complexity big oh notation notation.pptx

complexity big oh notation notation.pptx

• 1.
  Complexity Analysis COURSE NAME:DATA STRUCTURES AND ALGORITHM
• 2.
  ALGORITHMS • STEP BYSTEP PROCEDURE TO SOLVE A PROBLEM • USED IN DAILY LIFE PROBLEMS • IN CS, ALGORITHM IS DEFINED BY: 1- MUST HAVE SOME DATA TO OPERATE ON IT 2- MUST PRODUCE ATLEAST ONE RESULT 3- MUST TERMINATE AFTER FINITE NUMBER OF STEPS • MANY ALGORITHMS HAVE BEEN DESIGNED
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  • Searching algorithms •Sorting algorithms • Compression algorithms • Fast Fourier transforms( designed for digital signal processing) • Encoding algorithms (used for encryption of data) • Geometric algorithms( used for identification of geometric shapes) • Pattern matching algorithms ( comparing images and shapes)
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  Classification of algorithms •Depending on the strategy for solving a particular algorithms, algorithms can be classified as: • 1- Iterative (for example: sorting an array) • 2 Divide and Conquer • 3 Greedy algorithm ( an immediately available best solution at each step is chosen) e.g: which way of conveyance to choose immediately
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  Specification of algorithms •Natural language • Pseudo code • Natural language: words like read, write, subtract, divide, repeat, for while are used • Pseudo code: It is a blend of natural language and programming notation
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  Complexity • Time complexity •Space complexity • Input
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  • Worst-case Complexity:Maximum time required for program execution (Run slowest among all inputs) In the worst case analysis, we calculate upper bound on running time of an algorithm. • Average Complexity: Average time required for program execution. Gives the necessary information about algorithm’s behavior on random input. • Best Case: Minimum time required for program execution (Run Fastest among all inputs). It gives lower bound on running time of algorithm for any instances of input.
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  • Why WorstCase Analysis? The worst-case running time of an algorithm gives an upper bound on the running time for any input. Knowing it provides a guarantee that the algorithm will never take any longer.For some algorithms, the worst case occurs fairly often. For example, in searching a database for a particular piece of information, the searching algorithm’s worst case will often occur when the information is not present in the database. • The “average case” is often roughly as bad as the worst case.
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  Method Worst caseAverage case Selection sort N2 N2 Insertion sort N2 N2 Merge sort Nlogn Nlogn Quick sort n2 nlogn
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  • Asymptotic notationsgive time complexity as “fastest possible”, “slowest possible” or “average time”.
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  Asymptotic Notations • Followingare commonly used asymptotic notations used in calculating running time complexity of an algorithm. • 1. Big O Notationf(n)=O(g(n)) (read: f of n is big oh of g of n), if there exists a positive integer n0 and a positive number c such that |f(n)|≤c|g(n)|, for all n≥n0. • 2. Omega (Ω) Notationf(n)=Ω(g(n))(read: f of n is omega of g of n), if there exists a positive integer n0 and a positive integer c such that |f(n)|≥c|g(n)|, for all n≥n0. • 3. Theta (θ) Notationf(n)= Ө(g(n))(read: f of n is thita of g of n), if there exists a positive integer n0 and a positive integer c1 and c2 such that c1|g(n)| ≤ |f(n)| ≤c2|g(n)|, for all n≥n0.
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  Big oh notation •The Ο(n) is the formal way to express the upper bound of an algorithm's running time.It always indicates the maximum time required by an algorithm for all input values.That means Big - Oh notation describes the worst case of an algorithm time complexity.Definition:O(g(n)) = {f(n) : there exists positive constants c and n0 such that 0 ≤ f(n) ≤ c g(n) for all n ≥ n0.
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  • Omega ()notation (lower bound- best case )  -notation provides an asymptotic lower bound on a function.It  measures the best case time complexity or best amount of time an algorithm can possibly take to complete.Definition:
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  • It measuresthe Average case Complexity. 2. Theta (θ)notation(upper bound as well as lower bound – Average case )The θ(n) is the formal way to express both the lower bound and upper bound of an algorithm's running time. It measures the Average case Complexity.
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  NOTE: ALL BULLETPOINTS MENTIONED IN SLIDES SHOULD BE COVERED /LEARNT (STUDENTS MUST KNOW THEIR CONCEPTS)