GATE CS 2017 Set 1 Q.4 (CS Q.4): Asymptotic Analysis and Time/Space Complexity

Consider the following functions from positive integers to real numbers:

10, \quad \sqrt{n}, \quad n, \quad \log_2 n, \quad \frac{100}{n}

The CORRECT arrangement of the above functions in increasing order of asymptotic complexity is:

$\log_2 n, \; \frac{100}{n}, \; 10, \; \sqrt{n}, \; n$

$\frac{100}{n}, \; 10, \; \log_2 n, \; \sqrt{n}, \; n$

$10, \; \frac{100}{n}, \; \sqrt{n}, \; \log_2 n, \; n$

$\frac{100}{n}, \; \log_2 n, \; 10, \; \sqrt{n}, \; n$

Official answer and solution

Solution

Evaluate the asymptotic behavior of each function as $n \to \infty$ to establish their order of growth.

Let us examine the limit of each function as $n \to \infty$ :

$\frac{100}{n}$ : As $n \to \infty$ , $\frac{100}{n} \to 0$ . Its complexity is $O(1/n)$ , which decreases with $n$ and is asymptotically smaller than any positive constant.
$10$ : This is a constant function, $\Theta(1)$ .
$\log_2 n$ : This function grows unboundedly as $n \to \infty$ , but slower than any polynomial function $n^c$ with $c > 0$ .
$\sqrt{n} = n^{0.5}$ : This is a polynomial of degree $0.5$ .
$n = n^1$ : This is a linear function, which grows faster than $n^{0.5}$ .

Arranging these functions in increasing order of asymptotic complexity gives:

\frac{100}{n} < 10 < \log_2 n < \sqrt{n} < n

This corresponds to Option B.

Final answer

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