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Chapter 10: Problem 8

Write a CREW PRAM algorithm for determining for each element of an \(n\) -clement linked list if it is the middle \(\left(\int n / 2 \text { th }\right)\) element in \(\Theta(\lg n)\) time.

Short Answer

Expert verified
Divide and conquer technique is used in the algorithm leveraging CREW PRAM model's simultaneous read capability, enabling the processors to concurrently check their identifier with the current list length divided by 2. This approach halving the list's length each iteration, achieving \( \Theta(\lg n) \) time complexity.

Step by step solution

01

Title

Define the problem: we need to design an algorithm using a PRAM model to find the middle element in a linked list. The time complexity should be \( \Theta(\lg n) \).
02

Title

Initiate the processors: Each processor will need the position of its linked list element, with the first processor starting at position 1. Divide the work among all processes equally, giving each process a unique identifier within the range 1 to \( n \).
03

Title

Navigate to the middle element: Since processes read concurrently in a CREW PRAM model, have each processor check if its identifier equals \( \frac{n}{2} \). If this is the case, the processor has found the location of the middle element and can return it.
04

Title

Implement the time complexity constraint: To ensure the algorithm runs in \( \Theta(\lg n) \) time, divide the linked list into halves recursively. With each iteration, the processor checks if their identifier is equal to half the length of the new list. Terminate the recursion when the list length becomes 2, at this point the middle element is found.

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Most popular questions from this chapter

Write a CREW PRAM algorithm for adding all \(n\) numbers of a list in \(\Theta(\lg n)\) time.

Write a CRCW PRAM algorithm that uses \(n^{2}\) processors to multiply two \(n \times n\) matrices. Your algorithm should perform better than the standard \(\Theta\left(n^{3}\right)\) -time serial algorithm.

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

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

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