Problem Description

You are presented with a task of hiring k workers from an array of workers where each worker has a specified hiring cost. The workers are arranged in a 0-indexed array costs, with costs[i] representing the cost of hiring the ith worker. You are to follow specific hiring rules to minimize the total hiring cost.

Here are the hiring rules:

• You must conduct k hiring sessions, hiring one worker during each session.
• In each session, you can only consider the first candidates or the last candidates workers for hiring.
• You are to hire the worker with the lowest cost from these candidates. If there's a tie in the cost, you hire the worker with the smallest index.
• If there are fewer than candidates workers left, you select from among them based on the lowest cost, again breaking ties by index.
• Each worker can only be hired once.

After executing this hiring process exactly k times, your task is to calculate and return the total cost of hiring the k workers.

Intuition

Given the problem's constraints and the requirement to minimize the total hiring cost, it becomes evident that a greedy approach can be used to find the solution. The essence of this approach is to always pick the cheapest available worker during each hiring session.

An intuitive way to repeatedly find and hire the worker with the lowest cost is to use a min-heap, a data structure that allows us to efficiently retrieve and remove the minimum element. In each hiring session, the following operations are performed:

1. Push the costs and indices of the first candidates workers and the last candidates workers into a min-heap after filtering out workers who were previously hired.
2. Pop the worker with the lowest cost from the heap.
3. Include this cost in the running total of costs.
4. Adjust the range of workers considered for the next session to exclude the worker just hired.

To initialize this process:

• A heap is formed with the initial candidates from both ends of the array.
• The first hiring session is conducted by popping the cheapest worker from the heap.
• Following each hiring session, the heap is updated by adding the next candidate from the appropriate end of the array, as long as we have not surpassed the boundaries defined by the hired workers.

This process continues until all k workers have been hired. The cumulative total at the end of the k sessions is the total cost of hiring the workers.

Learn more about Two Pointers and Heap (Priority Queue) patterns.

Solution Approach

The solution leverages a min-heap data structure to efficiently manage the workers being considered for hiring. Here is a step-by-step breakdown of the algorithm based on the solution code provided:

1. Initialize the min-heap with candidate workers: Two pointers, i and j, are used to mark the boundary of the first and last candidates workers. The first candidates workers and the last candidates workers are pushed into the heap along with their indices. The heap is then heapified to establish the min-heap property.

   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)

2. Hire workers in k sessions: A total of k hiring sessions are to be conducted. In each session:

   • The worker with the lowest cost is popped from the heap using heappop().
   • The cost of the hired worker is added to the total cost ans.
   • Depending on the index of the worker hired, the pointers i and j are updated to ensure the correct range of candidates for the next hiring session.
   • If a worker from the first candidates (lower part of the range) is hired, the pointer i is incremented.
   • Conversely, if a worker from the last candidates (upper part of the range) is hired, the pointer j is decremented.
   • Provided there are still candidates between the updated i and j, a new worker is pushed onto the heap from the updated index position.

   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))

3. Return the total cost: Once all k sessions are complete and k workers are hired, the accumulated total cost (ans) is returned as the answer to the problem.

   return ans

This approach ensures that in each session, we are hiring the worker with the lowest cost from the eligible candidates by using the min-heap to efficiently find that worker. The heap is dynamically updated after each hiring to reflect the remaining candidates, and the use of pointers helps keep track of the updated range of candidates after each hiring session.

Example Walkthrough

Let's illustrate the solution approach with a small example. Assume we have an array of worker costs costs = [7, 3, 5, 1, 4, 8], we need to hire k = 3 workers, and we are allowed to consider candidates = 2 workers from each end of the array for hiring in each session.

Initial Step:

• Initialize the min-heap with the first candidates workers ([7, 3]) and the last candidates workers ([4, 8]). The heap, after heapifying, will contain (3, 1), (7, 0), (4, 4), (8, 5) – where each tuple consists of (cost of worker, index of the worker).

Session 1:

• Pop (3, 1), the worker with the lowest cost, from the heap. The total hiring cost ans is now 3.
• After hiring, i becomes 2 (candidates pointer moves by one) and the min-heap is updated with the new candidate (5, 2). No need to push from the end since j has not moved. The new heap after pushing and heapifying will be (4, 4), (7, 0), (5, 2), (8, 5).

Session 2:

• Pop (4, 4), the next worker with the lowest cost, from the heap. The total hiring cost ans is now 3 + 4 = 7.
• After hiring, j becomes 3 (candidates pointer moves by one from the end) and the min-heap is updated with a new candidate from the end which does not exist in this case since only one candidate (5) exists between i and j. The new heap remains unchanged as (7, 0), (5, 2), (8, 5).

Session 3:

• Pop (5, 2), the next worker with the lowest cost, from the heap. The total hiring cost ans is now 7 + 5 = 12.
• No more workers are pushed onto the heap as the hiring sessions required (k=3) have been completed.

Final Step:

• Return the total cost of hiring, which is 12.

This step-by-step process confirms that the presented solution approach will hire the workers with the lowest cost given the problem constraints, ensuring the total hiring cost is minimized while obeying the hiring rules.

Solution Implementation

Time and Space Complexity

Time Complexity

The main operations in the given code are:

1. Inserting elements into the min-heap: This happens twice, once for each section of the array costs determined by the parameter candidates. It is done for the first candidates and the last candidates. This results in 2*candidates heappush operations initially and the time complexity for this operation is O(candidates * log(candidates)) for each section since each heappush operation has a time complexity of O(log(n)) where n is the number of elements in the heap at that moment.
2. Heapifying the initial queue q with 2*candidates elements: Heapifying an array of n elements takes O(n) time, so for 2*candidates elements the complexity is O(candidates).
3. The for loop runs k times, and in each iteration, it does a heappop operation followed by a heappush if the condition if i < j holds. The heappop is O(log(n)) for each pop, and heappush is also O(log(n)). Since at most one heappush follows each heappop in the loop and n is the size of the heap which is at most 2*candidates, the total time complexity for this loop is O(k * log(candidates)).

Adding up these costs, the overall time complexity is O(candidates * log(candidates)) + O(candidates) + O(k * log(candidates)). Since O(candidates) is subsumed by O(candidates * log(candidates)), the simplified time complexity is O(candidates * log(candidates) + k * log(candidates)).

Space Complexity

The space complexity of the algorithm is determined by:

1. The min-heap q which has up to 2*candidates elements: So the space complexity for the heap is O(candidates).
2. The auxiliary space for the variables i, j, ans: which is O(1).

Therefore, the overall space complexity is O(candidates) for the heap storage.

Learn more about how to find time and space complexity quickly using problem constraints.