GATE CSE
First time here? Checkout the FAQ!
x
+2 votes
57 views

Which of the following two is correct?

  • If f(n) = Ο(g(n)) then h(f(n)) = Ο(h(g(n)))
  • If f(n) ≠ Ο(g(n)) then g(n) = Ο(f(n))
asked in Algorithms by Boss (8.7k points)   | 57 views
second only.

1 Answer

+2 votes
Best answer
Both of them are incorrect.

For first one, let's assume $f(n) = n$ and $g(n) = n^2$. Clearly, $f(n) = O(g(n))$. Now consider $h(n) = 1/n$.

$h(f(n)) = 1/f(n) = 1/n$ and $h(g(n)) = 1/g(n) = 1/n^2$. Now for no constants $c$ and $n_0$, $0 \leq h(f(n)) \leq ch(g(n))$ for all $n > n_0$. (You can confirm this by plotting both $1/n$ and $1/n^2$).

So first statement is incorrect.

For second statement.

CLRS says "Not all functions are asymptotically comparable." That is, for two functions $f(n)$ and $g(n)$, it may be the case
that neither $f(n) = O(g(n))$ nor $f(n) = \Omega(g(n))$ holds. An example would be $f(n) = n$ and $g(n) = n^{1+\sin{n}}$

Hence, both of the statements are incorrect.
answered by Loyal (3.3k points)  
edited by

please explain Not all functions are asymptotically comparable. bit more.

That statement means that it is possible for two functions $f(n)$ and $g(n)$ to exist such that, you can't apply the definitions of any of $O$, or, $\Omega$, or, $\Theta$, or, $o$, or, $\omega$ between them.

Considering the same $f(n) = n$ and $g(n) = n ^ {1 + \sin{n}}$, Since $1 + \sin{n}$ oscillates between 0 and 2, $g(n)$ oscillates between $n^0$ and $n^2$, hence it's not possible to find two constants, $c$ and $n_0$ such that $0 \leq f(n) \leq cg(n)$ for all $n > n_0$ or similarly for other definitions.

You can get more insights from here

Related questions

0 votes
1 answer
1
asked in Algorithms by Akriti sood Veteran (13.2k points)   | 47 views
+1 vote
0 answers
2
+2 votes
1 answer
3


Top Users Jun 2017
  1. Bikram

    3704 Points

  2. Arnab Bhadra

    1502 Points

  3. Hemant Parihar

    1502 Points

  4. Niraj Singh 2

    1481 Points

  5. junaid ahmad

    1432 Points

  6. Debashish Deka

    1384 Points

  7. Rupendra Choudhary

    1220 Points

  8. rahul sharma 5

    1220 Points

  9. Arjun

    1158 Points

  10. srestha

    1006 Points

Monthly Topper: Rs. 500 gift card
Top Users 2017 Jun 26 - Jul 02
  1. Arjun

    198 Points

  2. akankshadewangan24

    152 Points

  3. Debashish Deka

    138 Points

  4. Hira Thakur

    130 Points

  5. Soumya29

    104 Points


23,399 questions
30,111 answers
67,489 comments
28,424 users