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An algorithm specifies a series of steps that perform a particular computation or task. Algorithms were originally born as part of mathematics – the word “algorithm” comes from the Arabic writer Muḥammad ibn Mūsā al-Khwārizmī, – but currently the word is strongly associated with computer science. Throughout this book we’ll examine a number of different algorithms to perform a variety of tasks.

Algorithms resemble recipes. Recipes tell you how to accomplish a task by performing a number of steps. For example, to bake a cake the steps are: preheat the oven; mix flour, sugar, and eggs throughly; pour into a baking pan; and so forth.

However, “algorithm” is a technical term with a more specific meaning than “recipe”, and calling something an algorithm means that the following properties are all true:

Studying algorithms is a fundamental part of computer science. There are several different characteristics of an algorithm that are useful to know:

Most of these questions will be discussed for the algorithms covered in this book.

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Let’s look at a very simple algorithm called find_max() .

Problem: Given a list of positive numbers, return the largest number on the list.

Inputs: A list L of positive numbers. This list must contain at least one number. (Asking for the largest number in a list of no numbers is not a meaningful question.)

Outputs: A number n , which will be the largest number of the list.

Algorithm:

An implementation in Python:

Does this meet the criteria for being an algorithm?

There can be many different algorithms for solving the same problem. Here’s an alternative algorithm for find_max() :

Implementation:

Let’s ask our questions again.

Is it unambiguous? Yes. Each step is simple and easily translated into Python.

Does it have defined inputs and outputs? Yes.

Is it guaranteed to terminate? Yes. The algorithm obviously terminates if L is of length 1. If L has more than one element, find_max() is called with a list that’s one element shorter and the result is used in a computation.

Does the nested call to find_max() always terminate? Yes. Each time, find_max() is called with a list that’s shorter by one element, so eventually the list will be of length 1 and the nested calls will end.

Finally, does it produce the correct result? Yes. Here’s a sketch of a proof. [2]

Consider a list of length 1. In this case the largest number is also the only number on the list. find_max() returns this number, so it’s correct for lists of length 1.

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