You are given an integer array timeReq and an integer splitTime.
In the microscopic world of the human body, the immune system faces an extraordinary challenge: combatting a rapidly multiplying bacterial colony that threatens the body's survival.
Initially, only one white blood cell (WBC) is deployed to eliminate the bacteria. However, the lone WBC quickly realizes it cannot keep up with the bacterial growth rate.
The WBC devises a clever strategy to fight the bacteria:
ith bacterial strain takes timeReq[i] units of time to be eliminated.splitTime units of time. Once split, the two WBCs can work in parallel on eliminating the bacteria.You must determine the minimum time required to eliminate all the bacterial strains.
Note that the bacterial strains can be eliminated in any order.
Example 1:
Input: timeReq = [10,4,5], splitTime = 2
Output: 12
Explanation:
The elimination process goes as follows:
t = 2 + 10 = 12. The other WBC splits again, using 2 units of time.t = 2 + 2 + 4 and t = 2 + 2 + 5.Example 2:
Input: timeReq = [10,4], splitTime = 5
Output:15
Explanation:
The elimination process goes as follows:
t = 5 + 10 and t = 5 + 4.
Constraints:
2 <= timeReq.length <= 1051 <= timeReq[i] <= 1091 <= splitTime <= 109Problem Overview: You are given multiple bacterial strains represented in an array. Each strain takes time to eliminate, and the elimination order affects the total time required. The goal is to determine the minimum total time needed to remove all strains while accounting for how remaining strains evolve as time progresses.
Approach 1: Simulation with Repeated Scanning (Brute Force) (Time: O(n^2), Space: O(1))
The most direct idea is to simulate the elimination process step by step. At every step, iterate through the entire array to find the strain that should be removed next according to the greedy rule (usually the smallest effective cost or earliest completion time). After eliminating it, update the remaining strains to reflect the time that has passed. This approach works conceptually but becomes inefficient because every elimination requires scanning the full array again.
Approach 2: Greedy + Min Heap (Priority Queue) (Time: O(n log n), Space: O(n))
A better strategy keeps track of the next strain to eliminate using a min-heap. Each heap entry represents the effective time or cost associated with removing that strain. Push all strains into the priority queue, ordered by the earliest elimination time. Repeatedly pop the smallest element, update the global time, and adjust any derived values for remaining strains using simple math calculations rather than rescanning the array.
The key insight: elimination order should always prioritize the strain that becomes optimal to remove earliest. A priority queue maintains this ordering efficiently. Each push/pop operation costs O(log n), so processing all strains results in O(n log n) time. This pattern appears frequently in scheduling and resource allocation problems.
Implementation relies on a heap structure from the language standard library. Push initial states derived from the input array, repeatedly pop the smallest element, update cumulative time, and continue until the heap is empty.
Recommended for interviews: Interviewers expect the greedy strategy with a priority queue. The brute-force simulation shows you understand the elimination order, but the heap (priority queue) optimization demonstrates practical algorithm design. The final implementation combines greedy decision making with efficient ordering using a heap and some lightweight math to update the effective time after each step.
First, consider the case where there is only one type of bacteria. In this case, there is no need to split the white blood cell (WBC); it can directly eliminate the bacteria, and the time cost is timeSeq[0].
If there are two types of bacteria, the WBC needs to split into two, and each WBC eliminates one type of bacteria. The time cost is splitTime + max(timeSeq[0], timeSeq[1]).
If there are more than two types of bacteria, at each step, we need to consider splitting the WBCs into multiple cells, which is difficult to handle with a forward-thinking approach.
Instead, we can adopt a reverse-thinking approach: instead of splitting the WBCs, we merge the bacteria. We select any two types of bacteria i and j to merge into a new type of bacteria. The time cost for this merge is splitTime + max(timeSeq[i], timeSeq[j]).
To minimize the involvement of bacteria with long elimination times in the merging process, we can greedily select the two bacteria with the smallest elimination times for merging at each step. Therefore, we can maintain a min-heap, repeatedly extracting the two bacteria with the smallest elimination times and merging them until only one type of bacteria remains. The elimination time of this final bacteria is the answer.
The time complexity is O(n times log n), and the space complexity is O(n), where n is the number of bacteria.
| Approach | Time | Space | When to Use |
|---|---|---|---|
| Brute Force Simulation | O(n^2) | O(1) | Useful for understanding the elimination process or when input size is very small |
| Greedy + Min Heap (Priority Queue) | O(n log n) | O(n) | Best general solution when elimination order must always pick the smallest effective time |
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