Algorithms | Analysis of Algorithms | Question 13

Consider the following functions:

  f(n)   = 2^n
  g(n)   = n!
  h(n)   = n^logn 

Which of the following statements about the asymptotic behavior of f(n), g(n), and h(n) is true?
(A) f(n) = O(g(n)); g(n) = O(h(n))
(B) f(n) = \Omega(g(n)); g(n) = O(h(n))
(C) g(n) = O(f(n)); h(n) = O(f(n))
(D) h(n) = O(f(n)); g(n) = \Omega(f(n))
(A) A
(B) B
(C) C
(D) D


Answer: (D)

Explanation: According to order of growth: h(n) < f(n) < g(n) (g(n) is asymptotically greater than f(n) and f(n) is asymptotically greater than h(n) )
We can easily see above order by taking logs of the given 3 functions

   lognlogn < n < log(n!)  (logs of the given f(n), g(n) and h(n)).

Note that log(n!) = \theta(nlogn)

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