Algorithm

An algorithm is a finite set of procedural and elementary computational steps to transform a specific set of inputs to a specific set of outputs.

More technically, for a set of inputs $I$ and a set of outputs $O$ , an algorithm is a function $f : I \rightarrow O$ where inputs are mapped to outputs.

Algorithms are designed, analyzed and optimized on the theoretical basis of a model of computation and operate within those specifications.

Key Properties

Finiteness — must terminate after a finite number of steps
Definiteness — each step must be precisely defined
Input — zero or more well-defined inputs
Output — one or more well-defined outputs
Effectiveness — each operation must be basic enough to be done exactly in finite time

Complexity Classes

Algorithm efficiency is measured in terms of time and space as functions of input size $n$ :

Notation	Name	Example
$O(1)$	Constant	Array access
$O(\log n)$	Logarithmic	Binary search
$O(n)$	Linear	Linear scan
$O(n \log n)$	Linearithmic	Merge sort
$O(n^2)$	Quadratic	Bubble sort
$O(2^n)$	Exponential	Subset enumeration

Computational Problem — what algorithms solve
RAM Model of Computation — the abstract machine algorithms run on
Data Structure — how algorithms organize their working data

Key Properties

Complexity Classes

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