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Consider the Quicksort algorithm. Suppose there is a procedure for finding a pivot element which splits the list into two sub-lists each of which contains at least one-fifth of the elements. Let T(n) be the number of comparisons required to sort n elements. Then
- a)T(n) <= 2T(n/5) + n
- b)T(n) <= T(n/5) + T(4n/5) + n
- c)T(n) <= 2T(4n/5) + n
- d)T(n) <= 2T(n/2) + n
Correct answer is option 'B'. Can you explain this answer?
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Consider the Quicksort algorithm. Suppose there is a procedure for fin...
For the case where n/5 elements are in one subset, T(n/5) comparisons are needed for the first subset with n/5 elements, T(4n/5) is for the rest 4n/5 elements, and n is for finding the pivot. If there are more than n/5 elements in one set then other set will have less than 4n/5 elements and time complexity will be less than T(n/5) + T(4n/5) + n because recursion tree will be more balanced.
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Consider the Quicksort algorithm. Suppose there is a procedure for fin...
Understanding Quicksort and Pivot Selection
In the Quicksort algorithm, the choice of pivot is crucial for performance. In this scenario, we have a pivot selection method that guarantees that each sub-list contains at least one-fifth of the elements. This significantly influences the recurrence relation for the number of comparisons required to sort a list of size \(n\).
The Recurrence Relation
We denote \(T(n)\) as the number of comparisons needed to sort \(n\) elements. Given the pivot splits the list into two sub-lists, one with at least \(n/5\) elements and the other with at most \(4n/5\) elements, we can express the recurrence relation as follows:
- The first sub-list has a size of at least \(n/5\).
- The second sub-list has a size of at most \(4n/5\).
Thus, the number of comparisons can be captured by the relation:
Option B: T(n) <= T(n/5) + T(4n/5) + n
This relation indicates that:
- \(T(n/5)\) accounts for the time to sort the smaller sub-list.
- \(T(4n/5)\) accounts for the time to sort the larger sub-list.
- The \(n\) term represents the comparisons made during partitioning around the pivot.
Why Other Options are Incorrect
- Option A: \(T(n) <= 2T(n/5) + n\) is incorrect as it assumes both sub-lists are of size \(n/5\), which is not guaranteed.
- Option C: \(T(n) <= 2T(4n/5) + n\) is also incorrect because it does not account for the smaller sub-list.
- Option D: \(T(n) <= 2T(n/2) + n\) assumes a balanced split, which is not applicable here.
Conclusion
In summary, the correct recurrence relation that accurately reflects the behavior of the modified Quicksort with the specific pivot selection is:
Option B: T(n) <= T(n/5) + T(4n/5) + n
In the Quicksort algorithm, the choice of pivot is crucial for performance. In this scenario, we have a pivot selection method that guarantees that each sub-list contains at least one-fifth of the elements. This significantly influences the recurrence relation for the number of comparisons required to sort a list of size \(n\).
The Recurrence Relation
We denote \(T(n)\) as the number of comparisons needed to sort \(n\) elements. Given the pivot splits the list into two sub-lists, one with at least \(n/5\) elements and the other with at most \(4n/5\) elements, we can express the recurrence relation as follows:
- The first sub-list has a size of at least \(n/5\).
- The second sub-list has a size of at most \(4n/5\).
Thus, the number of comparisons can be captured by the relation:
Option B: T(n) <= T(n/5) + T(4n/5) + n
This relation indicates that:
- \(T(n/5)\) accounts for the time to sort the smaller sub-list.
- \(T(4n/5)\) accounts for the time to sort the larger sub-list.
- The \(n\) term represents the comparisons made during partitioning around the pivot.
Why Other Options are Incorrect
- Option A: \(T(n) <= 2T(n/5) + n\) is incorrect as it assumes both sub-lists are of size \(n/5\), which is not guaranteed.
- Option C: \(T(n) <= 2T(4n/5) + n\) is also incorrect because it does not account for the smaller sub-list.
- Option D: \(T(n) <= 2T(n/2) + n\) assumes a balanced split, which is not applicable here.
Conclusion
In summary, the correct recurrence relation that accurately reflects the behavior of the modified Quicksort with the specific pivot selection is:
Option B: T(n) <= T(n/5) + T(4n/5) + n
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Consider the Quicksort algorithm. Suppose there is a procedure for finding a pivot element which splits the list into two sub-lists each of which contains at least one-fifth of the elements. Let T(n) be the number of comparisons required to sort n elements. Thena)T(n) <= 2T(n/5) + nb)T(n) <= T(n/5) + T(4n/5) + nc)T(n) <= 2T(4n/5) + nd)T(n) <= 2T(n/2) + nCorrect answer is option 'B'. Can you explain this answer?
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Consider the Quicksort algorithm. Suppose there is a procedure for finding a pivot element which splits the list into two sub-lists each of which contains at least one-fifth of the elements. Let T(n) be the number of comparisons required to sort n elements. Thena)T(n) <= 2T(n/5) + nb)T(n) <= T(n/5) + T(4n/5) + nc)T(n) <= 2T(4n/5) + nd)T(n) <= 2T(n/2) + nCorrect answer is option 'B'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 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 Consider the Quicksort algorithm. Suppose there is a procedure for finding a pivot element which splits the list into two sub-lists each of which contains at least one-fifth of the elements. Let T(n) be the number of comparisons required to sort n elements. Thena)T(n) <= 2T(n/5) + nb)T(n) <= T(n/5) + T(4n/5) + nc)T(n) <= 2T(4n/5) + nd)T(n) <= 2T(n/2) + nCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the Quicksort algorithm. Suppose there is a procedure for finding a pivot element which splits the list into two sub-lists each of which contains at least one-fifth of the elements. Let T(n) be the number of comparisons required to sort n elements. Thena)T(n) <= 2T(n/5) + nb)T(n) <= T(n/5) + T(4n/5) + nc)T(n) <= 2T(4n/5) + nd)T(n) <= 2T(n/2) + nCorrect answer is option 'B'. Can you explain this answer?.
Consider the Quicksort algorithm. Suppose there is a procedure for finding a pivot element which splits the list into two sub-lists each of which contains at least one-fifth of the elements. Let T(n) be the number of comparisons required to sort n elements. Thena)T(n) <= 2T(n/5) + nb)T(n) <= T(n/5) + T(4n/5) + nc)T(n) <= 2T(4n/5) + nd)T(n) <= 2T(n/2) + nCorrect answer is option 'B'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 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 Consider the Quicksort algorithm. Suppose there is a procedure for finding a pivot element which splits the list into two sub-lists each of which contains at least one-fifth of the elements. Let T(n) be the number of comparisons required to sort n elements. Thena)T(n) <= 2T(n/5) + nb)T(n) <= T(n/5) + T(4n/5) + nc)T(n) <= 2T(4n/5) + nd)T(n) <= 2T(n/2) + nCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the Quicksort algorithm. Suppose there is a procedure for finding a pivot element which splits the list into two sub-lists each of which contains at least one-fifth of the elements. Let T(n) be the number of comparisons required to sort n elements. Thena)T(n) <= 2T(n/5) + nb)T(n) <= T(n/5) + T(4n/5) + nc)T(n) <= 2T(4n/5) + nd)T(n) <= 2T(n/2) + nCorrect answer is option 'B'. Can you explain this answer?.
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