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In quick sort, for sorting n elements, the (n/4)th smallest element is selected as pivot using an O(n) time algorithm. What is the worst case time complexity of the quick sort?
- a)θ(n)
- b)θ(nLogn)
- c)θ(n^2)
- d)θ(n^2 log n)
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
In quick sort, for sorting n elements, the (n/4)th smallest element is...
The recursion expression becomes: T(n) = T(n/4) + T(3n/4) + cn After solving the above recursion, we get 
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In quick sort, for sorting n elements, the (n/4)th smallest element is...
Understanding Quick Sort with a Good Pivot Selection
Quick Sort is a widely used sorting algorithm that can perform efficiently with the right choice of pivot. When the (n/4)th smallest element is selected as the pivot using an O(n) algorithm, it significantly improves the performance in the worst-case scenario.
Why the (n/4)th Element as Pivot?
- Selecting the (n/4)th element ensures that at least 1/4 of the elements are on either side of the pivot.
- This balanced partitioning means that even in the worst case, the recursion depth remains logarithmic.
Time Complexity Breakdown
- The recursive nature of Quick Sort requires sorting the left and right partitions.
- If the pivot divides the array into two parts where one part has at least 1/4 of the elements, the recursive relation can be expressed as:
T(n) = T(n/4) + T(3n/4) + O(n)
- The O(n) term comes from the partitioning process.
Solving the Recursion
- Using the Master Theorem or the recursive tree method, the depth of recursion is log(n).
- Each level of the recursion performs O(n) work due to partitioning.
Final Complexity
- The total time complexity sums up to O(n log n):
- Each of the log(n) levels performs O(n) work.
Thus, the worst-case time complexity of Quick Sort when using the (n/4)th element as a pivot is indeed:
Option B: θ(n log n)
Quick Sort is a widely used sorting algorithm that can perform efficiently with the right choice of pivot. When the (n/4)th smallest element is selected as the pivot using an O(n) algorithm, it significantly improves the performance in the worst-case scenario.
Why the (n/4)th Element as Pivot?
- Selecting the (n/4)th element ensures that at least 1/4 of the elements are on either side of the pivot.
- This balanced partitioning means that even in the worst case, the recursion depth remains logarithmic.
Time Complexity Breakdown
- The recursive nature of Quick Sort requires sorting the left and right partitions.
- If the pivot divides the array into two parts where one part has at least 1/4 of the elements, the recursive relation can be expressed as:
T(n) = T(n/4) + T(3n/4) + O(n)
- The O(n) term comes from the partitioning process.
Solving the Recursion
- Using the Master Theorem or the recursive tree method, the depth of recursion is log(n).
- Each level of the recursion performs O(n) work due to partitioning.
Final Complexity
- The total time complexity sums up to O(n log n):
- Each of the log(n) levels performs O(n) work.
Thus, the worst-case time complexity of Quick Sort when using the (n/4)th element as a pivot is indeed:
Option B: θ(n log n)
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In quick sort, for sorting n elements, the (n/4)th smallest element is selected as pivot using an O(n) time algorithm. What is the worst case time complexity of the quick sort?a)θ(n)b)θ(nLogn)c)θ(n^2)d)θ(n^2 log n)Correct answer is option 'B'. Can you explain this answer? for Computer Science Engineering (CSE) 2025 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about In quick sort, for sorting n elements, the (n/4)th smallest element is selected as pivot using an O(n) time algorithm. What is the worst case time complexity of the quick sort?a)θ(n)b)θ(nLogn)c)θ(n^2)d)θ(n^2 log n)Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In quick sort, for sorting n elements, the (n/4)th smallest element is selected as pivot using an O(n) time algorithm. What is the worst case time complexity of the quick sort?a)θ(n)b)θ(nLogn)c)θ(n^2)d)θ(n^2 log n)Correct answer is option 'B'. Can you explain this answer?.
In quick sort, for sorting n elements, the (n/4)th smallest element is selected as pivot using an O(n) time algorithm. What is the worst case time complexity of the quick sort?a)θ(n)b)θ(nLogn)c)θ(n^2)d)θ(n^2 log n)Correct answer is option 'B'. Can you explain this answer? for Computer Science Engineering (CSE) 2025 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about In quick sort, for sorting n elements, the (n/4)th smallest element is selected as pivot using an O(n) time algorithm. What is the worst case time complexity of the quick sort?a)θ(n)b)θ(nLogn)c)θ(n^2)d)θ(n^2 log n)Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In quick sort, for sorting n elements, the (n/4)th smallest element is selected as pivot using an O(n) time algorithm. What is the worst case time complexity of the quick sort?a)θ(n)b)θ(nLogn)c)θ(n^2)d)θ(n^2 log n)Correct answer is option 'B'. Can you explain this answer?.
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