Question

2. [01] (20 pts) (a) (10 pts) Use mathematical induction to show that when $n$ is an exact power of 2, the solution of the recurrence $T(n) = \begin{cases} 2, & n = 2 \\ 2T(\frac{n}{2}) + n, & n = 2^k, k > 1 \end{cases}$ is $T(n) = n \log n$. (b) (10 pts) Give a closed form for the following recurrence using asymptotic notation. In particular, you need to give a function $f$ such that $T(n) = \Theta(f(n))$ and justify your answer. $T(n) = 3T(\frac{n}{\sqrt{2}}) + n^4$

          2. [01] (20 pts)
(a) (10 pts) Use mathematical induction to show that when $n$ is an exact power of
2, the solution of the recurrence
$T(n) = \begin{cases} 2, & n = 2 \\ 2T(\frac{n}{2}) + n, & n = 2^k, k > 1 \end{cases}$
is $T(n) = n \log n$.
(b) (10 pts) Give a closed form for the following recurrence using asymptotic notation. In particular, you need to give a function $f$ such that $T(n) = \Theta(f(n))$ and
justify your answer.
$T(n) = 3T(\frac{n}{\sqrt{2}}) + n^4$

2. [01] (20 pts)
(a) (10 pts) Use mathematical induction to show that when n is an exact power of
2, the solution of the recurrence
T(n) = 2, n = 2
2T((n)/(2)) + n, n = 2^k, k > 1
is T(n) = n log n.
(b) (10 pts) Give a closed form for the following recurrence using asymptotic notation. In particular, you need to give a function f such that T(n) = Θ(f(n)) and
justify your answer.
T(n) = 3T((n)/(√(2))) + n^4

Added by Kelsey P.

Computer Science and Information Technology

Trishna Knowledge Systems 2018 Edition

Instant Answer

Solved by Expert Razan K

03/04/2022

Step 1

T(2) = 2 log 2 = 2 * 1 = 2 So the statement holds for the base case. Show more…

Show all steps

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(a) (10 pts) Use mathematical induction to show that when n is an exact power of 2, the solution of the recurrence T(n) = ( 2, n = 2 2T( n 2 ) + n, n = 2k , k > 1 is T(n) = n log n. (b) (10 pts) Give a closed form for the following recurrence using asymptotic notation. In particular, you need to give a function f such that T(n) = Θ(f(n)) and justify your answer. T(n) = 3T( n √ 2 ) + n 4 2. [O1] (20 pts) a) (10 pts) Use mathematical induction to show that when n is an exact power of 2.the solution of the recurrence n=2 T(n) 2T()+n,n=2k,k>1 is T(n) = n log n. (b) (10 pts) Give a closed form for the following recurrence using asymptotic nota tion. In particular, you need to give a function f such that T(n) = O(f(n)) and justify your answer. T(n=3T +n