Question

Consider the problem of adding $n$ numbers in 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 assume we have enough people? Justify your answer.

Short answer

To minimize the total time spent to get the final answer, we would need $n-1$ people with the minimum amount of time needed to find the answer being $\lceil {\log }_2(n)\rceil \ast t_a$.

Step-by-step solution

Analyze the Process

If $n$ numbers are added by a single person, it would require $(n-1)\ast t_a$ amount of time because it would require $n-1$ steps to add every number to one another.

Consider Parallel Execution

If we consider the maximum parallelism, every addition operation could be done at the same time. One might imagine passing pairs of numbers to different people, each individual would then calculate the sum of the pair at the same time. With this approach, processing time would reduce to $t_a$, but it would require $\frac{n}{2}$ people (assuming $n$ is even). However, this is not the end. After adding pairs of numbers, we will get $\frac{n}{2}$ new numbers that need to be added together. This process will repeat until we get the final result.

Calculate the Number of People Needed

For step 2, we can justify that we would need $n-1$ people: one person for each addition operation.

Calculate the Minimum Time Needed to Find the Answer

In terms of time, the process is halved at each step until only one number remains. Therefore, the total number of steps would require $\lceil {\log }_2(n)\rceil$ steps, thus, the total time is $\lceil {\log }_2(n)\rceil \ast t_a$.

Key concepts

Algorithm Efficiency

Algorithm efficiency is a crucial concept in computer science. It refers to how effectively a computational problem is solved using a specific algorithm. For our exercise, the goal is to add a list of numbers efficiently. When only one person is involved, the task is straightforward: you would add the numbers one after the other. This means you need to perform $n-1$ additions, where $n$ is the number of terms in the list. Since each addition takes $t_a$ time, the total time for a single person would be $(n-1)\ast t_a$.

However, when multiple people work in parallel, the efficiency improves drastically. Imagine if several people could each handle parts of the problem and work simultaneously. This method reduces the sequential nature and makes use of parallel algorithms. By leveraging parallel computing, we can minimize the time taken to solve the issue significantly.

Computational Complexity

Computational complexity deals with understanding the performance and resource usage of an algorithm.For the exercise, we explored the complexity in terms of time as we try to add numbers with minimal duration. When looking at computational complexity, we consider two key points:

• How many steps or operations are needed.
• The resources (in this case, people) required to complete these operations simultaneously.

To understand the time complexity for parallel execution, you can imagine dividing the numbers among groups of people who perform the required additions concurrently. Initially, $\frac{n}{2}$ people will add pairs of numbers, reducing the list's size by half. This collective process repeats, halving the number of results each time until just one final sum remains.

This is why, though the task requires multiple people, the computational process significantly narrows down. In terms of complexity, the solution transitions from a linear approach to a logarithmic one, indicated by $\lceil {\log }_2(n)\rceil$. This means the number of addition steps becomes proportional to the logarithm (base 2) of $n$, expressed as $\ast t_a$ for the time taken per step.

Divide and Conquer

Divide and conquer is a powerful strategy used in computer algorithms. It involves breaking down a large problem into smaller, more manageable parts. For adding a list of numbers, this approach allows us to divide the list into pairs, sum each pair, and repeat the process on the resulting sums until only one final sum remains.

To apply this in our problem, you would:

• Divide: Split the list of numbers into pairs.
• Conquer: Sum the individual pairs in parallel.
• Combine: Once the pairs are processed, further combine the results by repeating the divide and conquer process on the new list.

This strategy efficiently minimizes the computation time needed by reducing the problem size at each step, leveraging the simultaneous calculation ability of parallel processing. Therefore, it highlights why only $\lceil {\log }_2(n)\rceil$ time steps are necessary rather than $n-1$ as in a sequential approach. The key takeaway from divide and conquer is utilizing parallelism to handle multiple calculations at once, making it an efficient solution for such computational tasks.