Understanding the Problem
The Painter’s Partition Problem involves painting a sequence of boards, each with a certain length. The challenge is to assign these boards to a given number of painters such that each painter paints a contiguous segment of boards, and the time taken by the painter who paints the most is minimized.
For example, given board lengths [10, 20, 30, 40]
and 2
painters, the goal is to split the work so that both painters are efficiently utilized, and the maximum workload assigned to any painter is as small as possible. A simple split like [10, 20] | [30, 40]
leads to one painter painting boards totaling 30 and the other painting 70. But [10, 20, 30] | [40]
results in 60 and 40, which is better. Our goal is to find such an optimal partition.
Step-by-Step Solution with Example
step 1: Understand constraints and rules
Each painter must paint a **contiguous** section of the board. No painter can skip or pick randomly. All painters work in parallel, and the **goal is to minimize the maximum workload** assigned to any single painter.
step 2: Identify search space using Binary Search
The minimum time any painter must spend is the **max length** of a single board (since no board can be split). The maximum time is the **sum** of all boards (if one painter paints everything). So we search between max(boards)
and sum(boards)
.
step 3: Use Binary Search to find optimal time
We perform binary search in this range. For each mid-point (which is a candidate for maximum allowed workload), we check whether it's **possible to divide the boards** among the painters such that no one exceeds this limit.
step 4: Implement a helper function
This function simulates the partition. We go through each board and assign it to a painter. If adding a board would exceed the max allowed time, we assign it to the next painter. If we need more painters than allowed, the current mid value is too low and must be increased.
step 5: Work with the example [10, 20, 30, 40] and 2 painters
- Search range: 40 (max board) to 100 (sum).
- Mid = 70. Can we paint within 70 units? Try splitting: [10, 20, 30] = 60, [40] = 40 → valid.
- Try lower: Mid = 55. First group: [10, 20] = 30, next: [30] = 30, next: [40] = 40 → needs 3 painters → too many.
- Repeat until you find the **smallest maximum workload** that works, which is 60 in this case.
Edge Cases
- Empty board list: Nothing to paint. Return 0.
- Zero painters: No one to do the job. Return 0.
- One painter: Must paint everything. Return
sum(boards)
.
- Boards = Painters: Each painter paints one board. Return
max(boards)
.
- More painters than boards: Same as above; extra painters are unused. Return
max(boards)
.
Finally
This is a classic **load balancing** problem, and binary search is a powerful tool to find the optimal solution efficiently. Instead of brute-force checking every possible split (which is too slow), we search over the answer space intelligently using a helper function that simulates the partitioning process.
Time complexity is O(N log S), where N is the number of boards and S is the range of total paint time. This makes it scalable even for large inputs.
Comments
Loading comments...