Minimum Cost to Hire K Workers

Minimum Cost to Hire K Workers

1. What is this problem about?

The Minimum Cost to Hire K Workers is a challenging optimization problem that requires balancing two constraints: hiring exactly K workers and maintaining a fair wage distribution. Each worker has a specific quality and a minimum wage expectation. The rule is that if we hire a group of workers, we must pay them in proportion to their quality. Specifically, for any two workers in the group, the ratio of their wages must match the ratio of their qualities. Additionally, every hired worker must receive at least their minimum wage. The goal is to find the minimum total amount of money needed to hire exactly K workers while satisfying these conditions. This problem tests your ability to handle multiple constraints and find an optimal subset from a larger pool.

2. Why is this asked in interviews?

This problem is a favorite in interviews at top-tier tech companies like Google and Meta because it evaluates a candidate's proficiency in greedy algorithms and efficient data structure usage. The Minimum Cost to Hire K Workers interview question forces you to think about how to transform a complex set of rules into a manageable mathematical relationship. It also tests your ability to optimize a solution from a naive $O(N^2)$ or $O(N \log N \cdot N)$ approach to a more efficient $O(N \log N)$ using heaps. The combination of sorting and a priority queue is a classic "interview pattern" that demonstrates deep algorithmic understanding.

3. Algorithmic pattern used

The primary algorithmic pattern used here is the Greedy approach combined with a Max-Heap (Priority Queue). To minimize the total cost, we first observe that at least one worker in our set of K must be paid exactly their minimum wage. This worker's wage-to-quality ratio determines the ratio for the entire group. We sort all potential workers by their wage-to-quality ratio. As we iterate through the sorted list, we maintain a max-heap of the qualities of the workers we've considered. By keeping only the K workers with the smallest qualities, we minimize the total cost for the current ratio. This "Array, Sorting, Heap (Priority Queue), Greedy interview pattern" is essential for solving such constrained optimization problems.

4. Example explanation

Imagine you need to hire 2 workers (K=2). Worker A: Quality = 10, Min Wage = 70 (Ratio = 7.0) Worker B: Quality = 20, Min Wage = 100 (Ratio = 5.0) Worker C: Quality = 5, Min Wage = 30 (Ratio = 6.0)

First, we sort by ratio: B (5.0), C (6.0), A (7.0).

• If we use B's ratio (5.0):
  • B's wage = 20 * 5.0 = 100
  • C's wage = 5 * 5.0 = 25 (Less than C's min wage of 30, so this set is invalid if we only have B and C)
• If we use C's ratio (6.0):
  • B's wage = 20 * 6.0 = 120 (>= 100)
  • C's wage = 5 * 6.0 = 30 (>= 30)
  • Total = 150.
• If we use A's ratio (7.0), and we pick the two smallest qualities (C and A):
  • C's wage = 5 * 7.0 = 35 (>= 30)
  • A's wage = 10 * 7.0 = 70 (>= 70)
  • Total = 105. The minimum cost is 105.

5. Common mistakes candidates make

A frequent error in the Minimum Cost to Hire K Workers coding problem is failing to recognize that the total cost is calculated as (Sum of Qualities) * (Ratio of the highest-ratio worker in the set). Many candidates try to use dynamic programming or complex greedy choices without first simplifying the cost formula. Another mistake is not using a Max-Heap to efficiently track and remove the largest quality worker when the set size exceeds K. Without the heap, the solution becomes too slow for large inputs, leading to time limit exceeded errors during testing.

6. Interview preparation tip

When preparing for problems like this, focus on understanding how to "fix" one variable to simplify the rest. In this case, fixing the wage-to-quality ratio allows you to treat the problem as a simple selection of the smallest qualities. Practice identifying similar patterns where sorting by a specific metric (like ratio or density) is the key to a greedy solution. Mastering the use of a priority queue to maintain a running set of "best" elements is a vital skill for any competitive programming or technical interview scenario.