A-level Computing 2009/AQA/Problem Solving, Programming, Operating Systems, Databases and Networking/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 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) 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 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 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.