Asymptotic time order

Question

Comments

$(A)?$

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Plz explain with solution .

Answer 1

$f(n)=n^{\frac{1}{\sqrt{\log n}}}= e^{\log n^{\frac{1}{\sqrt{\log n}}}}=e^{\sqrt{\log n}}$

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$g(n)=\sqrt{\log n}=e^{\log \sqrt{\log n}}$

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$h(n)=n^{\frac{1}{100}}=e^{\log n^{\frac{1}{100}} }=e^{k \times \log n}$

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$g(n)\, < \,f(n) \, <h(n)$

Comments

can u tell me what is the wrong with this solution ?

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it will be better to derive the asymptotic function in  simplest term as you can .after you are done and if you have enough time,you can put large values to verify .don't put larger value to verify the answer in your first attempt, it might be risky and you may lose your marks :)

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Thanks .

But for all other questions my solution didn't went wrong by choosing a large number , but for this question only it went wrong . Why this happened ? Secondly, what is the wrong with my solution as depicted above .

Kindly help so that in future I could apply the things judiciously .

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you have done right, but i am not getting why you have written option $D$? you too are getting $A$

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For h(n) it is 10.24 not 1024 . So the larger value is f(n) and not h(n) . Little confused.

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I thought it as $1024$.As you know if $f(n)=O(g(n))\Rightarrow f(n) \leq c \times g(n) ,\text{where }n > n_0$

sometimes when you put larger value then you may lie  the within the range of $n_0$,perfect example is the case you did.That's the reason to break the asymptotic function in simpler terms.

Well you try with $2^{65,536}$ you will get you answer

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Thanks . It helped a lot.

But a liitle confusion . I have solved almost more than 95 percent of the problem using this method only and I got correct answer . But this case failed . Could u tell me the best approach so that it must be true in each and every case.

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i have told you  to simplify it in simpler terms see comment above

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Thank u :)

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But I think if you compare f(n) and h(n), f(n) is greater because it will have 1/(logn)^1/2 term whereas h(n) will contain a constant term 1/100

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I am facing problems in logarithmic conversions. Like how you did the last step of simplifying f(n). Please suggest resources to read about these formulae.