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Suppose we have a O(n) time algorithm that finds median of an unsorted array. Now consider a QuickSort implementation where we first find median using the above algorithm, then use median as pivot. What will be the worst case time complexity of this modified QuickSort.
- a)O(n^2 Logn)
- b)O(n^2)
- c)O(n Logn Logn)
- d)O(nLogn)
Correct answer is option 'D'. Can you explain this answer?
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Suppose we have a O(n) time algorithm that finds median of an unsorted...
If we use median as a pivot element, then the recurrence for all cases becomes T(n) = 2T(n/2) + O(n) The above recurrence can be solved using Master Method. It falls in case 2 of master method.
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Suppose we have a O(n) time algorithm that finds median of an unsorted...
Worst case time complexity of the modified QuickSort:
Explanation:
In the modified QuickSort algorithm, we first find the median of the unsorted array using the O(n) time algorithm. Then we use this median as the pivot for partitioning the array in the QuickSort algorithm.
Step 1: Finding the Median
Finding the median of an unsorted array takes O(n) time. This is because we can use the O(n) time algorithm to find the median in linear time. So, the first step of finding the median has a time complexity of O(n).
Step 2: Partitioning the Array
In the QuickSort algorithm, we partition the array around the pivot element. In the worst case scenario, if the pivot is always chosen as the median, the array will be evenly split into two halves. This means that each partitioning step will divide the array into two subarrays of roughly equal size. The worst case time complexity for partitioning is O(n) because we need to compare each element with the pivot and move them to the left or right subarray accordingly.
Step 3: Recursion
After partitioning the array, we recursively apply the QuickSort algorithm on the two subarrays. Since each partitioning step divides the array into two roughly equal halves, the size of the subarrays will be halved at each recursive call.
Time Complexity Analysis:
Let's denote the length of the array as n.
- In the first step, finding the median takes O(n) time.
- In the second step, partitioning the array around the median pivot takes O(n) time.
- In the third step, we recursively apply the QuickSort algorithm on two subarrays of size n/2.
The recurrence relation for the worst case time complexity of the modified QuickSort algorithm can be written as:
T(n) = T(n/2) + O(n)
Using the Master Theorem, we can solve this recurrence relation. In this case, the recurrence relation falls under Case 2 of the Master Theorem, which states that if the partitioning step divides the problem into two subproblems of size n/2, and the time complexity of the partitioning step is O(n), then the time complexity of the algorithm is O(n log n).
Therefore, the worst case time complexity of the modified QuickSort algorithm is O(n log n), which corresponds to option (d).
Explanation:
In the modified QuickSort algorithm, we first find the median of the unsorted array using the O(n) time algorithm. Then we use this median as the pivot for partitioning the array in the QuickSort algorithm.
Step 1: Finding the Median
Finding the median of an unsorted array takes O(n) time. This is because we can use the O(n) time algorithm to find the median in linear time. So, the first step of finding the median has a time complexity of O(n).
Step 2: Partitioning the Array
In the QuickSort algorithm, we partition the array around the pivot element. In the worst case scenario, if the pivot is always chosen as the median, the array will be evenly split into two halves. This means that each partitioning step will divide the array into two subarrays of roughly equal size. The worst case time complexity for partitioning is O(n) because we need to compare each element with the pivot and move them to the left or right subarray accordingly.
Step 3: Recursion
After partitioning the array, we recursively apply the QuickSort algorithm on the two subarrays. Since each partitioning step divides the array into two roughly equal halves, the size of the subarrays will be halved at each recursive call.
Time Complexity Analysis:
Let's denote the length of the array as n.
- In the first step, finding the median takes O(n) time.
- In the second step, partitioning the array around the median pivot takes O(n) time.
- In the third step, we recursively apply the QuickSort algorithm on two subarrays of size n/2.
The recurrence relation for the worst case time complexity of the modified QuickSort algorithm can be written as:
T(n) = T(n/2) + O(n)
Using the Master Theorem, we can solve this recurrence relation. In this case, the recurrence relation falls under Case 2 of the Master Theorem, which states that if the partitioning step divides the problem into two subproblems of size n/2, and the time complexity of the partitioning step is O(n), then the time complexity of the algorithm is O(n log n).
Therefore, the worst case time complexity of the modified QuickSort algorithm is O(n log n), which corresponds to option (d).
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