Problem Solving: Big O Notation

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UNIT 3 - ⇑ Problem Solving ⇑

Big O Notation Turing Machines →

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[edit | edit source]

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

Dim timer As New Stopwatch()
For x = 1 to 1000000000
   'count to one billion!
' 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)
   Code Output
Down arrow Hexagonal Icon.svg
Time Elapsed: 0:0:21:3249

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

optimised version

dim N as integer = 7483647
dim sum as double =  N * (1 + N) / 2

Order of Complexity[edit | edit source]

Notation Name Example
constant Determining if a number is even or odd; using a constant-size 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 n-bit 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 n-digit 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 Tree-adjoining 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 brute-force search
factorial Solving the travelling salesman problem via brute-force search; generating all unrestricted permutations of a poset; finding the determinant with expansion by minors.