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Chapter 11: Problem 10

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

Short Answer

Expert verified
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

01

THE ORIGINAL ARRAY

Let the given array be an array of \(n\) elements.
02

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.
03

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.
04

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.
05

GETTING THE FINAL SORTED ARRAY

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

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Most popular questions from this chapter

If we assume that one person can add two numbers in \(t_{a}\) time, how long will it take that person to add two \(n \times n\) matrices, if we consider the operation of addition as the basic operation? Justify your answer.

Write a CREW PRAM algorithm for determining for each element of an \(n\) -element linked list if it is the middle ( \([n / 2]\) th ) element in \(\theta\) ( \(\lg n\) ) time.

Write a PRAM algorithm that runs in \(\theta(\lg n)_{2}\) ) time for the problem of mergesorting. (Hint: Use \(n\) processors, and assign each processor to a key to determine the position of the key in the final list by binary searching.)

Consider the problem of adding \(n\) numbers in a list. If it takes \(t_{a}\) time for one person to add two numbers and it takes no time to pass the result of an addition from one person to another, how many people do we need to minimize the total time spent to get the final answer? What will be the minimum amount of time needed to find the answer, if we assume we have enough people? Justify your answer.

Write a sequential algorithm that implements the Tournament method to find the largest key in an array of \(n\) keys. Show that this algorithm is no more efficient than the standard sequential algorithm.
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