Problem Solving: Big O Notation
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From the Specification : BigO Notation

Big O Notation (also known as BigO 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[edit  edit source]
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 some 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[edit  edit source]
dim N as integer = 7483647
dim sum as double= 0
for i = 1 to N
sum = sum + i
loop
console.writeline(sum)
optimised version
dim N as integer = 7483647
dim sum as double = N * (1 + N) / 2
console.writeline(sum)
Order of Complexity[edit  edit source]
Notation  Name  Example 

constant  Determining if a number is even or odd; using a constantsize lookup table  
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.  
linear  Finding an item in an unsorted list or a malformed tree (worst case) or in an unsorted array; Adding two nbit integers by ripple carry.  
linearithmic, loglinear, or quasilinear  Performing a Fast Fourier transform; heapsort, quicksort (best and average case), or merge sort  
quadratic  Multiplying two ndigit numbers by a simple algorithm; bubble sort (worst case or naive implementation), Shell sort, quicksort (worst case), selection sort or insertion sort  
polynomial or algebraic  Treeadjoining grammar parsing; maximum matching for bipartite graphs  
exponential  Finding the (exact) solution to the travelling salesman problem using dynamic programming; determining if two logical statements are equivalent using bruteforce search  
factorial  Solving the travelling salesman problem via bruteforce search; generating all unrestricted permutations of a poset; finding the determinant with expansion by minors. 