Question

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.

Short answer

No, it is not possible for \(m\) people to compute the sum in less than \(\left[t_{e}(n-1)\right] / m\) time because addition is a sequential operation and cannot be effectively sped up by parallel processing.

Step-by-step solution

Understand the Concept of Sequential and Parallel Operations

Certain operations can be carried out in parallel, meaning they can be broken down into independent parts that can be executed simultaneously. Conversely, some operations are essentially sequential and cannot be speeded up by parallelizing. Addition of numbers is sequential in nature because you need the result of the previous operation (addition) to perform the next.

Analyze the Given Condition

According to the problem, it takes \(t_{d}(n-1)\) time for one person to add all \(n\) numbers. The question to consider is whether \(m\) people can compute the sum in less than \(\left[t_{e}(n-1)\right] / m\) time. For \(m\) people to add the \(n\) numbers more quickly than a single person, the addition task would need to be parallelizable. As established in Step 1, however, addition is inherently sequential in nature.

Formulate the Conclusion

Since addition is a sequential operation, distributing the work over \(m\) people does not speed up the process. Each new calculation must wait for the previous one to finish. Therefore, the time required to compute the sum would stay the same, even with multiple people working on it. Thus, it would not be possible for \(m\) people to compute the sum in less than \(\left[t_{e}(n-1)\right] / m\) time.

Key concepts

Sequential Operations

Sequential operations refer to tasks that must be completed in a specific order, one after another. A real-world example of this is adding numbers in a list. When adding numbers, the result of one addition is needed before the next can proceed. This characteristic makes the operation inherently sequential. Imagine trying to bake a cake. You can't add the frosting before the cake has baked. Similarly, when adding numbers, doing so out of order disrupts the process. It’s like needing the sum of the previous number before moving to the next. This sequential nature means that using multiple people or processors will not speed up the task because each must wait for the previous step to finish. So, even if you could divide the list and have each person add their part, you would still need to combine the results, which again would require sequential processing.

Parallel Operations

Parallel operations, on the other hand, involve tasks that can be broken down into discrete units, each capable of being executed simultaneously. This means different tasks or parts of a task can be managed independently, often resulting in time savings when using multiple processors or workers. Think about painting a wall with several people, each assigned to different sections. Here, each person paints independently, allowing the job to be completed faster than if just one person were to do it alone. In computing, tasks like sorting an array or processing individual elements in isolation can often benefit from parallel operations – each segment is sorted or computed separately and simultaneously. Parallelism shines when the tasks don't depend on the completion of the other, which is why it isn't applicable for the simple addition of numbers unless you change the problem paradigm.

Speedup and Performance

Speedup and performance refer to how much faster and efficient a task can be completed using multiple processors compared to just one. The main idea is that by executing tasks in parallel, time can be reduced significantly, leading to better performance. The concept of speedup is often defined as the ratio of time taken with one processor to time taken with multiple processors. However, due to the inherent dependency in sequential tasks, perfect or linear speedup is often not achievable. For instance, in the problem of adding numbers, regardless of the number of people or processors, each step waits on the result of the previous. Performance gains may be minimal, as efficiency is bound by the sequential dependencies. Understanding these concepts is crucial, especially when evaluating the feasibility of parallelizing a task. If dependencies are too strong, as in addition, parallel execution may offer little advantage.