Question

Master theorem: \begin{equation} T(n) = \begin{cases} c & \text{if } n < d\\ aT(n/b) + f(n) & \text{if } n \ge d \end{cases} \end{equation} 1. if $f(n)$ is $O(n^{\log_b a - \epsilon})$, then $T(n)$ is $\Theta(n^{\log_b a})$ 2. if $f(n)$ is $\Theta(n^{\log_b a} \log^k n)$, then $T(n)$ is $\Theta(n^{\log_b a} \log^{k+1} n)$ 3. if $f(n)$ is $\Omega(n^{\log_b a + \epsilon})$, then $T(n)$ is $\Theta(f(n))$, provided $af(n/b) \le \delta f(n)$ for some $\delta < 1$. Which of the three cases above applies to an algorithm, the running time of which is described using the following recurrence relation: T(n) = 2 T(n/2) + \log^2 n Type 1 for case 1, and so on. For case 2, also specify a k value after a space, e.g., "2 2" Type N/A if the Master theorem is not applicable to the recurrence relation Answer:

          Master theorem:
\begin{equation*}
T(n) = \begin{cases} c & \text{if } n < d\\ aT(n/b) + f(n) & \text{if } n \ge d \end{cases}
\end{equation*}
1. if $f(n)$ is $O(n^{\log_b a - \epsilon})$, then $T(n)$ is $\Theta(n^{\log_b a})$
2. if $f(n)$ is $\Theta(n^{\log_b a} \log^k n)$, then $T(n)$ is $\Theta(n^{\log_b a} \log^{k+1} n)$
3. if $f(n)$ is $\Omega(n^{\log_b a + \epsilon})$, then $T(n)$ is $\Theta(f(n))$,
provided $af(n/b) \le \delta f(n)$ for some $\delta < 1$.
Which of the three cases above applies to an algorithm, the running time of which is described
using the following recurrence relation:
T(n) = 2 T(n/2) + \log^2 n
Type 1 for case 1, and so on. For case 2, also specify a k value after a space, e.g., "2 2"
Type N/A if the Master theorem is not applicable to the recurrence relation
Answer:

Master theorem:

T(n) = c if n < d
aT(n/b) + f(n) if n ≥ d

1. if f(n) is O(n^ a - ϵ), then T(n) is Θ(n^ a)
2. if f(n) is Θ(n^ alog^k n), then T(n) is Θ(n^ alog^k+1 n)
3. if f(n) is Ω(n^ a + ϵ), then T(n) is Θ(f(n)),
provided af(n/b) ≤δ f(n) for some δ < 1.
Which of the three cases above applies to an algorithm, the running time of which is described
using the following recurrence relation:
T(n) = 2 T(n/2) + log^2 n
Type 1 for case 1, and so on. For case 2, also specify a k value after a space, e.g., "2 2"
Type N/A if the Master theorem is not applicable to the recurrence relation
Answer:

Added by Marc J.

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