Asymptotic Notation

Asymptotic Notations

• Asymptotic running time of an algorithm is defined in terms of functions.

• Asymptotic notation is useful describe the running time of the algorithm.

• Asymptotic notations give time complexity as “fastest possible”, “slowest possible” or “average time”.

• Bigh Oh (Ο) , Omega (Ω) and Theta (Θ) notations are useful to represent the asymptotic complexity of algorithms.

Big oh Notation

• “O” represents the Big oh notation.• “O” is a method of representing the upper

bound (longest amount of time) of algorithm’s running time.

f(n) is O(g(n)) ≡ C ∃ ∃ n0 n (n > ∀ n0 → f(n) ≤ C g(n))

• Let f(n) and g(n) be two non-negative functions. Let ‘n0‘and constant ‘c’ are two integers such that ‘n0‘ denotes some value of input n>n0. Similarly ‘c’ is some constant such that c>0.

f(n)<=c*g(n) then f(n) is O(g(n)).

In other words f(n) is less than g(n) if g(n) is multiple of some constant c.

Standard Method to Prove Big-Oh

• To prove that f(n) is O(g(n)):1. Choose n0 = 1.

2. Assuming n > 1, find/derive a C such that f(n)/g(n)≤C g(n)/g(n)= C

This shows that n > 1 implies f(n) ≤ C g(n).Keep in mind:• n > 1 implies 1 < n < n2 < n3 < . . .• “Increase” numerator to “simplify” fraction.

Big-Omega Notation

• “Ω” represents the Big Omega notation.• “Ω” is a method of representing the Lower

bound (minimum amount of time) of algorithm’s running time.

f(n) is Ω(g(n)) ≡ C ∃ ∃ n0 n (n > ∀ n0 → f(n) ≥ C g(n))

• Let f(n) and g(n) be two non-negative functions. Let ‘n0‘and constant ‘c’ are two integers such that ‘n0‘ denotes some value of input n>n0. Similarly ‘c’ is some constant such that c>0.

f(n) ≥ c*g(n) then f(n) is Ω(g(n)).

In other words f(n) is greater than g(n) if g(n) is multiple of some constant c.

Theta Notation

• “Θ” is a method of representing the running time between the Upper bound and Lower bound.

f(n) is Θ(g(n)) ≡ C∃ 1 C∃ 2 ∃ n0 n (n > ∀ n0 → C1g(n) ≤f(n) ≤ C2 g(n))

• Let f(n) and g(n) be two non-negative functions. Let ‘n0‘ and constants ‘c1’ and ‘c2‘ are integers such that ‘n0‘ denotes some value of input n>n0. Similarly ‘c1’ and ‘c2’ are some constants such that ‘c1’ >0 and ‘c2’>0.

C1g(n) ≤f(n) ≤ C2 g(n) then f(n) is Θ(g(n)).

In other words, function f(n)=Θ(g(n) iff g(n) is both an upper and lower bound of f(n).