Asymptoptic notations

Asymptoptic notations

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  By: Alisha Jindal AsymptoticNotations
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  Definition  Finite sequenceof logically related instructions to solve a computational problem  Expressions that are used to represent the complexity of an algorithm  Describes the behavior of time and space complexity for large instance characteristics and ignore for small values  Using asymptotic analysis, we can conclude best case, average case and worst case scenario of an algorithm.
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  Types of AsymptoticNotations Notations used in calculating running time complexity of an algorithm:  Big Oh Notation (O)  Big Omega Notation (Ω)  Theta Notation (θ)  Little Oh Notation (o)  Little Omega Notation (ω)
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  Big Oh Notation(O-notation)  It expresses the upper bound of an algorithm’s running time.  It measures the worst case time complexity or maximum amount of time an algorithm can take to complete.  It is used as a tight upper-bound on growth of algorithm
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  Big Oh Notation(O-notation) For a given function g(n), we denote by O(g(n)) the set of functions O(g(n))={f(n): there exist positive constants c and n0 such that 0 ≤ f(n) ≤ c.g(n) for all n ≥ n0}.
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  Examples Show that thefunction is tightly bound with big oh notation (a) f(n)=3n+5 (b) f(n)= 27n2+16n (c) f(n) = 2n +6n2 + 3n
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  Solutions (a) 3n+5 ≤3n+n ≤ 4n (c=4, n0 =5) So, f(n)= O(n) (b) As n2≥ 16n, 27n2 + 16n ≤ 27n2 + n2 ≤ 28n2 (c=28, n0 =16) So, f(n)=O(n2) (c) As n2≥ 3n 2n +6n2+3n ≤ 2n +6n2+n2 ≤ 2n +7n2 As 2n ≥ n2 (n ≥ 4) 2n +7n2 ≤ 2n +7*2n ≤ 8*2n (c=8, n0 =4) So, f(n)=O(2n)
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  Asymptotic Analysis ofRunning Time Comparing running time  Algorithm that runs in O(n) time is better than the algorithm that takes O(n2) time  Similarly, O(log n) is better than O(n)  Hierarchy of functions: 1< log n < n < nlogn < n2 < n3 < 2n
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  Big Omega Notation(Ω-notation)  It expresses the lower bound of an algorithm’s running time.  It measures the best case time complexity or minimum amount of time an algorithm can take to complete.  It is used as a tight lower-bound on growth of algorithm
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  Big Omega Notation(Ω-notation) For a given function g(n), we denote by Ω(g(n)) the set of functions Ω(g(n))={f(n): there exist positive constants c and n0 such that 0 ≤ c.g(n) ≤ f(n) for all n ≥ n0}.
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  Theta Notation (θ-notation) It encloses the function from above and below. Since it represents the upper and lower bound of an algorithm’s running time.  It is used for analyzing the average case time complexity of an algorithm
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  Theta Notation (θ-notation) Fora given function g(n), we denote by θ(g(n)) the set of functions θ(g(n))={f(n): there exist positive constants c1, c2 and n0 such that 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) for all n ≥ n0}.
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  Little-Oh Notation (o-notation) The functions in o(g) are the smaller function in O(g)  It is not asymptotically tight upper-bound  We define o(g(n)) as the set o(g(n))={f(n): there exist positive constants c and n0 such that 0 ≤ f(n) < cg(n) for all n ≥ n0}
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  Little-Omega Notation (ω- notation) The functions in ω(g) are the larger function of Ω(g)  It is not asymptotically tight lower-bound  We define ω(g(n)) as the set ω(g(n))={f(n): there exist positive constants c and n0 such that 0 ≤ cg(n) < f(n) for all n ≥ n0}
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  Analogy between functions Comparisonof two functions f and g and comparison of two real numbers a and b: f(n) = O(g(n)) ≅ a≤b f(n) = Ω(g(n)) ≅ a≥b f(n) = θ(g(n)) ≅ a=b f(n) = o(g(n)) ≅ a<b f(n) = ω(g(n))≅ a>b