Growth of Functions 16 Learning Objectives Understand the meaning - - PowerPoint PPT Presentation

Growth of Functions

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Learning Objectives

Understand the meaning of growth of functions. Measure the growth of the running time of an algorithm. Use the Big-Oh notation to compare the growth of two functions.

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Growth of Functions

1 2 3 4 5 6 7 8 9 10

g(n) f(n)

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O-notation

∃𝑑 > 0,𝑜0 > 0 0 ≤ 𝑔 𝑜 ≤ 𝑑𝑕 𝑜 𝑜 ≥ 𝑜0

g(n) is an asymptotic upper- bound for f(n)

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Ω-notation

∃𝑑 > 0, 𝑜0 > 0 0 ≤ 𝑑𝑕 𝑜 ≤ 𝑔 𝑜 𝑜 ≥ 𝑜0

g(n) is an asymptotic lower- bound for f(n)

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Θ-notation

∃𝑑1, 𝑑2 > 0, 𝑜0 > 0 0 ≤ 𝑑1𝑕 𝑜 ≤ 𝑔 𝑜 ≤ 𝑑2𝑕(𝑜) 𝑜 ≥ 𝑜0

g(n) is an asymptotic tight- bound for f(n)

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• -notation

∀𝑑 > 0 ∃𝑜0 > 0 0 ≤ 𝑔 𝑜 ≤ 𝑑𝑕 𝑜 𝑜 ≥ 𝑜0

g(n) is a non-tight asymptotic upper- bound for f(n) 𝑔 𝑜 = 𝑝(𝑕 𝑜 )

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ω-notation

∀𝑑 > 0 ∃𝑜0 > 0 0 ≤ 𝑑𝑕 𝑜 ≤ 𝑔 𝑜 𝑜 ≥ 𝑜0

g(n) is a non-tight asymptotic lower- bound for f(n) 𝑔 𝑜 = 𝜕(𝑕 𝑜 )

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Analogy to real numbers

Functions Real numbers 𝒈 𝒐 = 𝑷 𝒉 𝒐 𝑏 ≤ 𝑐 𝒈 𝒐 = Ω 𝒉 𝒐 𝑏 ≥ 𝑐 𝒈 𝒐 = Θ 𝒉 𝒐 𝑏 = 𝑐 𝒈 𝒐 = o 𝒉 𝒐 𝑏 < 𝑐 𝒈 𝒐 = ω 𝒉 𝒐 𝑏 > 𝑐

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Standard Classes of Functions

Constant: 𝑔 𝑜 = Θ 1 Logarithmic: 𝑔 𝑜 = Θ(lg 𝑜 ) Sublinear: 𝑔 𝑜 = 𝑝(𝑜) Linear: 𝑔 𝑜 = Θ 𝑜 Super-linear: 𝑔 𝑜 = 𝜕(𝑜) Quadratic: 𝑔 𝑜 = Θ(𝑜2) Polynomial: 𝑔 𝑜 = Θ(𝑜𝑙); k is a constant Exponential: 𝑔 𝑜 = Θ(𝑙𝑜); k is a constant

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Insertion Sort (Revisit)

n-times j-times

Θ(𝑜2)

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Using L'Hopital’s rule

Determine the relative growth rates by using L'Hopital's rule compute if 0: f(N) = o(g(N)) if constant  0: f(N) = (g(N)) if : g(N) = o(f(N)) limit oscillates: no relation

) ( ) ( lim N g N f

n  

Recursion

In math: A function is defined based on itself

Factorial: 𝑜! = (𝑜 − 1)! ∙ 𝑜, 0! = 1 Fibonacci: 𝐺(𝑜) = 𝐺(𝑜 − 1) + 𝐺(𝑜 − 2), 𝐺(0) = 𝐺(1) = 1

In programming: A function calls itself Question: Who is the recursion’s worst enemy?

int fib(int number) { if (number == 0) return 0; if (number == 1) return 1; return fib(number-1) + fib(number-2); }

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Function calls

main() { F1(…); } F1(…) { F2(…); } F2(…) { F3(…); }

Stack

Local variables Other things you do not want to know

F1 F2 F3

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