the permitted size,    // Max_len we will  not consider it.    else    {        t[n][Max_len]            = un_kp(price, length,                              Max_len, n - 1);    }     // Max_lene Max_lenill return the maximum    // value obtained, Max_lenhich is present    // at the nth roMax_len and Max_length column.    return t[n][Max_len];} /* Driver program totest above functions */int main(){    int price[] = { 1, 5, 8, 9, 10, 17, 17, 20 };    int n = sizeof(price) / sizeof(price[0]);    int length[n];    for (int i = 0; i < n; i++) {        length[i] = i + 1;    }    int Max_len = n;     // Function Call    cout << "Maximum obtained value  is "         << un_kp(price, length, n, Max_len) << endl;}
// C program for above approach#include <stdio.h>#include <stdlib.h> int max(int a, int b){  return (a > b) ? a : b;} // Global Array for the// purpose of memoization.int t[9][9]; // A recursive program, using ,// memoization, to implement the// rod cutting problem(Top-Down).int un_kp(int price[], int length[],                     int Max_len, int n){     // The maximum price will be zero,    // when either the length of the rod    // is zero or price is zero.    if (n == 0 || Max_len == 0)    {        return 0;    }     // If the length of the rod is less    // than the maximum length, Max_lene    // will consider it.Now depending    // upon the profit,    // either Max_lene  we will take it    // or discard it.    if (length[n - 1] <= Max_len)    {        t[n][Max_len]            = max(price[n - 1]                      + un_kp(price, length,                              Max_len - length[n - 1], n),                  un_kp(price, length, Max_len, n - 1));    }     // If the length of the rod is greater    // than the permitted size, Max_len    // we will  not consider it.    else    {        t[n][Max_len]            = un_kp(price, length,                             Max_len, n - 1);    }     // Max_lene Max_lenill return    // the maximum value obtained,    // Max_lenhich is present at the    // nth roMax_len and Max_length column.    return t[n][Max_len];} /* Driver program to test above functions */int main(){    int price[] = { 1, 5, 8, 9, 10, 17, 17, 20 };    int n = sizeof(price) / sizeof(price[0]);    int length[n];    for (int i = 0; i < n; i++)    {        length[i] = i + 1;    }    int Max_len = n;     // Function Call    printf("Maximum obtained value  is %d \n",           un_kp(price, length, n, Max_len));}
// Java program for above approachimport java.io.*; class GFG {     // Global Array for    // the purpose of memoization.    static int t[][] = new int[9][9];     // A recursive program, using ,    // memoization, to implement the    // rod cutting problem(Top-Down).    public static int un_kp(int price[], int length[],                            int Max_len, int n)    {         // The maximum price will be zero,        // when either the length of the rod        // is zero or price is zero.        if (n == 0 || Max_len == 0) {            return 0;        }         // If the length of the rod is less        // than the maximum length, Max_lene will        // consider it.Now depending        // upon the profit,        // either Max_lene  we will take        // it or discard it.        if (length[n - 1] <= Max_len) {            t[n][Max_len] = Math.max(                price[n - 1]                    + un_kp(price, length,                            Max_len - length[n - 1], n),                un_kp(price, length, Max_len, n - 1));        }         // If the length of the rod is        // greater than the permitted size,        // Max_len we will  not consider it.        else {            t[n][Max_len]                = un_kp(price, length, Max_len, n - 1);        }         // Max_lene Max_lenill return the maximum        // value obtained, Max_lenhich is present        // at the nth roMax_len and Max_length column.        return t[n][Max_len];    }     public static void main(String[] args)    {         int price[]            = new int[] { 1, 5, 8, 9, 10, 17, 17, 20 };        int n = price.length;        int length[] = new int[n];        for (int i = 0; i < n; i++) {            length[i] = i + 1;        }        int Max_len = n;        System.out.println(            "Maximum obtained value is "            + un_kp(price, length, n, Max_len));    }} // This code is contributed by rajsanghavi9.
# Python program for above approach # Global Array for# the purpose of memoization.t = [[0 for i in range(9)] for j in range(9)] # A recursive program, using ,# memoization, to implement the# rod cutting problem(Top-Down).def un_kp(price, length, Max_len, n):     # The maximum price will be zero,    # when either the length of the rod    # is zero or price is zero.    if (n == 0 or Max_len == 0):        return 0;          # If the length of the rod is less    # than the maximum length, Max_lene will    # consider it.Now depending    # upon the profit,    # either Max_lene we will take    # it or discard it.    if (length[n - 1] <= Max_len):        t[n][Max_len] = max(price[n - 1] + un_kp(price, length, Max_len - length[n - 1], n),                un_kp(price, length, Max_len, n - 1));          # If the length of the rod is    # greater than the permitted size,    # Max_len we will not consider it.    else:        t[n][Max_len] = un_kp(price, length, Max_len, n - 1);          # Max_lene Max_lenill return the maximum    # value obtained, Max_lenhich is present    # at the nth roMax_len and Max_length column.    return t[n][Max_len];  if __name__ == '__main__':     price = [1, 5, 8, 9, 10, 17, 17, 20 ];    n =len(price);    length = [0]*n;    for i in range(n):        length[i] = i + 1;         Max_len = n;    print("Maximum obtained value is " ,un_kp(price, length, n, Max_len)); # This code is contributed by gauravrajput1
// C# program for above approachusing System;public class GFG {   // Global Array for  // the purpose of memoization.  static int [,]t = new int[9,9];   // A recursive program, using ,  // memoization, to implement the  // rod cutting problem(Top-Down).  public static int un_kp(int []price, int []length, int Max_len, int n) {     // The maximum price will be zero,    // when either the length of the rod    // is zero or price is zero.    if (n == 0 || Max_len == 0) {      return 0;    }     // If the length of the rod is less    // than the maximum length, Max_lene will    // consider it.Now depending    // upon the profit,    // either Max_lene we will take    // it or discard it.    if (length[n - 1] <= Max_len) {      t[n,Max_len] = Math.Max(price[n - 1] + un_kp(price, length, Max_len - length[n - 1], n),                              un_kp(price, length, Max_len, n - 1));    }     // If the length of the rod is    // greater than the permitted size,    // Max_len we will not consider it.    else {      t[n,Max_len] = un_kp(price, length, Max_len, n - 1);    }     // Max_lene Max_lenill return the maximum    // value obtained, Max_lenhich is present    // at the nth roMax_len and Max_length column.    return t[n,Max_len];  }   // Driver code  public static void Main(String[] args) {     int []price = new int[] { 1, 5, 8, 9, 10, 17, 17, 20 };    int n = price.Length;    int []length = new int[n];    for (int i = 0; i < n; i++) {      length[i] = i + 1;    }    int Max_len = n;    Console.WriteLine("Maximum obtained value is " + un_kp(price, length, n, Max_len));  }} // This code is contributed by gauravrajput1.
<script> // Javascript program for above approach // Global Array for// the purpose of memoization.let t = new Array(9);for (var i = 0; i < t.length; i++) {    t[i] = new Array(2);} // A recursive program, using ,// memoization, to implement the// rod cutting problem(Top-Down).function un_kp(price, length, Max_len, n){     // The maximum price will be zero,    // when either the length of the rod    // is zero or price is zero.    if (n == 0 || Max_len == 0)    {        return 0;    }     // If the length of the rod is less    // than the maximum length, Max_lene will    // consider it.Now depending    // upon the profit,    // either Max_lene  we will take    // it or discard it.    if (length[n - 1] <= Max_len)    {        t[n][Max_len]            = Math.max(price[n - 1]                      + un_kp(price, length,                           Max_len - length[n - 1], n),                  un_kp(price, length, Max_len, n - 1));    }     // If the length of the rod is    // greater than the permitted size,    // Max_len we will  not consider it.    else    {        t[n][Max_len]            = un_kp(price, length,                              Max_len, n - 1);    }     // Max_lene Max_lenill return the maximum    // value obtained, Max_lenhich is present    // at the nth roMax_len and Max_length column.    return t[n][Max_len];}         // Driver code         let price = [ 1, 5, 8, 9, 10, 17, 17, 20 ];    let n = price.length;    let length = Array(n).fill(0);    for (let i = 0; i < n; i++) {        length[i] = i + 1;    }    let Max_len = n;     // Function Call    document.write("Maximum obtained value  is "         + un_kp(price, length, n, Max_len)); </script>
Maximum obtained value  is 22
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