## Solution

#### Approach 1: Greedy

Intuition

At least one worker will be paid their minimum wage expectation. If not, we could scale all payments down by some factor and still keep everyone earning more than their wage expectation.

Algorithm

For each captain worker that will be paid their minimum wage expectation, let's calculate the cost of hiring K workers where each point of quality is worth wage[captain] / quality[captain] dollars. With this approach, the remaining implementation is straightforward.

Note that this algorithm would not be efficient enough to pass larger test cases.

Complexity Analysis

• Time Complexity: , where is the number of workers.

• Space Complexity: .

#### Approach 2: Heap

Intuition

As in Approach #1, at least one worker is paid their minimum wage expectation.

Additionally, every worker has some minimum ratio of dollars to quality that they demand. For example, if wage[0] = 100 and quality[0] = 20, then the ratio for worker 0 is 5.0.

The key insight is to iterate over the ratio. Let's say we hire workers with a ratio R or lower. Then, we would want to know the K workers with the lowest quality, and the sum of that quality. We can use a heap to maintain these variables.

Algorithm

Maintain a max heap of quality. (We're using a minheap, with negative values.) We'll also maintain sumq, the sum of this heap.

For each worker in order of ratio, we know all currently considered workers have lower ratio. (This worker will be the 'captain', as described in Approach #1.) We calculate the candidate answer as this ratio times the sum of the smallest K workers in quality.

Complexity Analysis

• Time Complexity: , where is the number of workers.

• Space Complexity: .

Analysis written by: @awice.