Final answer:
Tractable growth rates are those that allow algorithms to complete in a reasonable time as input size grows. O(n log n), O(n³), and O(n² log n) are considered tractable, whereas O(2^n), O(n^n), and O(n!) are non-tractable because they grow too quickly.
Step-by-step explanation:
The tractability of a function often refers to how the complexity of an algorithm grows as the size of its input increases, which is important in areas such as computer science and engineering. When assessing which growth rates are tractable, we typically look for those that allow the completion of an algorithm in a reasonable amount of time as the input size grows.
- O(2^n): This represents an exponential growth rate which is generally not tractable because the time it takes to compute an algorithm increases too rapidly as the size of the input increases.
- O(n log n): This is considered a tractable growth rate and is often seen in efficient sorting algorithms.
- O(n^n): This signifies an extremely rapid growth that is non-tractable for even small input sizes.
- O(n!): Factorial growth is also non-tractable due to its incredibly fast rate of increase.
- O(n¹⁰): Polynomial growth rates are typically tractable; however, such a high-degree polynomial could be impractical for larger inputs.
- O(n³): This cubic growth rate is generally tractable, but may become impractical for very large inputs.
- O(n² log n): This growth is slightly more than quadratic and is generally tractable, but efficiency may decline as input sizes increase greatly.
In conclusion, O(n log n), O(n³), and O(n² log n) are considered tractable growth rates, while O(2^n), O(n^n), and O(n!) represent non-tractable growth rates due to their rapid increase relative to the size of the input.