# Problem Solving: Maths for big-O notation

 ← Comparing algorithms Maths for Big-O Notation Order of complexity →

 From the Specification : Big-O Notation Linear time, polynomial time, exponential time. Order of complexity.

Big O Notation (also known as Big-O Notation) is a mathematical way of describing the limiting behaviours of a function. In other words, it is a way of defining how efficient an algorithm is by how "fast" it will run.

## Timing

You can work out the time that an algorithm takes to run by timing it:

```Dim timer As New Stopwatch()
timer.Start()
For x = 1 to 1000000000
'count to one billion!
Next
timer.Stop()
' Get the elapsed time as a TimeSpan value.
Dim el As TimeSpan = stopWatch.Elapsed

' Format and display the TimeSpan value.
Dim formattedTime As String = String.Format("{0}:{1}:{2}.{3}", el.Hours, el.Minutes, el.Seconds, el.Milliseconds / 10)
Console.WriteLine( "Time Elapsed: " + formattedTime)
```

However, this isn't always suitable. What happens if you run some code on a 33 MHz processor, and the same code on a 3.4 GHz processor? Timing a function tells you a lot about the speed of a computer, and very little about the speed of an algorithm.

## Refining algorithms

We might want to write a program to calculate the sum of all the numbers between 0 and a variable `N`, where `N` in this case == 7483647. To solve this, you might write a solution that has a loop that cycles through each number all the way up to `N`, adding it to another variable `sum`. The code might look a little like thisː

```dim N as integer = 7483647
dim sum as double= 0
for i = 1 to N
sum = sum + i
loop
console.writeline(sum)
```

It certainly works, but as `N` gets larger the program runs more slowly. In fact, there is a relationship between the value of `N` and the speed of the algorithm. If you were to double `N`, the algorithm would take twice as long and if you were to treble `N`, the code would take three times as long. We can describe the speed of this relationship using big O notationː $O(n)\,$ . That is the speed of the agorithm is related to the number of items being processed, in our case `N`. This is known as linear time.

We can of course write this is a much faster wayː

```dim N as integer = 7483647
dim sum as double =  N * (1 + N) / 2
console.writeline(sum)
```

This version of the algorithm gets the same result, but doesn't use a loop. It completes in the same amount of time whatever value of `N` you give it. We can describe this algorithm using big O notation as beingː $O(1)\,$ . That is, it completes in a constant time.

## Order of Complexity

Notation Name Example
$O(1)\,$ constant Determining if a number is even or odd; using a constant-size lookup table
$O(\log n)\,$ logarithmic Finding an item in a sorted array with a binary search or a balanced search tree as well as all operations in a Binomial heap.
$O(n)\,$ linear Finding an item in an unsorted list or a malformed tree (worst case) or in an unsorted array; Adding two n-bit integers by ripple carry.
$O(n\log n)=O(\log n!)\,$ linearithmic, loglinear, or quasilinear Performing a Fast Fourier transform; heapsort, quicksort (best and average case), or merge sort
$O(n^{2})\,$ quadratic Multiplying two n-digit numbers by a simple algorithm; bubble sort (worst case or naive implementation), Shell sort, quicksort (worst case), selection sort or insertion sort
$O(n^{c}),\;c>1$ polynomial or algebraic Tree-adjoining grammar parsing; maximum matching for bipartite graphs
$O(c^{n}),\;c>1$ exponential Finding the (exact) solution to the travelling salesman problem using dynamic programming; determining if two logical statements are equivalent using brute-force search
$O(n!)\,$ factorial Solving the travelling salesman problem via brute-force search; generating all unrestricted permutations of a poset; finding the determinant with expansion by minors.