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Painting Fence Algorithm

Rod Cutting

Last Updated : 02 Dec, 2024

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Given a rod of length n inches and an array price[]. price[i] denotes the value of a piece of length i. The task is to determine the maximum value obtainable by cutting up the rod and selling the pieces.

Note: price[] is 1-indexed array.

Input: price[] = [1, 5, 8, 9, 10, 17, 17, 20]
Output: 22
Explanation: The maximum obtainable value is 22 by cutting in two pieces of lengths 2 and 6, i.e., 5 + 17 = 22.
Input : price[] = [3, 5, 8, 9, 10, 17, 17, 20]
Output : 24
Explanation : The maximum obtainable value is 24 by cutting the rod into 8 pieces of length 1, i.e, 8*price[1]= 8*3 = 24.
Input : price[] = [3]
Output : 3
Explanation: There is only 1 way to pick a piece of length 1.

Table of Content

Using Recursion - O(n^n) Time and O(n) Space

The recursive approach involves solving the problem by considering all possible ways to cut the rod into two pieces at every length j (where 1<=j<=i), calculating the profit for each cut, and finding the maximum profit among these options. At each step, the rod of length i is divided into two parts: j and i - j. The profit from this cut is the sum of the price of the piece of length j (given as price[j-1]) and the maximum profit obtainable from the remaining rod of length i-j (computed recursively). The base case for this recursion is when i equals 0, where the maximum profit is simply 0.
The recurrence relation will be:
cutRod(i) = max(price[j-1] + cutRod(i-j)) for (1<=j<=i)
Base Case:
If i == 0, return 0.

C++

// C++ program to find maximum // profit from rod of size n  #include <bits/stdc++.h> using namespace std;  int cutRodRecur(int i, vector<int> &price) {          // Base case     if (i==0) return 0;          int ans = 0;          // Find maximum value for each cut.     // Take value of rod of length j, and      // recursively find value of rod of      // length (i-j).     for (int j=1; j<=i; j++) {         ans = max(ans, price[j-1]+cutRodRecur(i-j, price));     }          return ans; }  int cutRod(vector<int> &price) {     int n = price.size();          return cutRodRecur(n, price); }  int main() {     vector<int> price =  { 1, 5, 8, 9, 10, 17, 17, 20};     cout << cutRod(price);     return 0; }

Java

// Java program to find maximum // profit from rod of size n  import java.util.*;  class GfG {      static int cutRodRecur(int i, int[] price) {                  // Base case         if (i == 0) return 0;                  int ans = 0;          // Find maximum value for each cut.         // Take value of rod of length j, and          // recursively find value of rod of          // length (i-j).         for (int j = 1; j <= i; j++) {             ans = Math.max(ans, price[j - 1] + cutRodRecur(i - j, price));         }          return ans;     }      static int cutRod(int[] price) {         int n = price.length;          return cutRodRecur(n, price);     }      public static void main(String[] args) {         int[] price = {1, 5, 8, 9, 10, 17, 17, 20};         System.out.println(cutRod(price));     } }

Python

# Python program to find maximum # profit from rod of size n   def cutRodRecur(i, price):          # Base case     if i == 0:         return 0          ans = 0      # Find maximum value for each cut.     # Take value of rod of length j, and      # recursively find value of rod of      # length (i-j).     for j in range(1, i + 1):         ans = max(ans, price[j - 1] + cutRodRecur(i - j, price))      return ans  def cutRod(price):     n = len(price)      return cutRodRecur(n, price)  if __name__ == "__main__":     price = [1, 5, 8, 9, 10, 17, 17, 20]     print(cutRod(price))

C#

// C# program to find maximum // profit from rod of size n   using System;  class GfG {      static int cutRodRecur(int i, int[] price) {                  // Base case         if (i == 0) return 0;                  int ans = 0;          // Find maximum value for each cut.         // Take value of rod of length j, and          // recursively find value of rod of          // length (i-j).         for (int j = 1; j <= i; j++) {             ans = Math.Max(ans, price[j - 1] + cutRodRecur(i - j, price));         }          return ans;     }      static int cutRod(int[] price) {         int n = price.Length;          return cutRodRecur(n, price);     }      static void Main(string[] args) {         int[] price = {1, 5, 8, 9, 10, 17, 17, 20};         Console.WriteLine(cutRod(price));     } }

JavaScript

// JavaScript program to find maximum // profit from rod of size n   function cutRodRecur(i, price) {          // Base case     if (i === 0) return 0;          let ans = 0;      // Find maximum value for each cut.     // Take value of rod of length j, and      // recursively find value of rod of      // length (i-j).     for (let j = 1; j <= i; j++) {         ans = Math.max(ans, price[j - 1] + cutRodRecur(i - j, price));     }      return ans; }  function cutRod(price) {     const n = price.length;      return cutRodRecur(n, price); }  const price = [1, 5, 8, 9, 10, 17, 17, 20]; console.log(cutRod(price));

Output

Using Top-Down DP (Memoization) - O(n^2) Time and O(n) Space

If we notice carefully, we can observe that the above recursive solution holds the following two properties of Dynamic Programming:

1. Optimal Substructure: Maximum profit at index i, i.e., cutRod(i), depends on the optimal solutions of the subproblems cutRod(i-1) , cutRod(i-2), ..., cutRod(0) . By comparing these optimal substructures, we can efficiently calculate the maximum profit at index i.
2. Overlapping Subproblems: While applying a recursive approach in this problem, we notice that certain subproblems are computed multiple times.

There is only 1 parameter: i that changes in the recursive solution. So we create a 1D array of size n for memoization.
We initialize this array as -1 to indicate nothing is computed initially.
Now we modify our recursive solution to first check if the value is -1, then only make recursive calls. This way, we avoid re-computations of the same subproblems.

C++

// C++ program to find maximum // profit from rod of size n  #include <bits/stdc++.h> using namespace std;  int cutRodRecur(int i, vector<int> &price, vector<int> &memo) {          // Base case     if (i==0) return 0;          // If value is memoized     if (memo[i-1]!=-1) return memo[i-1];          int ans = 0;          // Find maximum value for each cut.     // Take value of rod of length j, and      // recursively find value of rod of      // length (i-j).     for (int j=1; j<=i; j++) {         ans = max(ans, price[j-1]+cutRodRecur(i-j, price, memo));     }          return memo[i-1] = ans; }  int cutRod(vector<int> &price) {     int n = price.size();     vector<int> memo(price.size(), -1);     return cutRodRecur(n, price, memo); }  int main() {     vector<int> price =  { 1, 5, 8, 9, 10, 17, 17, 20};     cout << cutRod(price);     return 0; }

Java

// Java program to find maximum // profit from rod of size n   import java.util.*;  class GfG {      static int cutRodRecur(int i, int[] price, int[] memo) {                  // Base case         if (i == 0) return 0;                  // If value is memoized         if (memo[i - 1] != -1) return memo[i - 1];                  int ans = 0;          // Find maximum value for each cut.         // Take value of rod of length j, and          // recursively find value of rod of          // length (i-j).         for (int j = 1; j <= i; j++) {             ans = Math.max(ans, price[j - 1] + cutRodRecur(i - j, price, memo));         }          return memo[i - 1] = ans;     }      static int cutRod(int[] price) {         int n = price.length;         int[] memo = new int[n];         Arrays.fill(memo, -1);         return cutRodRecur(n, price, memo);     }      public static void main(String[] args) {         int[] price = {1, 5, 8, 9, 10, 17, 17, 20};         System.out.println(cutRod(price));     } }

Python

# Python program to find maximum # profit from rod of size n   def cutRodRecur(i, price, memo):          # Base case     if i == 0:         return 0          # If value is memoized     if memo[i - 1] != -1:         return memo[i - 1]          ans = 0      # Find maximum value for each cut.     # Take value of rod of length j, and      # recursively find value of rod of      # length (i-j).     for j in range(1, i + 1):         ans = max(ans, price[j - 1] + cutRodRecur(i - j, price, memo))      memo[i - 1] = ans     return ans  def cutRod(price):     n = len(price)     memo = [-1] * n     return cutRodRecur(n, price, memo)  if __name__ == "__main__":     price = [1, 5, 8, 9, 10, 17, 17, 20]     print(cutRod(price))

C#

// C# program to find maximum // profit from rod of size n   using System;  class GfG {      static int cutRodRecur(int i, int[] price, int[] memo) {                  // Base case         if (i == 0) return 0;                  // If value is memoized         if (memo[i - 1] != -1) return memo[i - 1];                  int ans = 0;          // Find maximum value for each cut.         // Take value of rod of length j, and          // recursively find value of rod of          // length (i-j).         for (int j = 1; j <= i; j++) {             ans = Math.Max(ans, price[j - 1] +             cutRodRecur(i - j, price, memo));         }          return memo[i - 1] = ans;     }      static int cutRod(int[] price) {         int n = price.Length;         int[] memo = new int[n];         Array.Fill(memo, -1);         return cutRodRecur(n, price, memo);     }      static void Main(string[] args) {         int[] price = {1, 5, 8, 9, 10, 17, 17, 20};         Console.WriteLine(cutRod(price));     } }

JavaScript

// JavaScript program to find maximum // profit from rod of size n   function cutRodRecur(i, price, memo) {          // Base case     if (i === 0) return 0;          // If value is memoized     if (memo[i - 1] !== -1) return memo[i - 1];          let ans = 0;      // Find maximum value for each cut.     // Take value of rod of length j, and      // recursively find value of rod of      // length (i-j).     for (let j = 1; j <= i; j++) {         ans = Math.max(ans, price[j - 1] + cutRodRecur(i - j, price, memo));     }      memo[i - 1] = ans;     return ans; }  function cutRod(price) {     const n = price.length;     const memo = Array(n).fill(-1);     return cutRodRecur(n, price, memo); }  const price = [1, 5, 8, 9, 10, 17, 17, 20]; console.log(cutRod(price));

Output

Using Bottom-Up DP (Tabulation) - O(n^2) Time and O(n) Space

The idea is to fill the dp table from bottom to up. For each rod of length i, starting from i = 1 to i = n, find the maximum value that can obtained by cutting it into two pieces of length j and (i-j).

C++

// C++ program to find maximum // profit from rod of size n  #include <bits/stdc++.h> using namespace std;  int cutRod(vector<int> &price) {     int n = price.size();     vector<int> dp(price.size()+1, 0);          // Find maximum value for all      // rod of length i.     for (int i=1; i<=n; i++) {         for (int j=1; j<=i; j++) {             dp[i] = max(dp[i], price[j-1]+dp[i-j]);         }     }          return dp[n]; }  int main() {     vector<int> price =  { 1, 5, 8, 9, 10, 17, 17, 20};     cout << cutRod(price);     return 0; }

Java

// Java program to find maximum // profit from rod of size n   import java.util.*;  class GfG {      static int cutRod(int[] price) {         int n = price.length;         int[] dp = new int[n + 1];          // Find maximum value for all          // rod of length i.         for (int i = 1; i <= n; i++) {             for (int j = 1; j <= i; j++) {                 dp[i] = Math.max(dp[i], price[j - 1] + dp[i - j]);             }         }          return dp[n];     }      public static void main(String[] args) {         int[] price = {1, 5, 8, 9, 10, 17, 17, 20};         System.out.println(cutRod(price));     } }

Python

# Python program to find maximum # profit from rod of size n   def cutRod(price):     n = len(price)     dp = [0] * (n + 1)      # Find maximum value for all      # rod of length i.     for i in range(1, n + 1):         for j in range(1, i + 1):             dp[i] = max(dp[i], price[j - 1] + dp[i - j])      return dp[n]  if __name__ == "__main__":     price = [1, 5, 8, 9, 10, 17, 17, 20]     print(cutRod(price))

C#

// C# program to find maximum // profit from rod of size n   using System;  class GfG {      static int cutRod(int[] price) {         int n = price.Length;         int[] dp = new int[n + 1];          // Find maximum value for all          // rod of length i.         for (int i = 1; i <= n; i++) {             for (int j = 1; j <= i; j++) {                 dp[i] = Math.Max(dp[i], price[j - 1] + dp[i - j]);             }         }          return dp[n];     }      static void Main(string[] args) {         int[] price = {1, 5, 8, 9, 10, 17, 17, 20};         Console.WriteLine(cutRod(price));     } }

JavaScript

// JavaScript program to find maximum // profit from rod of size n   function cutRod(price) {     const n = price.length;     const dp = Array(n + 1).fill(0);      // Find maximum value for all      // rod of length i.     for (let i = 1; i <= n; i++) {         for (let j = 1; j <= i; j++) {             dp[i] = Math.max(dp[i], price[j - 1] + dp[i - j]);         }     }      return dp[n]; }  const price = [1, 5, 8, 9, 10, 17, 17, 20]; console.log(cutRod(price));

Output

Using the idea of Unbounded Knapsack - O(n^2) time and O(n^2) space

This problem is very similar to the Unbounded Knapsack Problem, where there are multiple occurrences of the same item. Here we consider length of rod as capacity of knapsack. All lengths from 1 to n are considered as weights of items and prices as values of items.

C++

// C++ program to find maximum // profit from rod of size n  #include <bits/stdc++.h> using namespace std;  // Find the maximum value obtainable from rod  // of length j. int cutRodRecur(int i, int j, vector<int> &price,                  vector<vector<int>> &memo) {          // base case      if (i==0 || j==0) return 0;          // If value if memoized     if (memo[i][j]!=-1) return memo[i][j];          // There are two options:     // 1. Break it into (i) and (i-j) rods and      // take value of ith rod.          int take = 0;     if (i<=j) {         take = price[i-1] + cutRodRecur(i, j-i, price, memo);     }          // 2. Skip i'th length rod.     int noTake = cutRodRecur(i-1, j, price, memo);          return memo[i][j] = max(take, noTake); }  int cutRod(vector<int> &price) {     int n = price.size();     vector<vector<int>> memo(n+1, vector<int>(n+1, -1));          return cutRodRecur(n, n, price, memo); }  int main() {     vector<int> price =  { 1, 5, 8, 9, 10, 17, 17, 20};     cout << cutRod(price);     return 0; }

Java

// Java program to find maximum // profit from rod of size n   import java.util.Arrays;  class GfG {      // Find the maximum value obtainable from rod      // of length j.     static int cutRodRecur(int i, int j,                             int[] price, int[][] memo) {          // base case          if (i == 0 || j == 0) return 0;          // If value is memoized         if (memo[i][j] != -1) return memo[i][j];          // There are two options:         // 1. Break it into (i) and (i-j) rods and          // take value of ith rod.         int take = 0;         if (i <= j) {             take = price[i - 1] + cutRodRecur(i, j - i, price, memo);         }          // 2. Skip i'th length rod.         int noTake = cutRodRecur(i - 1, j, price, memo);          return memo[i][j] = Math.max(take, noTake);     }      static int cutRod(int[] price) {         int n = price.length;         int[][] memo = new int[n + 1][n + 1];         for (int[] row : memo) {             Arrays.fill(row, -1);         }          return cutRodRecur(n, n, price, memo);     }      public static void main(String[] args) {         int[] price = {1, 5, 8, 9, 10, 17, 17, 20};         System.out.println(cutRod(price));     } }

Python

# Python program to find maximum # profit from rod of size n   def cutRodRecur(i, j, price, memo):      # base case      if i == 0 or j == 0:         return 0      # If value is memoized     if memo[i][j] != -1:         return memo[i][j]      # There are two options:     # 1. Break it into (i) and (i-j) rods and      # take value of ith rod.     take = 0     if i <= j:         take = price[i - 1] + cutRodRecur(i, j - i, price, memo)      # 2. Skip i'th length rod.     noTake = cutRodRecur(i - 1, j, price, memo)      memo[i][j] = max(take, noTake)     return memo[i][j]  def cutRod(price):     n = len(price)     memo = [[-1 for _ in range(n + 1)] for _ in range(n + 1)]     return cutRodRecur(n, n, price, memo)  if __name__ == "__main__":     price = [1, 5, 8, 9, 10, 17, 17, 20]     print(cutRod(price))

C#

// C# program to find maximum // profit from rod of size n   using System;  class GfG {      // Find the maximum value obtainable from rod      // of length j.     static int cutRodRecur(int i, int j,                             int[] price, int[,] memo) {          // base case          if (i == 0 || j == 0) return 0;          // If value is memoized         if (memo[i, j] != -1) return memo[i, j];          // There are two options:         // 1. Break it into (i) and (i-j) rods and          // take value of ith rod.         int take = 0;         if (i <= j) {             take = price[i - 1] + cutRodRecur(i, j - i, price, memo);         }          // 2. Skip i'th length rod.         int noTake = cutRodRecur(i - 1, j, price, memo);          memo[i, j] = Math.Max(take, noTake);         return memo[i, j];     }      static int cutRod(int[] price) {         int n = price.Length;         int[,] memo = new int[n + 1, n + 1];         for (int i = 0; i <= n; i++) {             for (int j = 0; j <= n; j++) {                 memo[i, j] = -1;             }         }          return cutRodRecur(n, n, price, memo);     }      static void Main(string[] args) {         int[] price = {1, 5, 8, 9, 10, 17, 17, 20};         Console.WriteLine(cutRod(price));     } }

JavaScript

// JavaScript program to find maximum // profit from rod of size n   function cutRodRecur(i, j, price, memo) {      // base case      if (i === 0 || j === 0) return 0;      // If value is memoized     if (memo[i][j] !== -1) return memo[i][j];      // There are two options:     // 1. Break it into (i) and (i-j) rods and      // take value of ith rod.     let take = 0;     if (i <= j) {         take = price[i - 1] + cutRodRecur(i, j - i, price, memo);     }      // 2. Skip i'th length rod.     const noTake = cutRodRecur(i - 1, j, price, memo);      memo[i][j] = Math.max(take, noTake);     return memo[i][j]; }  function cutRod(price) {     const n = price.length;     const memo = Array.from({ length: n + 1 }, () => Array(n + 1).fill(-1));      return cutRodRecur(n, n, price, memo); }  const price = [1, 5, 8, 9, 10, 17, 17, 20]; console.log(cutRod(price));

Output

Painting Fence Algorithm

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