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Chapter 11: Problem 8

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.

Short Answer

Expert verified
The CREW PRAM algorithm repeatedly uses binary search for finding the middle element of a linked list. By dividing the space in half on each iteration, it achieves a log time complexity. The Concurrent Read, Exclusive Write (CREW) model is applied at each step.

Step by step solution

01

Initialize Variables and Structure

Initialize \( n \), the size of the linked list and \( mid \) variable to store the middle element. Set an array to represent the linked list, with indexes representing the processors.
02

Determine the Limits of the Search Space

Set the limits as \( left = 0 \) and \( right = n - 1 \).
03

Binary Search

Perform binary search to find the middle element. The algorithm would repeatedly divide the list into two halves and operate on the half that would contain the middle element.
04

Algorithm Iteration

On each iteration, calculate the middle index as \( mid = left + (right - left)/2 \). If \( mid \) is equal to \( [n/2] \), then it is the middle element, save it in variable \( mid \) and stop. Otherwise, if \( mid < [n/2] \), search in the right half by setting \( left = mid + 1 \). If \( mid > [n/2] \), search in the left half by setting \( right = mid - 1 \). Iterate until the middle element is found.
05

Concurrent Read, Exclusive Write (CREW)

All processors can concurrently read memory without collision (Concurrent Read) but only one processor can write to a memory location (Exclusive Write). Use this model for each step of the binary search algorithm.

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Most popular questions from this chapter

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 proble of adding two \(n \times n\) matrices. If it takes \(t_{a}\) time for one person to add two numbers, 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 that we have enough people? Justify your answers.

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.

If we have two people add \(n\) numbers in a list and it takes \(t_{a}\) time for one person to add two numbers, how long will it take the two people to add all \(n\) numbers in the list, if we consider the operation of addition as the basic operation and include \(t_{p}\) time for passing the result of an addition from one person to the other? Justify your answer.

Consider the problem of adding the numbers in a list of \(n\) numbers. If it takes \(t_{a}(n-1)\) time for one person to add all \(n\) numbers, is it possible for \(m\) people to compute the sum in less than \(\left[t_{a}(n\right.\) -1)]\(/ m\) time? Justify your answer.
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