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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.
1#include <stdio.h>
2#include <stdlib.h>
3
4int cmp(const void* a, const void* b) {
5 double diff = (*(double**)a)[0] - (*(double**)b)[0];
6 return (diff > 0) - (diff < 0);
7}
8
9double mincostToHireWorkers(int* quality, int* wage, int qualitySize, int k) {
10 double** workers = malloc(qualitySize * sizeof(double*));
11 for (int i = 0; i < qualitySize; i++) {
12 workers[i] = malloc(2 * sizeof(double));
13 workers[i][0] = 1.0 * wage[i] / quality[i];
14 workers[i][1] = quality[i];
15 }
16 qsort(workers, qualitySize, sizeof(double*), cmp);
17
18 double result = 1e9;
19 int heap[qualitySize];
20 int heapSize = 0, qualitySum = 0;
21
22 for (int i = 0; i < qualitySize; i++) {
23 qualitySum += workers[i][1];
24 heap[heapSize++] = workers[i][1];
25
26 if (heapSize > k) {
27 int maxq = 0, idx = -1;
28 for (int j = 0; j < heapSize; j++) {
29 if (heap[j] > maxq) {
30 maxq = heap[j];
31 idx = j;
32 }
33 }
34 qualitySum -= heap[idx];
35 for (int j = idx; j < heapSize - 1; j++) {
36 heap[j] = heap[j + 1];
37 }
38 heapSize--;
39 }
40
41 if (heapSize == k) {
42 double totalCost = qualitySum * workers[i][0];
43 if (totalCost < result) result = totalCost;
44 }
45 }
46
47 for (int i = 0; i < qualitySize; i++) free(workers[i]);
48 free(workers);
49
50 return result;
51}This C solution implements a basic priority queue (heap) to store qualities, keeping track of the max quality within the heap for efficiency. Workers are sorted by their wage-to-quality ratios. Tracking each possible group of k workers, the solution minimizes the cost by calculating the price to pay for the currently considered proportional quality sum.
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.
A Java solution calculates key worker ratios, sorts these ratios for an ordered evaluation. The subsequent enhancements of a priority queue ensure efficient candidate consideration based on quality sums for achieving minimum achievable costs.