2462. Total Cost to Hire K Workers
Description
You are given a 0indexed integer array costs
where costs[i]
is the cost of hiring the i^{th}
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 lastcandidates
workers. Break the tie by the smallest index. For example, if
costs = [3,2,7,7,1,2]
andcandidates = 2
, then in the first hiring session, we will choose the4^{th}
worker because they have the lowest cost[3,2,7,7,1,2]
.  In the second hiring session, we will choose
1^{st}
worker because they have the same lowest cost as4^{th}
worker but they have the smallest index[3,2,7,7,2]
. Please note that the indexing may be changed in the process.
 For example, if
 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 <= 10^{5}
1 <= costs[i] <= 10^{5}
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 $[j,..n1]$ 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 $costs$.







