Question

Suppose you are given 32 processors, each capable of finding the sum of two multidigit numbers in a millionth of a second. Describe how parallel processing techniques can be applied to find the sum of 64 numbers in only sixmillionths of a second. How much time does a single processor require to find this same sum?

Short answer

Parallel process takes 6 millionths of a second; a single processor takes 63 millionths.

Step-by-step solution

Understanding the Problem

We need to find the sum of 64 numbers using 32 processors, where each processor can compute the sum of two numbers in one-millionth of a second.

Initial Division of Numbers

First, divide the 64 numbers into 32 pairs. Each of the 32 processors will add one pair of numbers, taking one-millionth of a second.

First Aggregation Step

Now, we have 32 sums from the previous step. Pair these 32 sums into 16 pairs and use 16 processors to find their sums, which takes another millionth of a second.

Second Aggregation Step

Next, take the resulting 16 sums and pair them again into 8 pairs. Use 8 processors to add these pairs, needing another millionth of a second.

Third Aggregation Step

Then, take the 8 sums and form 4 pairs. Use 4 processors to add these pairs, taking yet another millionth of a second.

Fourth Aggregation Step

We now have 4 results. Pair them with 2 processors to form 2 sums, requiring one more millionth of a second.

Final Aggregation Step

Finally, add the last two sums together using 1 processor, taking one final millionth of a second.

Calculating the Total Time for Parallel Processing

The total time taken by parallel processing is the sum of each step's time: 1 + 1 + 1 + 1 + 1 + 1 = 6 millionths of a second.

Single Processor Calculation

A single processor would need to add each pair sequentially: with 63 sums required for 64 numbers, this would take 63 millionths of a second.

Key concepts

Processor Speed

Processor speed plays a crucial role in how quickly calculations can be completed. In our example, each processor can compute the sum of two numbers in one-millionth of a second. This incredible speed allows for rapid progression through calculations when using multiple processors together. Understanding processor speed starts with recognizing that it's about how many tasks a processor can perform in a given amount of time. In this case, each processor performs a task—adding two numbers—extremely quickly at 1 microsecond per operation. This high speed is what makes the concept of parallel processing so powerful when dealing with large computations.

Aggregation Steps

Aggregation steps are the keys in parallel processing that enable us to reduce a large set of numbers into a single sum efficiently. In the given exercise, we begin with 64 numbers. The 64 numbers are first broken into 32 pairs. Each of these pairs is summed by a separate processor.

This process is repeated through successive aggregation steps:

• From 32 sums (each from the first aggregation step), form 16 pairs and compute sums.
• From 16 new sums, form 8 pairs and compute sums again.
• Next, form 4 pairs from 8 sums and compute their sums.
• Pair the resulting 4 sums into 2 pairs and compute their sums.
• Finally, add the last two sums for the overall total sum.

Each aggregation step, therefore, systematically reduces the number of sums until only one remains. Each of these steps takes one-millionth of a second due to the speed of each processor.

Computational Efficiency

Computational efficiency in parallel processing is demonstrated by how quickly we can arrive at a solution compared to sequential processing. Parallel processing divides the task into smaller chunks that can be solved simultaneously. This not only saves time but also optimizes the computational resources by utilizing multiple processors concurrently. By taking advantage of parallel processing, computational efficiency is increased tremendously as each operation is executed in one-millionth of a second and all operations progress simultaneously. This combination of speed and concurrency helps us reach the final solution with significantly reduced time.

Single Processor Calculation

When using a single processor for such tasks, the approach is significantly different. The single processor needs to complete each addition one after the other, without the advantage of parallel steps. In the original problem, if a single processor were used, it would perform all 63 additive operations sequentially. To sum 64 numbers, we calculate 63 sums, as each sum reduces the total number of numbers by one, leading to just one final number—or total sum. Each operation—an addition taking a millionth of a second—would thus extend the required time to 63 millionths of a second. This showcases the vast difference in processing time between using a single processor and parallel processors, emphasizing how efficient parallel processing can transform computational tasks.