Find Time Required to Eliminate Bacterial Strains - Problem

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:

The ith bacterial strain takes timeReq[i] units of time to be eliminated.
A single WBC can eliminate only one bacterial strain. Afterwards, the WBC is exhausted and cannot perform any other tasks.
A WBC can split itself into two WBCs, but this requires splitTime units of time.
Once split, the two WBCs can work in parallel on eliminating the bacteria.
Only one WBC can work on a single bacterial strain. Multiple WBCs cannot attack one strain in parallel.

You must determine the minimum time required to eliminate all the bacterial strains. Note that the bacterial strains can be eliminated in any order.

Input & Output

Example 1 — Basic Case

$ Input: timeReq = [4, 2, 6], splitTime = 3

› Output: 9

💡 Note: Split the WBC (cost 3). At time 3, we have 2 WBCs. Assign strain 6 to WBC1 (completes at time 9) and strains 4,2 to WBC2 (completes at time 3+4+2=9). Total time is max(9,9) = 9.

Example 2 — No Splitting Needed

$ Input: timeReq = [1, 1, 1], splitTime = 10

› Output: 3

💡 Note: Split cost (10) is very high. Better to use single WBC: 1+1+1=3 total time.

Example 3 — Beneficial Splitting

$ Input: timeReq = [10, 10], splitTime = 1

› Output: 11

💡 Note: Split WBC (cost=1), then assign one strain to each WBC: max(1+10, 1+10) = 11. Better than single WBC taking 20 time.

Constraints

1 ≤ timeReq.length ≤ 10⁴
1 ≤ timeReq[i] ≤ 10⁴
1 ≤ splitTime ≤ 10⁴

Visualization

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Asked in

G Google 45 a Amazon 38 M Microsoft 32 f Meta 28

The key insight is to balance splitting costs against parallel execution benefits. The Greedy Priority Queue approach is optimal, using a min-heap to track WBC completion times and making greedy splitting decisions. It processes strains in descending order and splits WBCs when the splitting cost plus parallel execution time is less than sequential execution time, achieving O(n log n) complexity.

Common Approaches

✓ Brute Force Recursive (Exhaustive)

⏱️ Time: O(2^n) Space: O(n)

This approach explores every possible decision tree of splitting WBCs and assigning bacterial strains. For each state, it considers splitting the current WBCs or assigning strains to existing WBCs. It recursively computes the minimum time for all possibilities.

Greedy Priority Queue (Optimal)

⏱️ Time: O(n log n) Space: O(n)

This optimal approach uses a priority queue to simulate the WBC splitting and assignment process. It greedily decides when to split WBCs based on the benefit and uses a min-heap to always assign the next strain to the WBC that will be free earliest.

Hash Table Double Pass

⏱️ Time: O(n²) Space: O(n)

This approach uses hash tables to track strain-to-WBC assignments in two passes. First pass assigns strains greedily, second pass optimizes by considering splitting opportunities. It uses hash maps to maintain state and find optimal assignments efficiently.

Algorithm Steps — Algorithm Steps

Code -

solution.c — C

Time & Space Complexity

Time Complexity

⏱️

⚠ Quadratic Growth

Space Complexity

⚡ Linearithmic Space

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