Asymptotic Analysis in Data Structure using C

Asymptotic Analysis in Data Structure using C

• 1.
  Presented By: Deepshikha Haritwal MCA/25001/18
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   Meaning ofalgorithm  Algorithm analysis  Asymptotic analysis  Asymptotic notations
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   An algorithmmay be defined as a finite sequence of instructions each of which has a clear meaning and can be performed with a finite amount of effort in a finite length of time.  An algorithm has following properties:  Finiteness  Definiteness  Generality  Effectiveness  Input- Output
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   The performanceof algorithm can be measured on the basis of scale of time and space.  Time complexity of an algorithm or a program is a function of the running time of an algorithm or the program.  Space complexity of an algorithm or program is function of the space needed by algorithm or program to run to completion.
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   The timecomplexity of an algorithm could be computed by:  Posteriori analysis  Apriori analysis  Posteriori analysis calls for implementing the complete algorithms and executing them on computer for various instances of the problem.  Then the time taken by the execution of the programs for various instances of problem are noted and then compared.  Apriori analysis calls for mathematically determining the resources such as time and space as a function of parameter related to instances of problem.
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  In asymptotic Analysis,we evaluate the performance of an algorithm in terms of input size. Asymptotic analysis refers to computing the running time of any operation in mathematical units of computation.
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   The timerequired by an algorithm falls under three types −  Best Case − Minimum time required for program execution.  Average Case − Average time required for program execution.  Worst Case − Maximum time required for program execution.
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   Following arethe commonly used asymptotic notations to calculate the running time complexity of an algorithm:  Ο Notation  Ω Notation  θ Notation  Little o notation  Little omega notation
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   The notationΟ(n) is the formal way to express the upper bound of an algorithm's running time.  It measures the worst case time complexity or the longest amount of time an algorithm can possibly take to complete.  For example: the time complexity of Insertion sort is O(n^2).
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   O(g(n)) ={ f(n): there exist positive constants c and n0 such that 0 <= f(n) <= c*g(n) for all n >= n0}  Example : if f(n) = 16n3 + 78n2 + 12n , g(n)= n3 then f(n)=O(n3).
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   The notationΩ(n) is the formal way to express the lower bound of an algorithm's running time.  It measures the best case time complexity or the best amount of time an algorithm can possibly take to complete.  For example: the time complexity of Insertion Sort can be written as Ω(n).
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   Ω (g(n))= {f(n): there exist positive constants c and n0 such that 0 <= c*g(n) <= f(n) for all n >= n0}.  Example : if f(n)= 24n+9, g(n)= n then f(n)= Ω(n).
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   The notationθ(n) is the formal way to express both the lower bound and the upper bound of an algorithm's running time.  For example: If we use Θ notation to represent time complexity of Insertion sort, we have to use two statements for best and worst cases:  The worst case time complexity of Insertion Sort is Θ(n^2).  The best case time complexity of Insertion Sort is Θ(n).
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   Θ(g(n)) ={f(n): there exist positive constants c1, c2 and n0 such that 0 <= c1*g(n) <= f(n) <= c2*g(n) for all n >= n0}  Example : if f(n)= 28n+9, g(n)= n then f(n)= Θ (n). Since f(n)>28n and f(n)<=37n.
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   Little oprovides strict upper bound (equality condition is removed from Big O)  “Little-ο” (ο()) notation is used to describe an upper-bound that cannot be tight.  Definition :  Let f(n) and g(n) be functions that map positive integers to positive real numbers. We say that f(n) is ο(g(n)) if for any real constant c > 0, there exists an integer constant n0 ≥ 1 such that 0 ≤ f(n) < c*g(n).
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   little omegaprovides strict lower bound (equality condition removed from big omega).  We use ω notation to denote a lower bound that is not asymptotically tight.  Definition :  Let f(n) and g(n) be functions that map positive integers to positive real numbers. We say that f(n) is ω(g(n)) if for any real constant c > 0, there exists an integer constant n0 ≥ 1 such that f(n) > c * g(n) ≥ 0 for every integer n ≥ n0.