Problem Statement

The Rod Cutting Problem is an optimization problem that seeks to determine the best way to cut a rod into pieces in order to maximize profit. Given a rod of a certain length and a list of prices for each piece length, the goal is to find the maximum profit obtainable by cutting the rod into pieces.

Approach Explanation

This problem can be solved using dynamic programming. The idea is to use a table to store the maximum profit that can be achieved for each length of the rod.

Dynamic Programming Structure:

1. Define a DP array where dp[i] represents the maximum profit obtainable for a rod of length i.
2. Initialize the first element of the array to 0 since a rod of length 0 cannot yield any profit.
3. Iterate through lengths from 1 to n, and for each length, determine the maximum profit by trying every possible first cut.
4. The final answer will be in dp[n], where n is the length of the rod.

Time Complexity:

O(n^2), where n is the length of the rod.

Space Complexity:

O(n) for the DP array.

Python Code

def rod_cutting(prices, n):
    """
    Function to maximize profit by cutting a rod into pieces.

    Args:
    prices (List[int]): List of prices where the index represents the length of the rod.
    n (int): Length of the rod.

    Returns:
    int: Maximum profit obtainable by cutting the rod.
    """
    # Create a DP array initialized to 0
    dp = [0] * (n + 1)

    # Compute the maximum profit for each length
    for i in range(1, n + 1):
        max_profit = float('-inf')
        # Try every possible first cut
        for j in range(1, i + 1):
            max_profit = max(max_profit, prices[j - 1] + dp[i - j])
        dp[i] = max_profit

    return dp[n]


# Example usage
rod_length = 8
prices = [1, 5, 8, 9, 10, 17, 17, 20]  # Prices for lengths 1 to 8
max_profit = rod_cutting(prices, rod_length)

print("Maximum profit obtainable:", max_profit)
    

Explanation of the Program

Let’s break down the structure of the program:

1. Input:

The input consists of an array of prices corresponding to each length of the rod and the total length of the rod:

    rod_length = 8
    prices = [1, 5, 8, 9, 10, 17, 17, 20]

2. DP Array Initialization:

A DP array is created with length n + 1, initialized to 0. This array is used to store the maximum profit for each length of the rod.

3. Computing Maximum Profit:

The program iterates through each length from 1 to n. For each length, it tries every possible cut, calculating the profit for that cut and updating the maximum profit accordingly.

4. Final Result:

The maximum profit for a rod of length n can be found in dp[n].

Example Execution:

For the provided input, the output will display the maximum profit obtainable:

Maximum profit obtainable: 22

This indicates that the maximum profit that can be achieved by cutting a rod of length 8 is 22.