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54 votes
54 votes

Consider the following functions from positive integers to real numbers:

$10$, $\sqrt{n}$, $n$, $\log_{2}n$, $\frac{100}{n}$.
The CORRECT arrangement of the above functions in increasing order of asymptotic complexity is:

  1. $\log_{2}n$, $\frac{100}{n}$, $10$, $\sqrt{n}$, $n$
  2. $\frac{100}{n}$, $10$, $\log_{2}n$, $\sqrt{n}$, $n$
  3. $10$, $\frac{100}{n}$, $\sqrt{n}$, $\log_{2}n$, $n$
  4. $\frac{100}{n}$, $\log_{2}n$, $10$, $\sqrt{n}$, $n$
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12 Answers

Best answer
67 votes
67 votes

$10$ is constant.  $\therefore$ Growth rate is $0$.

$\sqrt{n}$  grows slower than linear but faster than $\log$. (Consider $\frac {\sqrt {n^2}}{\sqrt n} =\sqrt n$, whereas $\frac {\log \left(n^2\right)}{\log n} = 2$)

 $n$  : Growth rate is linear.

$\log_{2}n$  : Growth rate is logarithmic. For asymptotic growth, the base does not matter.

$\frac{100}{n}$  : Growth rate decreases with $n$.

So, correct answer is (B).

NOTE: Please never substitute large values of $n$ in such questions. If ever you do, at least do for $2$ such values and take ratio to get the growth rate or plot a graph. Remember ${1.01}^{n} \neq O\left(n^{100}\right)$.

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45 votes
45 votes
  • $10$, Constant
  • $\sqrt{n}$, Square root
  • $n$, Polynomial
  • $\log_{2}n$, Logarithmic
  • $\frac{100}{n}$. Constant division by polynomial (clearly less than constant for every value of n >100)

Now we know  Constant division by polynomial < Constant  < Logarithmic <Square root < Polynomial

So, correct order is $\frac{100}{n}$, $10$, $\log_{2}n$, $\sqrt{n}$, $n$

PS: In the above graph, for asymptotic notations any graph can be multiplied by a constant and so the $\textbf{SLOPE}$ is the important factor. If any plot is having higher slope, it'll eventually overtake the lower slope one at some value of $x$ and hence asymptotically larger than the other.

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13 votes
13 votes

Here I m using the term win but that I dont think is right..I have been using this for my own understanding purposes..

11 votes
11 votes

if we take a very large value of n suppose =21024

then we can follow the sequence with result

100/21024 <10 < log1024=1024 <sqrt(21024)  < 21024 

so following we see B is the correct answer

Answer:

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