GATE CS 2015 Set 1 Q.12 (CS Q.2): Asymptotic Analysis and Time/Space Complexity

Which one of the following is the recurrence equation for the worst case time complexity of the Quicksort algorithm for sorting $n$ ( $n \ge 2$ ) numbers? In the recurrence equations given in the options below, $c$ is a constant.

$T(n) = 2T(n/2) + cn$

$T(n) = T(n-1) + T(1) + cn$

$T(n) = 2T(n-1) + cn$

$T(n) = T(n/2) + cn$

Official answer and solution

Official answer: B
From: Official GATE 2015 answer key

Solution

Formulate the recurrence relation for Quicksort when the partition is maximally unbalanced at each recursive step.

In Quicksort, an array of $n$ elements is partitioned using a pivot. Partitioning takes linear time, $cn$ .

In the worst case (such as when the array is already sorted and the extreme element is chosen as pivot):

One partition contains $n - 1$ elements.
The other partition contains $0$ (or $1$ ) element.

Thus, the recurrence relation describing this behavior is:

T(n) = T(n-1) + T(1) + cn.

This matches option B.

Final answer

Common trap

Confusing worst-case Quicksort with best/average case $T(n) = 2T(n/2) + cn$ (option A).

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: 12 of 65, printed as CS Q.2
Section: Computer Science and Information Technology

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