Minimum Cost to Cut Board 🎯

Welcome to this comprehensive guide on the Minimum Cost to Cut Board problem! This lesson is perfect for both beginners and intermediate learners. Let's dive into the world of Data Structures and Algorithms, where we will solve a practical problem that is relevant to many real-world projects.

Introduction 📝

In this lesson, we will learn how to find the minimum cost to cut a board into given segments. This problem is a great example of using dynamic programming to solve an optimization problem.

Problem Statement 📝

Given a board of width W and an array C of size W+1, where C[i] denotes the cost of cutting the board at the i-th position, find the minimum cost to cut the board into G given segments.

Breaking it Down 💡

To solve this problem, we will use dynamic programming. Here's a high-level overview of our approach:

Initialize an array dp of size W+1 to store the minimum cost to cut the board up to the current position.
Iterate through the array C from right to left (index i from W to 0).
At each position, calculate the minimum cost to cut the board up to this position by considering two cases:
- Cutting the board at the current position, with the additional cost C[i].
- Not cutting the board at the current position, which means reusing the minimum cost from the previous position (i.e., dp[i+1]).
Choose the minimum cost between the two options and store it in dp[i].
After iterating through the entire array, the minimum cost to cut the board into G segments will be stored in dp[0].

Code Example ✅

Here's a Python code example for the Minimum Cost to Cut Board problem:

python

def minCost(W, G, C):
    dp = [float('inf')] * (W + 1)
    dp[0] = 0

    for i in range(W, -1, -1):
        dp[i] = min(dp[i], dp[i+1])
        if i + G <= W:
            dp[i] = min(dp[i], dp[i+G] + sum(C[i+1:i+G+1]))

    return dp[0]

In this code, we first initialize the dp array with infinite values to represent the minimum cost at each position. Then, we iterate through the array C from right to left and calculate the minimum cost at each position.

When iterating, we consider two cases:

If the current position (i) plus the number of segments (G) is less than or equal to the board width (W), we calculate the cost of cutting the board into G segments at the current position by adding the cost of the segments to the minimum cost of the previous position (i.e., dp[i+G] + sum(C[i+1:i+G+1])).
If the current position plus the number of segments is greater than the board width, we simply reuse the minimum cost from the previous position (i.e., dp[i+1]).

Quiz 💡

Conclusion 📝

Congratulations! You've now learned how to solve the Minimum Cost to Cut Board problem using dynamic programming. This problem is a great example of using this technique to find the optimal solution to an optimization problem.

Remember, practice makes perfect. Try solving this problem with different input values to reinforce your understanding of the concept.

Happy coding! 💻💪