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Chapter 10: Problem 16

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

Expert verified
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

01

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.
02

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.
03

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.

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Most popular questions from this chapter

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.

Write a PRAM algorithm for Quicksort using \(n\) processors to sort a list of \(n\) elements.

Consider the problem of adding \(n\) numbers of a list. If it takes \(t_{a}\) time for one person to add two numbers, and it takes no time to pass the result of an addition from one person to another, how many people do we need to minimize the total time spent to get the final answer? What will be the minimum amount of time needed to find the answer if we have enough people? Justify your answer.

Write a PRAM algorithm using \(n^{3}\) processors to multiply two \(n \times n\) matrices. Your algorithm should run in \(\Theta(\lg n)\) time.

Consider the problem of adding two \(n \times n\) matrices. If it takes \(t_{\alpha}\) time for one person to add two numbers, how many people do we need to minimize the total time spent to get the final answer? What will be the minimum amount of time needed to find the answer if we have enough people? Justify your answers.
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