**Prerequisite:** Asymptotic Notations

Assuming f(n), g(n) and h(n) be asymptotic functions the mathematical definitions are:

- If
**f(n) = Θ(g(n))**, then there exists positive constants c1, c2, n0 such that**0 ≤ c1.g(n) ≤ f(n) ≤ c2.g(n)**, for all n ≥ n0 - If
**f(n) = O(g(n))**, then there exists positive constants c, n0 such that**0 ≤ f(n) ≤ c.g(n)**, for all n ≥ n0 - If
**f(n) = Ω(g(n))**, then there exists positive constants c, n0 such that**0 ≤ c.g(n) ≤ f(n)**, for all n ≥ n0 - If
**f(n) = o(g(n))**, then there exists positive constants c, n0 such that**0 ≤ f(n) < c.g(n)**, for all n ≥ n0 - If
**f(n) = ω(g(n))**, then there exists positive constants c, n0 such that**0 ≤ c.g(n) < f(n)**, for all n ≥ n0

**Properties:**

**Reflexivity:**

If f(n) is given then**f(n) = O(f(n))***Example:*

If f(n) = n^{3}⇒ O(n^{3})

Similarly,f(n) = Ω(f(n)) f(n) = Θ(f(n))

**Symmetry:****f(n) = Θ(g(n)) if and only if g(n) = Θ(f(n))***Example:*

If f(n) = n^{2}and g(n) = n^{2}then f(n) = Θ(n^{2}) and g(n) = Θ(n^{2})*Proof:**Necessary part:*

f(n) = Θ(g(n)) ⇒ g(n) = Θ(f(n))

By the definition of Θ, there exists positive constants c1, c2, no such that c1.g(n) ≤ f(n) ≤ c2.g(n) for all n ≥ no

⇒ g(n) ≤ (1/c1).f(n) and g(n) ≥ (1/c2).f(n)

⇒ (1/c2).f(n) ≤ g(n) ≤ (1/c1).f(n)

Since c1 and c2 are positive constants, 1/c1 and 1/c2 are well defined. Therefore, by the definition of Θ, g(n) = Θ(f(n))*Sufficiency part:*

g(n) = Θ(f(n)) ⇒ f(n) = Θ(g(n))

By the definition of Θ, there exists positive constants c1, c2, no such that c1.f(n) ≤ g(n) ≤ c2.f(n) for all n ≥ no

⇒ f(n) ≤ (1/c1).g(n) and f(n) ≥ (1/c2).g(n)

⇒ (1/c2).g(n) ≤ f(n) ≤ (1/c1).g(n)

By the definition of Theta(Θ), f(n) = Θ(g(n))

**Transistivity:****f(n) = O(g(n)) and g(n) = O(h(n)) ⇒ f(n) = O(h(n))***Example:*

If f(n) = n, g(n) = n^{2}and h(n) = n^{3}

⇒ n is O(n^{2}) and n^{2}is O(n^{3}) then n is O(n^{3})*Proof:*

f(n) = O(g(n)) and g(n) = O(h(n)) ⇒ f(n) = O(h(n))

By the definition of Big-Oh(O), there exists positive constants c, no such that f(n) ≤ c.g(n) for all n ≥ no

⇒ f(n) ≤ c1.g(n)

⇒ g(n) ≤ c2.h(n)

⇒ f(n) ≤ c1.c2h(n)

⇒ f(n) ≤ c.h(n), where, c = c1.c2 By the definition, f(n) = O(h(n))

Similarly,f(n) = Θ(g(n)) and g(n) = Θ(h(n)) ⇒ f(n) = Θ(h(n)) f(n) = Ω(g(n)) and g(n) = Ω(h(n)) ⇒ f(n) = Ω(h(n)) f(n) = o(g(n)) and g(n) = o(h(n)) ⇒ f(n) = o(h(n)) f(n) = ω(g(n)) and g(n) = ω(h(n)) ⇒ f(n) = ω(h(n))

**Transpose Symmetry:****f(n) = O(g(n)) if and only if g(n) = Ω(f(n))***Example:*

If f(n) = n and g(n) = n^{2}then n is O(n^{2}) and n^{2}is Ω(n)*Proof:**Necessary part:*

f(n) = O(g(n)) ⇒ g(n) = Ω(f(n))

By the definition of Big-Oh (O) ⇒ f(n) ≤ c.g(n) for some positive constant c ⇒ g(n) ≥ (1/c).f(n)

By the definition of Omega (Ω), g(n) = Ω(f(n))*Sufficiency part:*

g(n) = Ω(f(n)) ⇒ f(n) = O(g(n))

By the definition of Omega (Ω), for some positive constant c ⇒ g(n) ≥ c.f(n) ⇒ f(n) ≤ (1/c).g(n)

By the definition of Big-Oh(O), f(n) = O(g(n))

Similarly,

f(n) = o(g(n)) if and only if g(n) = ω(f(n))

- Since these properties hold for asymptotic notations, analogies can be drawn between functions f(n) and g(n) and two real numbers a and b.
*g(n) = O(f(n)) is similar to a ≤ b**g(n) = Ω(f(n)) is similar to a ≥ b**g(n) = Θ(f(n)) is similar to a = b**g(n) = o(f(n)) is similar to a < b**g(n) = ω(f(n)) is similar to a > b*

**Observations:***max(f(n), g(n)) = Θ(f(n) + g(n))**Proof:*

Without loss of generality, assume f(n) ≤ g(n), ⇒ max(f(n), g(n)) = g(n)

Consider, g(n) ≤ max(f(n), g(n)) ≤ g(n)

⇒ g(n) ≤ max(f(n), g(n)) ≤ f(n) + g(n)

⇒ g(n)/2 + g(n)/2 ≤ max(f(n), g(n)) ≤ f(n) + g(n)

From what we assumed, we can write

⇒ f(n)/2 + g(n)/2 ≤ max(f(n), g(n)) ≤ f(n) + g(n)

⇒ (f(n) + g(n))/2 ≤ max(f(n), g(n)) ≤ f(n) + g(n)

By the definition of Θ, max(f(n), g(n)) = Θ(f(n) + g(n))*O(f(n)) + O(g(n)) = O(max(f(n), g(n)))**Proof:*

Without loss of generality, assume f(n) ≤ g(n)

⇒ O(f(n)) + O(g(n)) = c1.f(n) + c2.g(n)

From what we assumed, we can write

O(f(n)) + O(g(n)) ≤ c1.g(n) + c2.g(n)

≤ (c1 + c2) g(n)

≤ c.g(n)

≤ c.max(f(n), g(n))

By the definition of Big-Oh(O),

O(f(n)) + O(g(n)) = O(max(f(n), g(n)))

**Note:**

- If
, c ∈ R+ then*lim*_{n→∞}f(n)/g(n) = c*f(n) = Θ(g(n))* - If
, c ∈ R (c can be 0) then*lim*_{n→∞}f(n)/g(n) ≤ c*f(n) = O(g(n))* - If
, then*lim*_{n→∞}f(n)/g(n) = 0and*f(n) = O(g(n))**g(n) = O(f(n))* - If
, c ∈ R (c can be ∞) then*lim*_{n→∞}f(n)/g(n) ≥ c*f(n) = Ω(g(n))* - If
, then*lim*_{n→∞}f(n)/g(n) = ∞and*f(n) = Ω(g(n))**g(n) = Ω(f(n))*

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