在渐近符号中，为什么我们不使用所有可能的 function 来描述我们的 function 的增长率？

Question

if f(n) = 3n + 8,
for this we say or prove that f(n) = Ω(n)
Why we not use Ω(1) or Ω(logn) or.... for describing growth rate of our function?

Answer 1

In the context of studying the complexity of algorithms, the Ω asymptotic bound can serve at least two purposes:

• check if there is any chance of finding an algorithm with an acceptable complexity;

• check if we have found an optimal algorithm, ie such that its O bound matches the known Ω bound.

For theses purposes, a tight bound is preferable (mandatory).

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Also note that f(n)=Ω(n) implies f(n)=Ω(log(n)), f(n)=Ω(1) and all lower growth rates, and we needn't repeat that.

Answer 2

You can actually do that. Check the Big Omega notationhere and let's take Ω(log n) as an example. We have:

f(n) = 3n + 8 = Ω(log n)

because:

(according to the 1914 Hardy-Littlewood definition)

or:

(according to the Knuth definition).

For the definition of liminf and limsup symbols (with pictures) please check here .

Perhaps what was really meant is Θ (Big Theta), that is, both O() and Ω() simultaneously.