#857

Hard

auto_awesomeGreedy choice plus invariant validation

LeetCode Problem Workspace

Minimum Cost to Hire K Workers

Find the minimum cost to hire exactly k workers based on quality and wage expectations in this challenging greedy problem.

array greedy sorting heap priority queue

Problem Statement

You are given two integer arrays, quality and wage, where quality[i] represents the quality of the ith worker and wage[i] represents the minimum wage expectation of that worker. You need to hire exactly k workers, and the cost to hire these workers is determined by the following rule: the wage paid to each worker must be proportional to their quality.

To minimize the cost of hiring k workers, you must pay the workers according to the rule mentioned above. The total cost to hire k workers is the highest wage among the chosen workers, and each worker's wage is adjusted proportionally to their quality. Return the least amount of money needed to hire exactly k workers. The result should be accurate within a 10^-5 tolerance.

Examples

Example 1

Input: quality = [10,20,5], wage = [70,50,30], k = 2

Output: 105.00000

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

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

Solution Approach

Greedy Approach with Sorting

To minimize the cost, sort the workers by their wage-to-quality ratio. For each worker, calculate the potential cost if they are included in the group, ensuring that the highest wage is proportional to their quality. This greedy strategy ensures that we always select workers whose wage expectations align well with their qualities.

Priority Queue for Optimal Quality Selection

Using a min-heap (priority queue), we can efficiently keep track of the k workers with the smallest qualities while maintaining the total cost. As we progress through the sorted workers, we add their qualities to the heap and compute the total cost, adjusting the group of workers when necessary.

Invariant Validation for Correctness

After sorting and selecting the workers, we validate the greedy choices using the invariant that each worker's wage must be proportional to their quality. This guarantees that the total cost for any valid set of workers is minimized, ensuring correctness for the solution.

Complexity Analysis

Metric	Value
Time	O(n \log n + n \log k)
Space	O(n + k)

The time complexity of the solution is O(n log n + n log k), where n is the number of workers, as we first sort the workers by their wage-to-quality ratio and then perform heap operations to manage the k smallest qualities. The space complexity is O(n + k), accounting for the storage of the workers' data and the priority queue.

What Interviewers Usually Probe

Look for understanding of greedy algorithms and their application to optimization problems.
Check if the candidate can handle edge cases with varying worker qualities and wage expectations.
Evaluate the ability to use sorting and priority queues efficiently in solving complex problems.

Common Pitfalls or Variants

Common pitfalls

Failing to properly maintain the invariant between wage and quality ratios, which can lead to incorrect cost calculations.
Overlooking the need to use a priority queue for efficient selection of workers, resulting in inefficient solutions.
Confusing the total wage with the total cost, failing to account for the proportional wage adjustment based on quality.

Follow-up variants

Modify the problem to hire more or fewer workers, testing the algorithm's scalability.
Change the constraints to larger arrays, examining the efficiency of the solution under higher input sizes.
Add additional restrictions on worker quality or wage ratios to test if the greedy strategy still holds.

FAQ

What is the key pattern used to solve the Minimum Cost to Hire K Workers problem?

The problem follows the greedy choice pattern, where we select workers based on their wage-to-quality ratio and use a priority queue to manage the smallest qualities.

How does the priority queue help in the Minimum Cost to Hire K Workers problem?

The priority queue helps by efficiently tracking the k workers with the lowest qualities while maintaining the total cost, ensuring optimal worker selection.

What is the time complexity of the solution for the Minimum Cost to Hire K Workers problem?

The time complexity is O(n log n + n log k), where n is the number of workers, and k is the number of workers to be hired.

What are the common pitfalls when solving the Minimum Cost to Hire K Workers problem?

Common pitfalls include failing to maintain the wage-quality invariant, using inefficient selection methods, and misunderstanding the cost calculation.

What are some variations of the Minimum Cost to Hire K Workers problem?

Variations include adjusting the number of workers hired, changing input sizes, and adding new restrictions on worker qualities and wage ratios.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def mincostToHireWorkers(
        self, quality: List[int], wage: List[int], k: int
    ) -> float:
        t = sorted(zip(quality, wage), key=lambda x: x[1] / x[0])
        ans, tot = inf, 0
        h = []
        for q, w in t:
            tot += q
            heappush(h, -q)
            if len(h) == k:
                ans = min(ans, w / q * tot)
                tot += heappop(h)
        return ans

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