# Algorithms

## Table of contents

## Introduction

An algorithm is a procedure that takes input and produces output that accomplishes a specific task [1, P. 3].

An algorithm should solve a general, well-specified problem. A specification for an algorithm should include the complete set of instances it will operate on, and the algorithm’s output after running on one of these instances [1, P. 3].

A good algorithm is:

- Correct
- Efficient
- Easy to implement

It’s not always possible to meet these goals [1, P. 4].

An algorithm is commonly expressed in one of the following forms:

- English
- Pseudocode
- A programming language

## Reasoning about correctness

The most important property of an algorithm is that it’s correct.

One way to demonstrate the correctness of an algorithm is a mathematical proof.

A mathematical proof consists of four parts:

- A statement of what you’re trying to achieve
- A set of assumptions you take to be true
- A chain of reasoning that takes you from the assumptions to the statement you are attempting to prove
- A little square (∎) or QED to denote the end of the proof

### Problems and properties

In order to demonstrate correctness, your problem must be well-specified.

Problem specifications have two parts:

- The set of allowed input instances
- The required properties of the algorithm output

You should avoid asking ill-defined questions. Asking “what is the best path?” is ill-defined: what does *best* mean? You should be more specific, for example: “which is the *fastest* path to take?” [1, P. 13].

### Demonstrating incorrectness

You can prove an algorithm incorrect by providing an input that produces the incorrect output. These are called **counter-examples** [1, P. 13].

Good counter-examples are verifiable and simple [1, P. 13].

### Induction and recursion

Failure to prove an algorithm as incorrect does not make it correct. **Mathematical induction** is a common method to prove the correctness of an algorithm.

The way to prove a predicate (P) through induction is to:

- Prove the case P(0)
- Assume that P(k) is true
- Prove P(k+1)

*see this video teaching proof by induction for further explanation*

### Sigma notation

Summations are common in algorithm analysis.

Sigma notation is a way of expressing summation formulas. For example, the sum of 1 to n in sigma notation is:

*see this video explaining sigma notation for further explanation*

## Modelling the problem

Modelling is the process of formulating your application in terms of well-defined, well-understood problems. Modelling can eliminate the need to create your own algorithm, since you can rephrase your problem to use a pre-written algorithm [1, P. 19].

Most problems are real-world problems. For example, you might need to create a system to route traffic. Algorithms don’t work on real-world objects, they work abstractions, like a graph. In order to write effective algorithms you must learn how to describe your problems in terms of abstract strictures [1, P. 19].

In order to model a problem, you should have a solid understanding of the data structures available to you.

## Conclusion

Algorithms are procedures that accomplish a specific task.

You can determine the correctness of an algorithm using a mathematical proof, one way of doing this is by using mathematical induction.

Algorithms work on abstract objects. In order to write algorithms for real-world problems, you need to model your real-world problems in terms of abstract objects that an algorithm can work on.

## References

- [1] S. Skiena,
*The Algorithm Design Manual*, 2nd ed. Springer, 2008.