GATE CS 2014 Set 1 Q.52 (CS Q.42): Asymptotic Analysis and Time/Space Complexity

Consider the following pseudo code. What is the total number of multiplications to be performed?

D = 2
for i = 1 to n do
    for j = i to n do
        for k = j + 1 to n do
            D = D * 3

Half of the product of the 3 consecutive integers.

One-third of the product of the 3 consecutive integers.

One-sixth of the product of the 3 consecutive integers.

None of the above.

Official answer and solution

Official answer: C
From: Official GATE 2014 answer key

Solution

Count the number of iterations of the three nested loops by analyzing the loop index inequalities.

In the given pseudocode:

Loop 1: i runs from $1$ to $n$ .
Loop 2: j runs from i to $n$ .
Loop 3: k runs from $j + 1$ to $n$ .
Inside the innermost loop, one multiplication D = D * 3 is performed per iteration.

The number of multiplications is the number of integer triples $(i, j, k)$ satisfying:

1 \le i \le j < k \le n

Let us split this condition into two disjoint cases based on whether $i < j$ or $i = j$ :

Case $i < j < k$ :
This corresponds to choosing 3 distinct indices from $\{1, 2, \dots, n\}$ :
$\text{Count}_1 = \binom{n}{3} = \frac{n(n-1)(n-2)}{6}$
Case $i = j < k$ :
Here $i$ and $j$ are identical, so we choose 2 distinct values (the common value $i = j$ , and $k$ ):
$\text{Count}_2 = \binom{n}{2} = \frac{n(n-1)}{2} = \frac{3n(n-1)}{6}$

Total number of multiplications:

\text{Total} = \binom{n}{3} + \binom{n}{2} = \frac{n(n-1)(n-2) + 3n(n-1)}{6}

\text{Total} = \frac{n(n-1)[(n-2) + 3]}{6} = \frac{(n-1)n(n+1)}{6}

Notice that $(n-1)$ , $n$ , and $(n+1)$ are three consecutive integers.
Therefore, the total number of multiplications is 'one-sixth of the product of the 3 consecutive integers' (Option C).

Final answer

Common trap

Assuming $i < j < k$ strictly, which gives only $\binom{n}{3}$ , forgetting that $j$ starts at $i$ .

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

Source

Paper: GATE 2014 Computer Science and Information Technology (CS) Session 1
Question: 52 of 65, printed as CS Q.42
Section: Computer Science and Information Technology

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