Question

Describe a randomized version of in-place quick-sort in pseudo-code.

Short answer

Randomized in-place quick-sort uses a random pivot, partitions the array around it, and recursively sorts the sub-arrays.

Step-by-step solution

Understand Quick-Sort

Quick-Sort is a divide-and-conquer algorithm that sorts an array by selecting a 'pivot' element and partitioning the array around the pivot. Elements smaller than the pivot end up to its left, and elements larger to its right. Then, the algorithm recursively sorts the sub-arrays.

Randomized Pivot Selection

In a randomized version of Quick-Sort, the pivot is chosen randomly from the array. This helps in avoiding the worst-case performance that occurs when the pivot is always poorly chosen.

Define the Partition Function

The partition function rearranges the elements of the array such that elements less than the pivot are on the left, and those greater than or equal are on the right. It returns the index of the pivot after partition.

Write the Pseudo-Code

Combine the logic into the following randomized in-place Quick-Sort pseudo-code:```function randomizedQuickSort(A, low, high): if low < high: // Randomly select a pivot index pivotIndex = randomPivot(low, high) // Swap pivot with the end element swap(A[pivotIndex], A[high]) // Perform partition and get pivot index pivotNewIndex = partition(A, low, high) // Recursively sort elements before and after partition randomizedQuickSort(A, low, pivotNewIndex - 1) randomizedQuickSort(A, pivotNewIndex + 1, high)function randomPivot(low, high): return random number between low and highfunction partition(A, low, high): pivot = A[high] i = low - 1 for j = low to high - 1: if A[j] < pivot: i = i + 1 swap(A[i], A[j]) swap(A[i + 1], A[high]) return i + 1```The `randomPivot` function generates a random index between `low` and `high`.

Key concepts

In-place Quick-Sort

The term 'in-place' quick-sort refers to how the algorithm operates directly within the input array without requiring additional storage. This is highly efficient because it uses only a constant amount of extra space. The algorithm rearranges elements by swapping them within the same array.

Here are the steps involved in in-place quick-sort:

• Choose a pivot element.
• Partition the array around the pivot.
• Recursively apply the same steps to the sub-arrays.

Using this method minimizes space complexity and is generally faster on average compared to other sorting algorithms.

Divide-and-Conquer

Quick-Sort uses a divide-and-conquer approach to sorting. This means it breaks down the problem into smaller, more manageable sub-problems, solves each one individually, and combines the results.

The main steps in the divide-and-conquer strategy for quick-sort are:

• Dividing: The array is divided into two sub-arrays based on a pivot element.
• Conquering: Each sub-array is sorted independently using the same algorithm.
• Combining: Since each sub-array is already sorted, no additional work is required for combining.

This strategy ensures that each step reduces the complexity of the problem, making quick-sort both efficient and effective.

Pivot Selection

Pivot selection is a key part of the quick-sort algorithm. The pivot is the element around which the rest of the array is partitioned.

In the randomized version of quick-sort, we choose the pivot element randomly rather than using a fixed position. This helps in avoiding the worst-case scenarios, which can happen if the pivot is always the largest or smallest element.

Here's how to select a pivot randomly:

• Select a random index between the starting and ending indices of the array segment.
• Swap the pivot element with the last element of the segment to facilitate partitioning.

This random selection minimizes the chances of encountering the worst-case time complexity, thus making the sort more reliable.

Partition Function

The partition function is responsible for rearranging the elements in the array such that all elements less than the pivot are on its left, and all elements greater than or equal are on its right.

Here’s a step-by-step explanation of how the partition function works:

• Choose the last element in the segment as the pivot (after swapping with the randomly selected pivot).
• Initialize an index to track the position of smaller elements.
• Traverse the array and, for each element smaller than the pivot, increase the tracker index and swap the element with the element at the tracker index.
• After the traversal, swap the pivot element back to its proper position by placing it right after the last smaller element.

This operation ensures that, for the selected pivot, all elements to its left are smaller and those to its right are larger, thereby effectively partitioning the array.