There exist positive constants c1, c2, and n0 such that
0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) for all n ≥ n0.
We say g(n) is an asymptotically tight bound for f(n) if f(n) = Θ(g(n)). Informally we can say, as far as time complexity or order of growth, these functions are equal.
f(n) = O(g(n))
There exist positive constants c and n0 such that
0 ≤ f(n) ≤ cg(n) for all n ≥ n0.
We say g(n) is an asymptotic upper bound for f(n) if f(n) = O(g(n)).
f(n) = Ω(g(n))
There exist positive constants c and n0 such that
0 ≤ cg(n) ≤ f(n) for all n ≥ n0.
We say g(n) is an asymptotic lower bound for f(n) if f(n) = Ω(g(n)).
f(n) = o(g(n))
For any positive constant c > 0, there exists a constant
n0 > 0 such that 0 ≤ f(n) < cg(n) for all n ≥ n0.
We say f(n) is asymptotically smaller than g(n) if f(n) = o(g(n)). We can also express this as the following limit
f(n) = ω(g(n))
For any positive constant c > 0, there exists a constant
n0 > 0 such that 0 ≤ cg(n) < f(n) for all n ≥ n0.
We say f(n) is asymptotically bigger than g(n) if f(n) = ω(g(n)). We can also express this as the following limit.
f(n) = Θ(g(n)) and g(n) = Θ(h(n)) ⇒ f(n) = Θ(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))
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))
f(n) = Θ(f(n))
f(n) = O(f(n))
f(n) = Ω(f(n))
f(n) = Θ(g(n)) iff g(n) = Θ(f(n))
f(n) = O(g(n)) iff g(n) = Ω(f(n))
f(n) = o(g(n)) iff g(n) = ω(f(n))
f(n) = O(g(n)) ≈ a ≤ b
f(n) = Ω(g(n)) ≈ a ≥ b
f(n) = Θ(g(n)) ≈ a = b
f(n) = o(g(n)) ≈ a < b
f(n) = ω(g(n)) ≈ a > b
From the definitions of Θ, O, and Ω notations we can conclude the following:
* For any two functions f(n) and g(n), we have f(n) = Θ(g(n)) iff f(n) = O(g(n)) and f(n) = Ω(g(n)).
* If f(n) = O(g(n)) then g(n) = Ω(f(n)) and vice versa.
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