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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...
Explanation:
The Quicksort algorithm is a divide and conquer algorithm that recursively divides a list into sub-lists and sorts them. The choice of the pivot element plays a crucial role in the efficiency of the algorithm.
Given:
- The pivot element splits the list into two sub-lists, each containing at least one-fifth of the elements.
- T(n) is the number of comparisons required to sort n elements.
Using the given information, we can analyze the time complexity of the algorithm:
1. Finding the pivot element:
- In the worst case, finding the pivot element takes O(n) time, as we need to compare each element with the pivot element.
- However, since the pivot element splits the list into two sub-lists, each containing at least one-fifth of the elements, the size of the sub-lists will be (4n/5) and (n/5).
- Therefore, finding the pivot element can be considered as a constant time operation, i.e., O(1).
2. Partitioning the list:
- Partitioning the list around the pivot element takes O(n) time, as we need to compare each element with the pivot element and move them to the appropriate side of the pivot.
- The partitioning step divides the list into two sub-lists, each containing at least one-fifth of the elements.
3. Recursive calls:
- After partitioning, we recursively call the Quicksort algorithm on the two sub-lists.
- The size of the first sub-list is (4n/5) and the size of the second sub-list is (n/5).
- Therefore, the time complexity of the Quicksort algorithm can be expressed as a recurrence relation:
T(n) = T(4n/5) + T(n/5) + O(n)
Using the recurrence relation, we can solve for T(n):
- T(n) = T(4n/5) + T(n/5) + O(n)
- T(n) = T(n/5) + O(n) + T(4n/5) [rearranging terms]
- T(n) = T(n/5) + T(4n/5) + O(n) [rearranging terms]
- T(n) = T(n/5) + T(4n/5) + O(n) [rearranging terms]
Comparing the above equation with the given options:
- Option a) T(n) = 2T(n/5) n
- This option does not match the given recurrence relation. Therefore, it is incorrect.
- Option b) T(n) = T(n/5) T(4n/5)
- This option matches the given recurrence relation. Therefore, it is correct.
- Option c) T(n) = 2T(4n/5)
- This option does not match the given recurrence relation. Therefore, it is incorrect.
- Option d) T(n) = 2T(n/2) n
- This option does not match the given recurrence relation. Therefore, it is incorrect.
Hence, the correct answer is option b) T(n) = T(n/5) T(
The Quicksort algorithm is a divide and conquer algorithm that recursively divides a list into sub-lists and sorts them. The choice of the pivot element plays a crucial role in the efficiency of the algorithm.
Given:
- The pivot element splits the list into two sub-lists, each containing at least one-fifth of the elements.
- T(n) is the number of comparisons required to sort n elements.
Using the given information, we can analyze the time complexity of the algorithm:
1. Finding the pivot element:
- In the worst case, finding the pivot element takes O(n) time, as we need to compare each element with the pivot element.
- However, since the pivot element splits the list into two sub-lists, each containing at least one-fifth of the elements, the size of the sub-lists will be (4n/5) and (n/5).
- Therefore, finding the pivot element can be considered as a constant time operation, i.e., O(1).
2. Partitioning the list:
- Partitioning the list around the pivot element takes O(n) time, as we need to compare each element with the pivot element and move them to the appropriate side of the pivot.
- The partitioning step divides the list into two sub-lists, each containing at least one-fifth of the elements.
3. Recursive calls:
- After partitioning, we recursively call the Quicksort algorithm on the two sub-lists.
- The size of the first sub-list is (4n/5) and the size of the second sub-list is (n/5).
- Therefore, the time complexity of the Quicksort algorithm can be expressed as a recurrence relation:
T(n) = T(4n/5) + T(n/5) + O(n)
Using the recurrence relation, we can solve for T(n):
- T(n) = T(4n/5) + T(n/5) + O(n)
- T(n) = T(n/5) + O(n) + T(4n/5) [rearranging terms]
- T(n) = T(n/5) + T(4n/5) + O(n) [rearranging terms]
- T(n) = T(n/5) + T(4n/5) + O(n) [rearranging terms]
Comparing the above equation with the given options:
- Option a) T(n) = 2T(n/5) n
- This option does not match the given recurrence relation. Therefore, it is incorrect.
- Option b) T(n) = T(n/5) T(4n/5)
- This option matches the given recurrence relation. Therefore, it is correct.
- Option c) T(n) = 2T(4n/5)
- This option does not match the given recurrence relation. Therefore, it is incorrect.
- Option d) T(n) = 2T(n/2) n
- This option does not match the given recurrence relation. Therefore, it is incorrect.
Hence, the correct answer is option b) T(n) = T(n/5) T(
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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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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?, a detailed solution 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? has been provided alongside types of 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? theory, EduRev gives you an
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