variable information presented as a symbol, but they’re not required to use variables.
Formula: A combination of numbers and symbols used to express information or ideas. Formulas normally present mathematical or logical concepts, such as defining the Greatest Common Divisor (GCD) of two integers (the video at https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-factors-and-multiples/cc-6th-gcf/v/greatest-common-divisor tells how this works). Generally, they show the relationship between two or more variables.
Algorithm: A sequence of steps used to solve a problem. The sequence presents a unique method of addressing an issue by providing a particular solution. An algorithm need not represent mathematical or logical concepts, even though the presentations in this book often do fall into those categories because people most commonly use algorithms in this manner. In order for a process to represent an algorithm, it must be:Finite: The algorithm must eventually solve the problem. This book discusses problems with a known solution so that you can evaluate whether an algorithm solves the problem correctly.Well-defined: The series of steps must be precise and present steps that are understandable. Especially because computers are involved in algorithm use, the computer must be able to understand the steps to create a usable algorithm.Effective: An algorithm must solve all cases of the problem for which someone defined it. An algorithm should always solve the problem it has to solve. Even though you should anticipate some failures, the incidence of failure is rare and occurs only in situations that are acceptable for the intended algorithm use.
With these definitions in mind, the following sections help to clarify the precise nature of algorithms. The goal isn’t to provide a precise definition for algorithms, but rather to help you understand how algorithms fit into the grand scheme of things so that you can develop your own understanding of what algorithms are and why they’re so important.
The right way to make toast: Defining algorithm uses
An algorithm always presents a series of steps and doesn’t necessarily perform these steps to solve a math formula. The scope of algorithms is incredibly large. You can find algorithms that solve problems in science, medicine, finance, industrial production and supply, and communication. Algorithms provide support for all parts of a person’s daily life. Anytime a sequence of actions achieving something in our life is finite, well-defined, and effective, you can view it as an algorithm. For example, you can turn even something as trivial and simple as making toast into an algorithm. In fact, the making toast procedure often appears in computer science classes, as discussed at http://brianaspinall.com/now-thats-how-you-make-toast-using-computer-algorithms/.
Unfortunately, the algorithm on the site is flawed. The instructor never removes the bread from the wrapper and never plugs the toaster in, so the result is damaged plain bread still in its wrapper stuffed into a nonfunctional toaster (see the discussion at http://blog.johnmuellerbooks.com/2013/03/04/procedures-in-technical-writing/ for details). Even so, the idea is the correct one, yet it requires some slight, but essential, adjustments to make the algorithm finite and effective.
One of the most common uses of algorithms is as a means of solving formulas. For example, when working with the GCD of two integer values, you can perform the task manually by listing each of the factors for the two integers and then selecting the greatest factor that is common to both. For example, GCD (20, 25) is 5 because 5 is the largest number that divides evenly into both 20 and 25. However, processing every GCD manually is time consuming and error prone, so the Greek mathematician Euclid created a better algorithm to perform the task. You can see the Euclidean method demonstrated at https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/the-euclidean-algorithm.
However, a single formula, which is a presentation of symbols and numbers used to express information or ideas, can have multiple solutions, each of which is an algorithm. In the case of GCD, another common algorithm is one created by Derrick Henry Lehmer (https://www.imsc.res.in/~kapil/crypto/notes/node11.html). Because you can solve any formula multiple ways, people spend a great deal of time comparing algorithms to determine which one works best in a given situation. (See a comparison of Euclid to Lehmer at http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.31.693&rep=rep1&type=pdf.)
Because our society and its accompanying technology are changing quickly, we need algorithms that can keep the pace. Scientific achievements such as sequencing the human genome were possible in our age because scientists found algorithms that run fast enough to complete the task. Measuring which algorithm is better in a given situation, or in an average usage situation, is really serious stuff and is a topic of discussion among computer scientists.
When it comes to computer science, the same algorithm can have multiple presentations; why do it one way when you can invent multiple methods just for fun? For example, you can present the Euclidean algorithm in both recursive and iterative forms, as explained at http://cs.stackexchange.com/questions/1447/what-is-most-efficient-for-gcd. In short, algorithms present a method of solving formulas, but it would be a mistake to say that just one acceptable algorithm exists for any given formula or that only one acceptable presentation of an algorithm exists. Using algorithms to solve problems of various sorts has a long history — it isn’t something that has just happened.
Even if you limit your gaze to computer science, data science, artificial intelligence, and other technical areas, you find many kinds of algorithms — too many for a single book. For example, The Art of Computer Programming, by Donald E. Knuth (Addison-Wesley), spans 3,168 pages in four volumes (see http://www.amazon.com/exec/obidos/ASIN/0321751043/datacservip0f-20/) and still doesn’t manage to cover the topic (the author intended to write more volumes). However, here are some interesting uses for you to consider:
Searching: Locating information or verifying that the information you see is the information you want is an essential task. Without this ability, you couldn’t perform many tasks online, such as finding the website on the Internet selling the perfect coffee pot for your office. These algorithms change constantly, as shown by Google’s recent change in its algorithm (https://www.youaretech.com/blog/2021/1/26/webpage-experience-a-major-google-algorithm-update-in-2021nbsp).
Sorting: Determining which order to use to present information is important because most people today suffer from information overload, and putting information in order is one way to reduce the onrush of data. Imagine going to Amazon, finding that more than a thousand coffee pots are for sale there, and yet not being able to sort them in order of price or the most positive review. Moreover, many complex algorithms require data in the proper order to work dependably, so ordering is an important requisite for solving more problems.
Transforming: Converting one sort of data to another sort of data is critical to understanding and using the data effectively. For example, you might understand imperial weights just fine, but all your sources use the metric system. Converting
