Difference between revisions of "ECE PhD QE CE 2013 Problem1.1" - Rhea

Revision as of 17:25, 20 July 2017

Solution 1

(a) First, let us change the variable. 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 in 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 an counter example: Let $ f(n) = n $ and $ g(n)=\frac{n}{2} $. Then </math>f(n) = O(g(n))</math>.

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)}) $.

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