Properties of Asymptotic Notations

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

  1. 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
  2. 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
  3. 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
  4. 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
  5. 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:

  1. Reflexivity:
    If f(n) is given then



    f(n) = O(f(n))

    Example:
    If f(n) = n3 ⇒ O(n3)
    Similarly,

    f(n) = Ω(f(n)) 
    f(n) = Θ(f(n)) 
  2. Symmetry:
    f(n) = Θ(g(n)) if and only if g(n) = Θ(f(n))

    Example:
    If f(n) = n2 and g(n) = n2 then f(n) = Θ(n2) and g(n) = Θ(n2)
    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))
  3. Transistivity:
    f(n) = O(g(n)) and g(n) = O(h(n)) ⇒ f(n) = O(h(n))

    Example:
    If f(n) = n, g(n) = n2 and h(n) = n3
    ⇒ n is O(n2) and n2 is O(n3) then n is O(n3)
    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))
  4. Transpose Symmetry:
    f(n) = O(g(n)) if and only if g(n) = Ω(f(n))

    Example:
    If f(n) = n and g(n) = n2 then n is O(n2) and n2 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)) 
  5. 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
  6. 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))

  7. 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:

  1. If lim n→∞ f(n)/g(n) = c, c ∈ R+ then f(n) = Θ(g(n))
  2. If lim n→∞ f(n)/g(n) ≤ c, c ∈ R (c can be 0) then f(n) = O(g(n))
  3. If lim n→∞ f(n)/g(n) = 0, then f(n) = O(g(n)) and g(n) = O(f(n))
  4. If lim n→∞ f(n)/g(n) ≥ c, c ∈ R (c can be ∞) then f(n) = Ω(g(n))
  5. If lim n→∞ f(n)/g(n) = ∞, then f(n) = Ω(g(n))and g(n) = Ω(f(n))



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