Minimum Cost to Hire K Workers
Problem Description
There are n workers. You are given two integer arrays quality and wage where quality[i] is the quality of the ith worker and wage[i] is the minimum wage expectation for the ith worker.
We want to hire exactly k workers to form a paid group. To hire a group of k workers, we must pay them according to the following rules:
- Every worker in the paid group must be paid at least their minimum wage expectation.
- In the group, each worker's pay must be directly proportional to their quality. This means if a worker’s quality is double that of another worker in the group, then they must be paid twice as much as the other worker.
Given the integer k, return the least amount of money needed to form a paid group satisfying the above conditions. Answers within 10-5 of the actual answer will be accepted.
Example 1:
Input: quality = [10,20,5], wage = [70,50,30], k = 2
Output: 105.00000
Explanation: We pay 70 to 0th worker and 35 to 2nd worker.
Example 2:
Input: quality = [3,1,10,10,1], wage = [4,8,2,2,7], k = 3
Output: 30.66667Explanation: We pay 4 to 0th worker, 13.33333 to 2nd and 3rd workers separately.
Constraints:
- n == quality.length == wage.length
- 1 <= k <= n <= 104
- 1 <= quality[i], wage[i] <= 104
Solve it on LeetCode
Approaches
1. Brute Force Approach
Intuition:
The problem of hiring K workers at the minimum cost can be broken down into finding the optimal group of K workers such that the ratio (quality to wage ratio) aligns across all selected workers. This can be achieved by evaluating all possible combinations of K workers, which although infeasible for large inputs, provides a necessary understanding of the problem structure.
Approach:
- First, calculate the ratio of wage to quality for every worker and sort these ratios.
- Evaluate all potential combinations of workers, selecting and ensuring that an optimal ratio is met for any chosen group of K workers. The aim is to minimize the total wage while ensuring no worker in the group is underpaid based on their specified ratio.
- This method is computationally expensive because it examines a large number of combinations, but serves as a direct application of understanding the constraint problem.
Code:
- Time Complexity: O(n^2 log n) due to sorting and brute force combinations.
- Space Complexity: O(n) for storing the ratio-quality pairs.
2. Heap Based Approach
Intuition:
To hire K workers while minimizing the cost, consider the smallest ratio of wage-to-quality and always select the lowest quality workers to achieve optimal cost. By applying a max-heap on quality, we can evaluate the minimal ratios while keeping control over the maximum quality sum to pay the minimal cost.
Approach:
- Calculate and sort workers based on their wage-to-quality ratio.
- Use a max-heap to efficiently manage the top K lowest quality workers.
- For each worker, if the heap size exceeds K, remove the largest quality element, then calculate the minimal possible cost using fixed ratio across these K workers.
Code:
- Time Complexity: O(n log n) due to sort and heap operations.
- Space Complexity: O(K) for managing K elements in the max-heap.