GATE CS 2015 Set 1 Q.41 (CS Q.31): Asymptotic Analysis and Time/Space Complexity

Consider the following C function:

int fun1(int n) {
    int i, j, k, p, q = 0;
    for (i = 1; i < n; ++i) {
        p = 0;
        for (j = n; j > 1; j = j / 2)
            ++p;
        for (k = 1; k < p; k = k * 2)
            ++q;
    }
    return q;
}

Which one of the following most closely approximates the return value of the function fun1?

$n^3$

$n(\log n)^2$

$n \log n$

$n \log(\log n)$

Official answer and solution

Official answer: D
From: Official GATE 2015 answer key

Solution

Trace the step counts of nested loops in terms of input size n.

Analyzing the nested loops of fun1(n):

Outer loop: for (i = 1; i < n; ++i) executes $n - 1 = \Theta(n)$ times.
First inner loop: for (j = n; j > 1; j = j / 2) ++p;
This loop repeatedly divides $j$ by 2 starting from $n$ until $j \le 1$ .
The number of iterations is $p = \lfloor \log_2 n \rfloor = \Theta(\log n)$ .
Second inner loop: for (k = 1; k < p; k = k * 2) ++q;
This loop starts with $k = 1$ and doubles $k$ until $k \ge p$ .
The number of increments to $q$ in one outer iteration is $\lfloor \log_2 p \rfloor = \log_2(\log_2 n) = \Theta(\log \log n)$ .
Total return value:
Summing $q$ over all $n$ iterations of the outer loop:
$q \approx n \times \log_2(\log_2 n) = \Theta(n \log(\log n)).$

This matches option D.

Final answer

Common trap

Confusing the doubling of $k$ with linear incrementation, which would erroneously yield $n \log n$ .

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2015 Computer Science and Information Technology (CS) Session 1
Question: 41 of 65, printed as CS Q.31
Section: Computer Science and Information Technology

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