Question

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

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

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.

Determine the Limits of the Search Space

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

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.

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.

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.

Key concepts

Understanding Linked Lists

A linked list is a fundamental data structure used to store a collection of elements. Each element, or 'node', contains two parts: the data itself and a reference (or pointer) to the next node in the sequence. This structure is particularly useful because:

• It allows for efficient insertion and deletion of elements.
• The size of the list can dynamically increase or decrease.

However, linked lists do not support direct access to elements based on their index, unlike arrays. This means that to reach the middle element, you need to traverse the list step-by-step, starting from the head node. This traversal property makes linked lists both powerful and a bit tricky to work with in certain algorithms, such as the CREW PRAM method for parallel computation.

Exploring Binary Search

Binary search is an efficient algorithm for finding an element within a sorted array. It works by repeatedly dividing the search interval in half. Binary search can be utilized here despite linked lists usually not providing direct indexing:

• At each step, determine the middle element of the current range.
• Check if this element matches the target criteria.
• If it doesn't match, decide which half of the list to continue searching in.

The algorithm trims the search space logarithmically, making it highly efficient, with a time complexity of \(\theta(\lg n)\). When adapted to parallel computing, binary search helps numerous processors simultaneously work towards finding the middle element.

Introduction to Parallel Computing

Parallel computing involves splitting a problem into sub-tasks that can be processed concurrently. The CREW PRAM (Concurrent Read, Exclusive Write Parallel Random Access Machine) model is particularly important in this context:

• Multiple processors can read from the same memory location simultaneously (Concurrent Read).
• Exclusive Write ensures that only one processor can write to a memory location at a time.

In our algorithm, parallel computing allows several processors to simultaneously examine different parts of the list, speeding up the search process for the middle element. This approach not only enhances efficiency but also maximizes resource utilization.

Measuring Algorithm Efficiency

Algorithm efficiency often refers to how well an algorithm performs in terms of time and space complexity. The CREW PRAM algorithm for managing linked lists is efficient due to:

• Its use of binary search, which reduces time complexity to \(\theta(\lg n)\).
• The parallel processing model, which minimizes the computation time by allowing multiple operations simultaneously.

Efficiency is crucial in high-performance computing, particularly when dealing with large data sets. By optimizing both the algorithmic approach and leveraging parallel computing, the goal is to achieve significant reductions in execution time while maintaining accuracy and reliability.