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:
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: 30.66667 Explanation: We pay 4 to 0th worker, 13.33333 to 2nd and 3rd workers separately.
Constraints:
n == quality.length == wage.length1 <= k <= n <= 1041 <= quality[i], wage[i] <= 104The key insight for #857 Minimum Cost to Hire K Workers is understanding the wage-to-quality ratio requirement. If a group of workers is hired together, they must all be paid according to the highest wage/quality ratio among them while maintaining proportional pay based on their quality.
A common strategy is to compute the wage/quality ratio for each worker and sort workers by this ratio. As you iterate through the sorted list, treat the current worker’s ratio as the potential group pay rate. To minimize total cost, you want the smallest possible total quality among exactly k workers. A max heap (priority queue) helps maintain the k workers with the smallest combined quality by removing the largest quality when the group exceeds size k.
Each time the heap reaches size k, compute the potential cost using the current ratio and the sum of qualities. This greedy + heap approach efficiently evaluates optimal groups. The overall time complexity is O(n log n) due to sorting and heap operations, while space complexity is O(k).
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Greedy with Sorting and Max Heap | O(n log n) | O(k) |
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This approach uses a heap to dynamically maintain workers while iterating over them sorted by wage-to-quality ratio. The goal is to keep the sum of qualities of the selected workers minimal while ensuring all conditions are satisfied. We sort all workers by their wage-to-quality ratio because for each worker, to satisfy both their minimum wage and relative payment constraints, each selected worker must be paid at least this ratio times their quality.
Time Complexity: O(n log n) due to sorting, and O(n log k) for heap operations.
Space Complexity: O(n) for the sorted list of workers or the heap.
1import heapq
2
3class Solution:
4 def mincostToHireWorkers(self, quality, wage, k):
5 workers = sorted((w / q,
The solution begins by sorting each worker by their wage-to-quality ratio. It then uses a max-heap to maintain the k workers with the smallest sum of qualities. For each worker evaluated, we add their quality to the max heap. If the max heap exceeds k in size, we remove the largest quality to ensure we're considering the smallest k qualities. The current cost calculated by multiplying the present worker's rate with the sum of qualities in the heap is considered as a potential minimum.
This approach focuses on sorting workers by the ratio of their wage expectation to quality. By calculating this ratio, we can use it as a reference point to determine the required payment for every possible group of k workers. Selecting the right workers and calculating the minimum payment by understanding proportional wages can be done by iterating over sorted worker lists using two indices, capturing the required details and performing the necessary calculations.
Time Complexity: O(n log n) due to sorting, and O(n log n) due to managing heap.
Space Complexity: O(n) for worker list and heap storage.
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Yes, this problem is considered a strong interview-level question because it combines greedy reasoning, sorting, and heap usage. Variations of this problem or similar optimization problems frequently appear in FAANG and other top tech interviews.
A max heap (priority queue) is the most useful data structure for this problem. It helps efficiently track and remove the worker with the largest quality when maintaining a group of k workers with the smallest total quality.
The optimal approach uses a greedy strategy combined with sorting and a heap. Workers are sorted by their wage-to-quality ratio, and a max heap is used to maintain the k workers with the smallest total quality while evaluating possible hiring groups.
Sorting by wage-to-quality ratio ensures that when a worker is considered, their ratio becomes the minimum acceptable pay rate for the group. This allows the algorithm to evaluate valid groups while minimizing the total quality sum and overall cost.
Here, the Python code sorts workers by their ratio of wage to quality. A max-heap manages the quality selection, and when a potential solution with k workers is found, the resultant cost is calculated. The approach makes efficient use of sorted ratios and quality management to derive the answer.