Question

Describe how the average of a collection of numbers can be computed more rapidly with a multiprocessor machine than a single processor machine.

Short answer

A multiprocessor can compute the average faster by dividing and paralleling the sum calculation across processors.

Step-by-step solution

Introduction to Averaging

To find the average of a collection of numbers, you sum all the numbers and divide the sum by the count of the numbers. For a single processor, this operation is sequential: first finding the sum and then dividing it by the count.

Divide the Numbers

In a multiprocessor system, the collection of numbers can be divided into smaller subgroups, where each processor is responsible for computing the sum of its assigned subgroup.

Parallel Summation

Each processor independently computes the sum of the numbers in its subgroup simultaneously. This parallel computation significantly reduces the total time taken to compute the overall sum compared to a single processor working sequentially.

Aggregate Results

Once each processor has computed the sum of its subgroup, the results are combined to form the total sum of the collection. This aggregation step involves another processor (or the same ones) quickly adding these interim sums.

Compute the Average

Finally, a single processor can divide the total sum by the number of elements to find the average. Since the summation was performed in parallel, only this operation is done sequentially.

Key concepts

Multiprocessing

Multiprocessing leverages multiple processors to handle tasks concurrently, as opposed to performing them sequentially on a single processor. In the realm of computing, this approach is highly advantageous for processes that can be divided into smaller, independent sections.
This method allows multiple processing units to work simultaneously on different parts of a task, thereby accelerating the overall process. Imagine having several cooks in a kitchen, each preparing a part of a meal. By working together at the same time, they can complete the meal faster than a single cook.
In the context of averaging a collection of numbers, using multiprocessing means you split the task of summing numbers among multiple processors. Each processor completes its portion in parallel with the others, drastically cutting down the time required for this step. Once the partial sums are completed, they can be quickly totaled, allowing for efficient computation of the average.

Averaging Algorithm

An averaging algorithm involves determining the mean of a set of numbers, which is achieved by summing all the numbers and then dividing by the total count of these numbers.
Typically, with a single processor, these operations are performed sequentially: first summing and then dividing. However, when utilizing a multiprocessor system, this algorithm can be optimized for faster execution.
The collection of numbers is split into several subgroups, with each subgroup handled by a different processor. Each processor calculates the sum of its respective subgroup at the same time, thanks to the parallel nature of multiprocessor systems.

• Each processor performs the summation concurrently.
• The results are then combined from all subgroups, forming the overall sum.
• Finally, one processor does the simple task of dividing the total sum by the number of elements to produce the average.

This division and parallel processing result in a significant reduction in computation time.

Data Processing Efficiency

Data processing efficiency is crucial in any computational task, especially in big data scenarios or when speedy calculations are desired. Efficiency indicates how effectively a task uses resources – here, processing power and time.
By distributing the workload among several processors, as seen in parallel computing, data processing efficiency can be greatly enhanced. Multiprocessing not only minimizes the necessary time but also optimally utilizes each processor's capabilities.
In the example of averaging a set of numbers, dividing the task among multiple processors and performing summation in parallel yield a faster overall process.

• Each processor is fully used during its task, ensuring no processor remains idle.
• The reduced individual workloads lead to a quicker summarization of the data.
• Ultimately, time savings translate to efficiency, allowing other computational tasks to begin sooner.

Emphasizing data processing efficiency allows systems to provide faster results and handle larger datasets effectively.