Problem Description
There are buckets buckets of liquid, where exactly one of the buckets is poisonous. To figure out which one is poisonous, you feed some number of (poor) pigs the liquid to see whether they will die or not. Unfortunately, you only have minutesToTest minutes to determine which bucket is poisonous. You can feed the pigs according to these steps: choose some live pigs to feed, for each pig, choose which buckets to feed it, wait for minutesToDie minutes, and determine which pigs have died after the wait. Given buckets, minutesToDie, and minutesToTest, return the minimum number of pigs needed to figure out which bucket is poisonous within the allotted time.
Key Insights
- Each pig can provide multiple outcomes based on whether it dies or survives after being fed different buckets.
- The number of tests that can be performed is limited by the time available and the time it takes for pigs to die.
- The number of outcomes from a certain number of pigs is exponential, depending on the number of pigs and the time available for testing.
Space and Time Complexity
Time Complexity: O(log(buckets)) - The complexity is determined by the number of pigs needed to cover all buckets in the allowed time. Space Complexity: O(1) - We only need a constant amount of space to store variables for calculations.
Solution
To solve this problem, we can use a mathematical approach based on the fact that we can represent the outcome of the pigs dying or surviving as a base system. Each pig can represent a digit in that base system, and the total number of outcomes can be derived from the number of pigs and the time constraints. We calculate how many pigs are needed to cover all possible outcomes based on the number of buckets and the time available for testing.
Let k be the number of pigs. The total number of outcomes after t minutes is (t / minutesToDie) + 1, as we can test multiple times. We need to satisfy the condition:
(buckets <= (t / minutesToDie + 1) ^ k)
We can iterate through increasing values of k until we find the minimum number that satisfies this condition.