Question

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.

Short answer

The minimum number of people needed for the task would be n-1 and the minimum time needed, given optimal parallelization and if it takes $t_a$ time to add two numbers, would be ${\log }_{2}n\ast t_a$.

Step-by-step solution

Understanding the Problem

First, it's important to visualize the problem. Imagine having a line of people, each holding a number. In the first 'round', every two contiguous persons add their numbers together, passing the sum to one of them. This process continues until only one number (the sum of all numbers) is left.

Illustrating Parallel Addition

Here is what the process looks like for four numbers (a, b, c, d), where every letter represents a number: a+b = e; c+d = f; and then in the second 'round', e+f, and so on. You can imagine it like a binary tree: first, you have n leaves (the numbers), which collapse into n/2 nodes (first round), then n/4 nodes (second round), and so on, until reaching the root (final result).

Determining Number of People

Since each addition operation requires two people (or 'numbers' in this case), and the problem describes a scenario with optimal parallelization (meaning multiple operations can occur simultaneously), the number of people needed would be n-1. This is because the number of additions required to add all the numbers is one less than the total number of numbers to be added. In each round, people 'pair up' to perform an addition, reducing the number of 'numbers' (or leaf nodes) by half.

Calculating Minimum Time

As for the time to complete the task, if each addition operation takes time $t_a$ and there are n numbers, the time taken would be ${\log }_{2}n\ast t_a$. This is calculated based on the binary tree-like structure of the operation, where each 'round' of operation is equivalent to one level of the binary tree. Essentially this means that in each round, the total number of 'numbers' remaining gets halved, hence the logarithmic time calculation.

Key concepts

Parallel Computing

Parallel computing refers to the practice of simultaneously using multiple processors or computers to perform computational tasks more efficiently. By dividing a large problem into smaller, manageable parts, different parts can be processed at the same time. Think of it like a group project where each person works on a section of the assignment; together, the whole project gets done faster.

In your exercise of adding numbers, parallel computing is utilized when each pair of numbers is added by different people at the same time. This simultaneous action is what makes the whole process much more efficient as opposed to having a single person do all the additions sequentially. The reduction in overall computation time showcases the strength of parallel computing in tasks that can be divided into independent chunks. Each chunk can be processed in parallel, leading to a significant decrease in the total amount of time needed to complete the task.

Binary Tree

A binary tree is a hierarchical structure in computing, where each node has a maximum of two children. This tree starts with a single root and branches out to the left and the right. In a binary tree representation of the numbers addition problem, the numbers at the bottom of the tree represent the initial list of numbers. When two numbers are paired and added, they form a new 'node' higher up in the tree.

In each round of addition, you're effectively moving up one level of the tree. The bottom-most level consists of the original numbers, and as you add numbers in pairs, each sum forms part of the next level. The process continues until you are left with a single number at the root, which represents the sum of all the numbers. This concept not only aids in understanding the addition process but also visualizes why the process is logarithmic, which we'll discuss next.

Logarithmic Time Complexity

Logarithmic time complexity, denoted as O(log n), is one of the fundamental concepts used to describe the efficiency of algorithms. It indicates that the time required to complete a task increases logarithmically as the size of the input increases. In simpler terms, each time the amount of data doubles, the number of operations required only increases by one step.

In the context of the numbers addition exercise, as you add pairs of numbers in each 'round', the number of resulting sums (and thereby the 'pieces' of the problem you're solving) halves. You're reducing the problem size by a factor of two each time, which is why the number of rounds it takes to add all the numbers is proportional to the logarithm of n, written as ${\log }_{2}n$. This represents the height of the binary tree - the number of levels you must progress through to reach the final sum.

Addition of Numbers

Addition is one of the most basic operations in mathematics, involving combining numbers to get a sum. In the context of your exercise, you are exploring how to add a list of numbers in the quickest possible way. It's an excellent example of how a simple operation, when scaled, can give birth to questions about optimal procedures and efficiency.

Traditionally, if you were to add these numbers sequentially, it would take much longer as you could only add two numbers at any given moment. However, by using the method described in the exercise, you can add multiple pairs of numbers at the same time, drastically reducing the time taken. This process is not only about addition; it's about finding the most time-efficient way to perform this fundamental operation, which is often a key consideration in computing and algorithm design.