Painting the Walls - Video Solutions

Problem Overview: You are given two arrays cost and time. Hiring a paid painter for wall i costs cost[i] and takes time[i] units, during which a free painter can paint other walls. The goal is to minimize the total cost required to paint all walls.

Approach 1: Dynamic Programming (O(n²) time, O(n) space)

This problem behaves like a knapsack-style decision process. When you hire a paid painter for wall i, that painter finishes one wall and the free painter can complete up to time[i] additional walls while the paid work happens. That means a single paid job effectively covers 1 + time[i] walls. Use dynamic programming where dp[j] represents the minimum cost required to finish j walls. Iterate through each wall and update the DP state backwards, similar to a 0/1 knapsack transition. The update becomes dp[min(n, j + time[i] + 1)] = min(dp[min(n, j + time[i] + 1)], dp[j] + cost[i]). This ensures each paid painter expands coverage efficiently while minimizing cost. Time complexity is O(n²) because each wall updates up to n states, and space complexity is O(n). This approach directly models the scheduling benefit of the free painter and guarantees the optimal result.

The DP formulation is a classic resource coverage optimization problem and commonly appears in dynamic programming interview questions. The input arrays are processed sequentially, making the state transition straightforward while keeping memory usage small.

Approach 2: Greedy Strategy with Priority Selection (O(n log n) time, O(n) space)

A heuristic approach tries to prioritize walls that provide the most free painting coverage relative to their cost. Compute an efficiency metric such as (time[i] + 1) / cost[i] and process walls that give the largest coverage benefit first. Use sorting or a priority queue to repeatedly pick the wall that yields the most additional painted walls per unit cost. Each selection reduces the remaining number of walls that must be handled by paid painters. This greedy strategy runs in O(n log n) time due to sorting or heap operations and uses O(n) space.

Greedy works well in many practical cases because larger time[i] values effectively amortize costs across more walls. However, it does not guarantee optimality for all inputs since the benefit of one painter depends on combinations with others. For guaranteed correctness, dynamic programming remains the safer option.

The greedy reasoning relies on evaluating tradeoffs between cost and coverage, which is a common technique in array optimization problems and interview-style greedy decisions.

Recommended for interviews: The dynamic programming solution is the expected approach. Interviewers want to see you convert the free-time effect into a state transition similar to knapsack coverage. Discussing a greedy idea shows good intuition about cost efficiency, but implementing the DP demonstrates stronger algorithmic control and guarantees the optimal answer.