Painting the Walls

Problem Description

You are given two 0-indexed integer arrays, cost and time, of size n representing the costs and the time taken to paint n different walls respectively. There are two painters available:

• A paid painter that paints the ith wall in time[i] units of time and takes cost[i] units of money.
• A free painter that paints any wall in 1 unit of time at a cost of 0. But the free painter can only be used if the paid painter is already occupied.

Return the minimum amount of money required to paint the n walls.

Key Insights

• The paid painter can be used for walls that have a higher cost compared to the time it takes to paint them.
• The free painter can be utilized to cover walls while the paid painter is busy, allowing for cost optimization.
• The problem can be modeled using dynamic programming to keep track of minimum costs over time.

Space and Time Complexity

Time Complexity: O(n^2)
Space Complexity: O(n)

Solution

To solve the problem, we can use a dynamic programming approach. We maintain an array dp where dp[i] represents the minimum cost to paint the first i walls.

1. Initialize dp[0] to 0 since no walls require no cost.
2. For each wall i, we iterate over all previous walls j to determine if we can use the free painter while the paid painter is painting wall j.
3. We calculate the cost of using the paid painter for wall i and the total time taken, ensuring that the free painter can handle the remaining walls within the time taken by the paid painter.
4. Update the dp[i] with the minimum cost found.

This approach effectively balances the usage of both painters while minimizing the overall cost.

Code Solutions