Randomized Quicksort

Preview

1. What is randomized Quicksort and how does it differ from the traditional implementation of Quicksort?

2. How can the partitioning step of randomized Quicksort be implemented to ensure that the running time is O(n log n) with high probability?

3. What is the main advantage of randomized Quicksort over other sorting algorithms?

4. How does the choice of pivot element affect the performance of randomized Quicksort?

5. Can randomized Quicksort be modified to handle duplicate elements in the input array without sacrificing its running time complexity?

Explain

Randomized Quicksort is a sorting algorithm that partitions an array by picking a random pivot element and sorting the smaller elements on the left and the larger elements on the right. It is a randomized version of the Quicksort algorithm, which picks the first or the last element as the pivot.

Steps of Randomized Quicksort Algorithm:

1. Select a random element from the array as the pivot.

2. Partition the array around the pivot element by comparing every other element to it. All the smaller elements go to the left of the pivot and all the larger elements go to the right.

3. Recursively apply the algorithm to the left and the right partition.

4. The base case for the recursion is when there is only one element in the subarray.

Example:

Suppose we have an array [7, 2, 8, 1, 4, 6, 9, 3, 5]. We can apply the Randomized Quicksort algorithm to sort the array as follows:

1. Pick a random pivot. Let’s say the pivot is 6.

2. Partition the array around the pivot: [2, 1, 4, 3, 5], [7, 8, 9]

3. Recursively apply the algorithm to the left and the right partition.

• On the left partition, pick a new pivot randomly. Let’s say the pivot is 4.

• Partition the left partition: [2, 1, 3], [4, 5]

• Recursively apply the algorithm to the left subpartition and the right subpartition.

• Continue this process until the base case is reached.

• On the right partition, pick a new pivot randomly. Let’s say the pivot is 8.

• Partition the right partition: [7], [8, 9]

• Recursively apply the algorithm to the left subpartition and the right subpartition.

• Continue this process until the base case is reached.

4. Finally, combine the sorted subarrays to get the sorted array. [1, 2, 3, 4, 5, 6, 7, 8, 9]

Keypoint

1. Randomized Quicksort is an efficient sorting algorithm that uses the divide-and-conquer strategy.
2. It selects a pivot element randomly and partitions the array into two sub-arrays.
3. One sub-array contains elements that are less than or equal to the pivot, while the other sub-array contains elements greater than the pivot.
4. The algorithm recursively sorts the sub-arrays using the same process.
5. The randomized selection of the pivot element ensures that the probability of worst-case performance is reduced.
6. The time complexity of Randomized Quicksort is O(n log n) on average, and O(n^2) in the worst case.
7. Randomized Quicksort is widely used in practice due to its average-case performance and easy implementation.
8. It is also a popular choice for sorting large data sets or in applications where sorting needs to be done quickly.

Review

1. What is the time complexity of Randomized Quicksort in the worst case?
   Answer: The time complexity of Randomized Quicksort in the worst case is O(n^2), where n is the size of the input array.

2. How does Randomized Quicksort differ from regular Quicksort?
   Answer: Randomized Quicksort differs from regular Quicksort in that it randomly selects a pivot element, rather than always choosing the first or last element in the array.

3. Is Randomized Quicksort a stable sorting algorithm?
   Answer: No, Randomized Quicksort is not a stable sorting algorithm. This means that the relative order of elements with equal values may change during the sorting process.

4. What is the average case time complexity of Randomized Quicksort?
   Answer: The average case time complexity of Randomized Quicksort is O(n log n), where n is the size of the input array.

5. How can you implement Randomized Quicksort to efficiently handle duplicate elements in the input array?
   Answer: One way to implement Randomized Quicksort to handle duplicate elements is to partition the array into three parts: elements smaller than the pivot, elements equal to the pivot, and elements greater than the pivot. This can be done recursively on the smaller and greater partitions, while leaving the “equal” partition unchanged.