Quicksort and Divide-and-Conquer Analysis Practice Questions for GATE DA: 20+ Solved Questions with Step-by-Step Solutions

Solve 20+ Quicksort and Divide-and-Conquer Analysis practice questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Quicksort and Divide-and-Conquer

Orientation

Chapter Roadmap: Quicksort and Divide-and-Conquer

Welcome to the chapter on Quicksort and Divide-and-Conquer Analysis. We will first master the mechanics of Quicksort partitioning and learn how to precisely count swaps. Then, we will transition into the mathematical analysis of expected recurrence relations.

Step 1: Quicksort Partitioning and Swap Count

Master the Lomuto and Hoare schemes. Trace arrays, count exact swaps, and handle edge cases like sorted or reverse-sorted inputs.

Step 2: Expected Recurrence Analysis

Formulate and solve the recurrence relation for randomized Quicksort to prove the expected $O (n lo g n)$ time complexity.

The Heart of Quicksort: Partitioning

Concept

The Heart of Quicksort: Partitioning

The fundamental principle of Quicksort relies on the Partitioning Step:

Choose a Pivot: Select an element from the array (e.g., the last, first, or a random element).

Partition: Rearrange the array such that:

Every element in the left subarray is $\leq$ pivot.
Every element in the right subarray is $\geq$ pivot.
The pivot is placed in its exact final sorted index.

Recurse: Recursively apply Quicksort to the left and right subarrays.

The partitioning step takes $O (n)$ time. The overall time complexity depends on how balanced the resulting subarrays are, which is dictated by the pivot choice and the specific partitioning scheme used.

Quicksort and Divide-and-Conquer Analysis: Solved Questions with Step-by-Step Explanations (5 Problems)

Question 1 · Programming, Data Structures and Algorithms MCQ

In the expected time recurrence relation for randomized Quicksort, what does the linear term $O (n)$ represent?

A.
The time to recursively sort the left subarray
B.
The time to recursively sort the right subarray
C.
The time required for the partitioning step itself
D.
The time to randomly select the pivot

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct recall question about the components of the expected time recurrence relation for Quicksort.

Step 1: Recall the structure of the recurrence relation: $T (n) = \frac{1}{n} \sum [T (k) + T (n - k - 1)] + O (n)$ .

Step 2: The terms $T (k)$ and $T (n - k - 1)$ represent the expected time to recursively sort the left and right subarrays.

Step 3: The $O (n)$ term is added outside the summation. It represents the work done at the current level of recursion before making the recursive calls.

Step 4: This work is the partitioning step, which scans the array and rearranges elements, taking linear time $O (n)$ .

Answer: C

Question 2 · Programming, Data Structures and Algorithms MCQ

What is the primary goal of the partitioning step in the Quicksort algorithm?

A.
To find the minimum element in the current subarray.
B.
To place the pivot element in its correct sorted position, with all smaller elements before it and all larger elements after it.
C.
To divide the array into two subarrays of exactly equal size.
D.
To sort the entire array in a single linear pass without recursion.

Correct Answer:

Step-by-Step Solution

Key idea: This is a definition question, recognizable by the request to identify the fundamental purpose of a specific algorithm step.

Step 1: Recall the definition of Quicksort partitioning. The core intuition is to pick a pivot and rearrange the array.

Step 2: Identify the rearrangement goal. All elements smaller than the pivot must come before it, and all elements larger must come after it.

Step 3: Conclude the result. Once this is done, the pivot is in its final sorted position, and the algorithm can recursively sort the left and right subarrays.

Answer: B

Question 3 · Programming, Data Structures and Algorithms MCQ

If the standard Lomuto partition scheme (using the last element as the pivot) is applied to an array that is already sorted in ascending order, what is the immediate result of the first partition step?

A.
The pivot is placed at the beginning of the array.
B.
The array is divided into two perfectly equal halves.
C.
The pivot remains at the end of the array, which is its correct final sorted position.
D.
The algorithm terminates immediately without performing any comparisons.

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct application question testing the behavior of the Lomuto scheme on a specific edge-case input.

Step 1: Identify the input: an array already sorted in ascending order.

Step 2: Identify the pivot: the last element, which is the maximum element in the subarray.

Step 3: Trace the partition logic. Since every element in the array is less than or equal to the maximum element (the pivot), the condition $A [j] \leq pivot$ is always true.

Step 4: Conclude the result. The boundary pointer $i$ will simply track the scanning pointer $j$ , resulting in self-swaps. Finally, the pivot is swapped with itself at the end. The pivot remains at the end, which is its correct final sorted position.

Answer: C

Question 4 · Programming, Data Structures and Algorithms MCQ

In the expected recurrence analysis of randomized Quicksort, what is the probability that any specific element is chosen as the pivot from a subarray of size $n$ ?

A.
$1/2$
B.
$1/ n$
C.
$1/ (n - 1)$
D.
$1/ lo g n$

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct recall question about the fundamental probability distribution used in randomized Quicksort.

Step 1: Recall the definition of randomized Quicksort. The pivot is chosen uniformly at random from the current subarray.

Step 2: "Uniformly at random" means every element in the subarray has an equal chance of being selected.

Step 3: Since there are $n$ elements in the subarray, the probability of selecting any specific element is exactly $1/ n$ .

Answer: B

Question 5 · Programming, Data Structures and Algorithms MCQ

In the standard Lomuto partition scheme, which element is conventionally chosen as the pivot for a given subarray?

A.
The first element of the subarray.
B.
The middle element of the subarray.
C.
The last element of the subarray.
D.
A randomly selected element from the subarray.

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct recall question about the mechanics of a specific partitioning scheme.

Step 1: Identify the scheme mentioned in the question: Lomuto partition scheme.

Step 2: Recall the standard implementation of the Lomuto scheme. It conventionally selects the last element of the current subarray (at index $r$ ) as the pivot.

Step 3: Match this fact with the given options.

Answer: C

Quicksort and Divide-and-Conquer Analysis Practice Questions for GATE DA: 20+ Solved Questions with Step-by-Step Solutions

Chapter Roadmap: Quicksort and Divide-and-Conquer

Chapter Roadmap: Quicksort and Divide-and-Conquer

Step 1: Quicksort Partitioning and Swap Count

Step 2: Expected Recurrence Analysis

The Heart of Quicksort: Partitioning

The Heart of Quicksort: Partitioning

Quicksort and Divide-and-Conquer Analysis: Solved Questions with Step-by-Step Explanations (5 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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