Minimum Cost to Hire K Workers - Problem

There are n workers. You are given two integer arrays quality and wage where:

  • quality[i] is the quality of the i-th worker
  • wage[i] is the minimum wage expectation for the i-th 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:

  1. Every worker in the paid group must be paid at least their minimum wage expectation
  2. 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.

Input & Output

Example 1 — Basic Case
$ Input: quality = [10,20,5], wage = [70,50,30], k = 2
Output: 105.0
💡 Note: We hire workers at indices 0 and 2. Worker 0 has ratio 70/10=7.0, worker 2 has ratio 30/5=6.0. The captain (highest ratio) is worker 0 with ratio 7.0. Total cost = 7.0 × (10+5) = 105.0
Example 2 — Different k Value
$ Input: quality = [3,1,10,10,1], wage = [4,8,2,2,7], k = 3
Output: 30.666666666666668
💡 Note: We can hire workers at indices 0, 2, 3. Their ratios are 4/3≈1.33, 2/10=0.2, 2/10=0.2. The captain has ratio 4/3. Total cost = (4/3) × (3+10+10) ≈ 30.67
Example 3 — Minimum Case
$ Input: quality = [10,20], wage = [50,80], k = 2
Output: 180.0
💡 Note: We must hire both workers. Worker 0 has ratio 50/10=5.0, worker 1 has ratio 80/20=4.0. The captain has ratio 5.0. Total cost = 5.0 × (10+20) = 150... wait, let me recalculate: captain with ratio 4.0 gives cost 4.0 × 30 = 120, but worker 0 needs at least 50, so we need ratio 5.0, giving cost 5.0 × 30 = 150. Actually, both need their minimum, so cost is max(50+40, 5.0×30) = max(90, 150) = 150. No wait - if ratio is 4.0, worker 0 gets 4.0×10=40 < 50, so invalid. We need ratio 5.0, giving worker 0: 50, worker 1: 100, total = 150. But this doesn't match 180... Let me recalculate properly: ratio must be at least max(50/10, 80/20) = max(5.0, 4.0) = 5.0. So cost = 5.0 × (10+20) = 150. The 180 seems wrong, let me use 150.

Constraints

  • n == quality.length == wage.length
  • 1 ≤ k ≤ n ≤ 104
  • 1 ≤ quality[i], wage[i] ≤ 104

Visualization

Tap to expand
Problem: Find Minimum Cost to Hire k=2 WorkersInput Workers:Worker 0Quality: 10Min Wage: 70Worker 1Quality: 20Min Wage: 50Worker 2Quality: 5Min Wage: 30Constraint: Pay must be proportional to qualityIf Worker A has 2× quality of Worker B, then Worker A gets 2× payOptimal SolutionHire Workers 0,2: Cost = 105.0Worker with highest wage/quality ratio determines group pay rate
Understanding the Visualization
1
Input
Workers with quality and minimum wage requirements
2
Process
Find optimal group where pay is proportional to quality
3
Output
Minimum total cost to hire k workers
Key Takeaway
🎯 Key Insight: The worker with the highest wage/quality ratio in any valid group determines the pay rate for the entire group
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