Computer Engineering(CE)
Question 1: Algorithms
August 2013
Solution 1
(a) First, let us change the variables. Let $ n = 2^{m} $, so equivalently, we have $ m = \log_2 n $. Thus, $ \sqrt[]{n} = 2^{\frac{m}{2}} $.
Then we have: $ T(2^m) = 2 T(2^{\frac{m}{2}}) + \log {2^m} = 2 T(2^{\frac{m}{2}}) + m $. We denote the running time in terms of $ m $ is $ S(m) $, so $ S(m) = T(2^m) $, where $ m = \log n $. so we have $ S(m) = 2S(\frac{m}{2})+ m $.
Now this recurrence can be written in the form of $ T(m) = aT(\frac{m}{b})+ f(m) $, where $ a=2 $, $ b=2 $, and $ f(m)=m $.
$ f(m) = m = \Theta(n^{\log _{b}{a}}) = \Theta(n) $. So the second case of master's theorem applies, we have $ S(k) = \Theta(k^{\log _{b}{a}} \log k) = \Theta(k \log k) $.
Replace back with $ T(2^m) =S(m) $, and $ m = \log_2 n $, we have $ T(n) = \Theta((\log n) (\log \log n)) $.
For the given recurrence, we replace n with $ 2^m $ and denote the running time as $ S(m) $. Thus,we have $ S(m) = T(2^m) = 2 T(2^{\frac{m}{2}}) + m $
(b) $ 3^{f(n)} $ is NOT $ O(3^{g(n)} $). Here is a counter example:
Let $ f(n) = n $ and $ g(n)=\frac{n}{2} $. Then $ f(n) = O(g(n)) $. Now, $ 3^{f(n)}=3^n $, $ f(3^{f(n)})=O(3^n) $; however, $ O(3^{g(n)})=O(3^{\frac{n}{2}}) $. So $ f(3^{f(n)}) \neq O(3^{g(n)}) $.
Solution 2
(a) Assume $ T(n) = O(\log n) $, so
$ \begin{equation} \begin{aligned} T(\sqrt[]{n}) &= O(\log \sqrt[]{n} ) \\ &= O(\frac{1}{2}\log n) \end{aligned} \end{equation} $
So
,
$ \begin{equation} \begin{aligned} T(n) &= 2 T(\sqrt[]{n}) + \log n \\ &= O(\log n ) + \log n \\ &= O(\log n) \end{aligned} \end{equation} $
(b) $ f(n) $ is $ O(\log n) $, then
$ f(n) <= g(n) $.
So,
$ \begin{equation} 3^{f(n)} <= 3^{g(n)} \end{equation} $
So, $ 3^{f(n)} $ is $ O(3^{g(n)}) $
