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?
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).
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.
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.
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