Question

Write a PRAM algorithm for Quicksort using \(n\) processors to sort a list of \(n\) elements.

Short answer

A PRAM algorithm for Quicksort is implemented by firstly selecting a pivot and partitioning the array into lists of elements greater and less than or equal to pivot. Then recursively apply Quicksort to the separate lists, using \(n/2\) processors for each half. The sorted array is obtained by combining the sorted smaller elements list, pivot, and sorted larger elements list.

Step-by-step solution

THE ORIGINAL ARRAY

Let the given array be an array of \(n\) elements.

CHOOSING PIVOT

Firstly, a pivot element is selected. Usually, the rightmost or the leftmost element is chosen as the pivot. For simplicity, let's assume that the rightmost element is chosen as the pivot.

PARTITION PROCESS

All processors are used in the parallel partition step. Each processor \(i\) compares array element \(arr[i]\) with the pivot. Two separate lists are created - one for elements less than or equal to the pivot and another for elements greater than the pivot.

RECURSIVE SORT

Now, the PRAM algorithm recursively applies Quicksort to two separate lists (partitions) - using \(n/2\) processors for each half thereby using total of \(n\) processors.

GETTING THE FINAL SORTED ARRAY

The smaller elements list + pivot + larger elements list gives the sorted array.

Key concepts

Parallel Computing

Parallel computing involves performing multiple calculations simultaneously. It's a way to push beyond the limitations of single-processor systems by distributing tasks across many processors in a computational model. This is particularly useful when dealing with large data sets, such as sorting a long list of numbers. In the context of the PRAM (Parallel Random Access Machine) Quicksort algorithm, parallel computing allows each processor to take an active role in sorting a list. This technique greatly reduces the time needed to sort as multiple elements are compared and sorted in parallel. By leveraging multiple processors, problems can be broken down into smaller, independent tasks that can be solved simultaneously.

Algorithm Design

Designing an algorithm such as PRAM Quicksort requires a strategic approach. Algorithm design is not just about finding the solution but understanding the process that leads there. Key elements include identifying the tasks that can be executed in parallel, optimizing the steps, and ensuring that the algorithm is efficient. For the PRAM Quicksort, this means breaking the problem into parts that can be handled by separate processors. Choosing a pivot is critical, as it sets the stage for how the array will be divided and conquered. After dividing the list, the algorithm recursively sorts the smaller partitions, using intelligent resource allocation. These design choices are aimed at minimizing the time complexity and maximizing performance.

Sorting Algorithms

Sorting algorithms are methods used to arrange data in a particular order, such as ascending or descending. They are fundamental in computer science due to their wide range of applications. QuickSort is one of these key algorithms because of its efficiency and speed. Typically, sorting involves repetitive comparisons and swaps. However, when sorting with a PRAM algorithm, these tasks are distributed among processors. By executing simultaneous comparisons and movements of data, the sorting process becomes much faster. PRAM Quicksort capitalizes on the strengths of QuickSort while introducing parallelism to leverage the power of multiple processors, providing a scalable solution to sorting large datasets effectively.

Recursion

Recursion is a fundamental programming concept where a function calls itself to solve smaller sub-problems. In the context of the PRAM Quicksort algorithm, recursion is employed to divide the problem into sub-problems, which involve sorting parts of the array. By choosing a pivot and partitioning the array, each smaller array is further broken down with recursive calls. This leads to a situation where different processors handle different levels of the recursion, enabling efficient parallel processing. Recursion continues until the sub-arrays have reached a base case, typically when their length is one or zero, signaling an already sorted state. Using recursion, complex problems are handled in a structured, easily manageable way.