Question

Select the correct inequality for the asymptotic order of growth of the function n²ⁿ – 2ⁿ?

1) n²ⁿ = O(2ⁿ)
2) n²ⁿ = Ω(2ⁿ)
3) n²ⁿ = Θ(2ⁿ)
4) n²ⁿ = o(2ⁿ)

Answer 1

The correct inequality for the asymptotic growth of the function n²ⁿ − 2ⁿ is n²ⁿ = Ω(2ⁿ), which indicates that n²ⁿ grows at least as fast as 2ⁿ.

Therefore, the correct option is 2) n²ⁿ = Ω(2ⁿ).

Step-by-step explanation:

In the context of algorithm analysis, it is often crucial to understand the asymptotic order of growth of functions. This helps in predicting how the running time of an algorithm increases with the size of the input. The function in question is n²ⁿ − 2ⁿ, and we are to choose the correct inequality that describes the asymptotic behavior of this function relative to 2ⁿ.

Let's evaluate the options given:

1. n²ⁿ = O(2ⁿ) states that n²ⁿ is upper bounded by 2ⁿ, which is incorrect as n²ⁿ grows faster than 2ⁿ.
2. n²ⁿ = Ω(2ⁿ) indicates that n²ⁿ grows at least as fast as 2ⁿ, which is the correct inequality in this case.
3. n²ⁿ = Θ(2ⁿ) states that n²ⁿ grows at the same rate as 2ⁿ, which is also incorrect.
4. n²ⁿ = o(2ⁿ) suggests that n²ⁿ grows slower than 2ⁿ, which, again, is not accurate.

Therefore, the correct option is 2), where n²ⁿ = Ω(2ⁿ). As n grows, the exponential term n²ⁿ will outstrip the growth of 2ⁿ, making this the best representation of the function's growth.