# 2462. Total Cost to Hire K Workers

## Description

You are given a 0-indexed integer array costs where costs[i] is the cost of hiring the ith worker.

You are also given two integers k and candidates. We want to hire exactly k workers according to the following rules:

• You will run k sessions and hire exactly one worker in each session.
• In each hiring session, choose the worker with the lowest cost from either the first candidates workers or the last candidates workers. Break the tie by the smallest index.
• For example, if costs = [3,2,7,7,1,2] and candidates = 2, then in the first hiring session, we will choose the 4th worker because they have the lowest cost [3,2,7,7,1,2].
• In the second hiring session, we will choose 1st worker because they have the same lowest cost as 4th worker but they have the smallest index [3,2,7,7,2]. Please note that the indexing may be changed in the process.
• If there are fewer than candidates workers remaining, choose the worker with the lowest cost among them. Break the tie by the smallest index.
• A worker can only be chosen once.

Return the total cost to hire exactly k workers.

Example 1:

Input: costs = [17,12,10,2,7,2,11,20,8], k = 3, candidates = 4
Output: 11
Explanation: We hire 3 workers in total. The total cost is initially 0.
- In the first hiring round we choose the worker from [17,12,10,2,7,2,11,20,8]. The lowest cost is 2, and we break the tie by the smallest index, which is 3. The total cost = 0 + 2 = 2.
- In the second hiring round we choose the worker from [17,12,10,7,2,11,20,8]. The lowest cost is 2 (index 4). The total cost = 2 + 2 = 4.
- In the third hiring round we choose the worker from [17,12,10,7,11,20,8]. The lowest cost is 7 (index 3). The total cost = 4 + 7 = 11. Notice that the worker with index 3 was common in the first and last four workers.
The total hiring cost is 11.


Example 2:

Input: costs = [1,2,4,1], k = 3, candidates = 3
Output: 4
Explanation: We hire 3 workers in total. The total cost is initially 0.
- In the first hiring round we choose the worker from [1,2,4,1]. The lowest cost is 1, and we break the tie by the smallest index, which is 0. The total cost = 0 + 1 = 1. Notice that workers with index 1 and 2 are common in the first and last 3 workers.
- In the second hiring round we choose the worker from [2,4,1]. The lowest cost is 1 (index 2). The total cost = 1 + 1 = 2.
- In the third hiring round there are less than three candidates. We choose the worker from the remaining workers [2,4]. The lowest cost is 2 (index 0). The total cost = 2 + 2 = 4.
The total hiring cost is 4.


Constraints:

• 1 <= costs.length <= 105
• 1 <= costs[i] <= 105
• 1 <= k, candidates <= costs.length

## Solutions

Solution 1: Priority Queue (Min Heap)

We maintain a priority queue (min heap) for the current candidate workers, and use variables $i$ and $j$ to mark the minimum index of the frontmost worker and the minimum index of the rearmost worker. Initially, $i = \text{candidates} - 1$ and $j = n - \text{candidates}$.

First, we put the costs of the first $candidates$ workers into the priority queue, then we put the costs of the last $candidates$ workers into the priority queue. Before putting them in, we need to check whether they are already in the priority queue according to $i$ or $j$. If they are, we don’t need to put them in again.

We loop $k$ times, each time taking out the worker with the smallest cost from the priority queue and accumulating the cost. If the index $x$ of the current worker is in the index range $[0,..i]$ of the frontmost workers, we move $i$ one step to the right, and then check whether we need to put the cost of the worker corresponding to $i$ into the priority queue; if the index is in the index range $[j,..n-1]$ of the rearmost workers, we move $j$ one step to the left, and then check whether we need to put the cost of the worker corresponding to $j$ into the priority queue.

After the traversal ends, we return the accumulated cost as the answer.

The time complexity is $O(n \times \log n)$, where $n$ is the length of the array $costs$.

• class Solution {
public long totalCost(int[] costs, int k, int candidates) {
PriorityQueue<int[]> q = new PriorityQueue<>((a, b) -> {
if (a[0] == b[0]) {
return a[1] - b[1];
}
return a[0] - b[0];
});
int n = costs.length;
int i = candidates - 1, j = n - candidates;
for (int h = 0; h < candidates; ++h) {
q.offer(new int[] {costs[h], h});
}
for (int h = n - candidates; h < n; ++h) {
if (h > i) {
q.offer(new int[] {costs[h], h});
}
}
long ans = 0;
while (k-- > 0) {
var e = q.poll();
int c = e[0], x = e[1];
ans += c;
if (x <= i) {
if (++i < j) {
q.offer(new int[] {costs[i], i});
}
}
if (x >= j) {
if (--j > i) {
q.offer(new int[] {costs[j], j});
}
}
}
return ans;
}
}

• using pii = pair<int, int>;

class Solution {
public:
long long totalCost(vector<int>& costs, int k, int candidates) {
priority_queue<pii, vector<pii>, greater<pii>> q;
int n = costs.size();
int i = candidates - 1, j = n - candidates;
for (int h = 0; h < candidates; ++h) q.push({costs[h], h});
for (int h = n - candidates; h < n; ++h)
if (h > i) q.push({costs[h], h});
long long ans = 0;
while (k--) {
auto [c, x] = q.top();
q.pop();
ans += c;
if (x <= i) {
if (++i < j) {
q.push({costs[i], i});
}
}
if (x >= j) {
if (--j > i) {
q.push({costs[j], j});
}
}
}
return ans;
}
};

• class Solution:
def totalCost(self, costs: List[int], k: int, candidates: int) -> int:
q = []
n = len(costs)
i, j = candidates - 1, n - candidates
for h in range(candidates):
q.append((costs[h], h))
for h in range(n - candidates, n):
if h > i:
q.append((costs[h], h))
heapify(q)
ans = 0
for _ in range(k):
c, x = heappop(q)
ans += c
if x <= i:
i += 1
if i < j:
heappush(q, (costs[i], i))
if x >= j:
j -= 1
if i < j:
heappush(q, (costs[j], j))
return ans


• func totalCost(costs []int, k int, candidates int) int64 {
q := hp{}
n := len(costs)
i, j := candidates-1, n-candidates
for h := 0; h < candidates; h++ {
heap.Push(&q, pair{costs[h], h})
}
for h := n - candidates; h < n; h++ {
if h > i {
heap.Push(&q, pair{costs[h], h})
}
}
ans := 0
for k > 0 {
p := heap.Pop(&q).(pair)
c, x := p.c, p.x
ans += c
if x <= i {
i++
if i < j {
heap.Push(&q, pair{costs[i], i})
}
}
if x >= j {
j--
if i < j {
heap.Push(&q, pair{costs[j], j})
}
}
k--
}
return int64(ans)
}

type pair struct{ c, x int }
type hp []pair

func (h hp) Len() int           { return len(h) }
func (h hp) Less(i, j int) bool { return h[i].c < h[j].c || h[i].c == h[j].c && h[i].x < h[j].x }
func (h hp) Swap(i, j int)      { h[i], h[j] = h[j], h[i] }
func (h *hp) Push(v any)        { *h = append(*h, v.(pair)) }
func (h *hp) Pop() any          { a := *h; v := a[len(a)-1]; *h = a[:len(a)-1]; return v }

• function totalCost(costs: number[], k: number, candidates: number): number {
const n = costs.length;
if (candidates * 2 >= n) {
costs.sort((a, b) => a - b);
return costs.slice(0, k).reduce((acc, x) => acc + x, 0);
}
const pq = new PriorityQueue({
compare: (a, b) => (a[0] === b[0] ? a[1] - b[1] : a[0] - b[0]),
});
for (let i = 0; i < candidates; ++i) {
pq.enqueue([costs[i], i]);
pq.enqueue([costs[n - i - 1], n - i - 1]);
}
let [l, r] = [candidates, n - candidates - 1];
let ans = 0;
while (k--) {
const [cost, i] = pq.dequeue()!;
ans += cost;
if (l > r) {
continue;
}
if (i < l) {
pq.enqueue([costs[l], l++]);
} else {
pq.enqueue([costs[r], r--]);
}
}
return ans;
}