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

Growth of Functions 16

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. 17

Growth of Functions g(n) f(n) 1 2 3 4 5 6 7 8 9 10 18

O -notation ∃𝑑 > 0 , 𝑜 0 > 0 0 ≤ 𝑔 𝑜 ≤ 𝑑𝑕 𝑜 𝑜 ≥ 𝑜 0 g(n) is an asymptotic upper- bound for f(n) 19

Ω -notation ∃𝑑 > 0, 𝑜 0 > 0 0 ≤ 𝑑𝑕 𝑜 ≤ 𝑔 𝑜 𝑜 ≥ 𝑜 0 g(n) is an asymptotic lower- bound for f(n) 20

Θ -notation ∃𝑑 1 , 𝑑 2 > 0, 𝑜 0 > 0 0 ≤ 𝑑 1 𝑕 𝑜 ≤ 𝑔 𝑜 ≤ 𝑑 2 𝑕(𝑜) 𝑜 ≥ 𝑜 0 g(n) is an asymptotic tight- bound for f(n) 21

o-notation ∀𝑑 > 0 ∃𝑜 0 > 0 0 ≤ 𝑔 𝑜 ≤ 𝑑𝑕 𝑜 𝑜 ≥ 𝑜 0 g(n) is a non-tight 𝑔 𝑜 = 𝑝(𝑕 𝑜 ) asymptotic upper- bound for f(n) 22

ω -notation ∀𝑑 > 0 ∃𝑜 0 > 0 0 ≤ 𝑑𝑕 𝑜 ≤ 𝑔 𝑜 𝑜 ≥ 𝑜 0 g(n) is a non-tight asymptotic lower- bound for f(n) 𝑔 𝑜 = 𝜕(𝑕 𝑜 ) 23

Analogy to real numbers Functions Real numbers 𝑏 ≤ 𝑐 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 24

Standard Classes of Functions 𝑔 𝑜 = Θ 1 Constant: Logarithmic: 𝑔 𝑜 = Θ(lg 𝑜 ) 𝑔 𝑜 = 𝑝(𝑜) Sublinear: 𝑔 𝑜 = Θ 𝑜 Linear: Super-linear: 𝑔 𝑜 = 𝜕(𝑜) 𝑔 𝑜 = Θ(𝑜 2 ) Quadratic: 𝑔 𝑜 = Θ(𝑜 𝑙 ) ; k is a constant Polynomial: Exponential: 𝑔 𝑜 = Θ(𝑙 𝑜 ) ; k is a constant 25

Insertion Sort (Revisit) n-times j-times Θ(𝑜 2 ) 26

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

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 int fib(int number) { if (number == 0) return 0; if (number == 1) return 1; return fib(number-1) + fib(number-2); } Question: Who is the recursion’s worst enemy? 28

Function calls main() { F1(…); } F3 F1(…) { F2(…); F2 } F2(…) { Other things you do not want to know F3(…); F1 } Local variables Stack 29