Asymptotic lower bound
In algorithmic analysis, an asymptotic lower bound is a function that serves as the lower bound of another function when the argument tends to infinity. Usually the notation Ω(g(x)) is used to refer to functions bounded below by the function g(x).
More formally it is defined:
- Ω Ω (g(x))={f(x):existc,x0positive constants such thatРусский Русский x:x0≤ ≤ x:0≤ ≤ cg(x)≤ ≤ f(x)!{displaystyle Omega (g(x)))=left{{begin{matrixf}(x):{mbox{existen }c,x_{0}{mbox{mbox{ positive constants such as}}forall x:x_{0}leq x:0leq cg(x)leq
A function f(x) belongs to Ω(g(x)) when there is a positive constant c from a value x0{displaystyle x_{0}}, cg(x){displaystyle cg(x)} Not over. f(x). It means the function f is higher than g from a given value except for a constant factor.
The asymptotic lower bound is useful in Computational Complexity Theory for calculating the best-case complexity for algorithms.
Even though Ω(g(x)) is defined as a set, it is customary to write f(x)=Ω(g(x)) instead of f(x) 한 Ω(g(x)). Often, a function is also spoken by appointing only its expression, as in x2 instead of h(x)=x2as long as it is clear what the function parameter is within the expression. In the graph a schematic example of how it behaves cg(x){displaystyle cg(x)} with regard to f(x) When x tends to infinity.
The asymptotic fitted bound (Θ notation) is related to the asymptotic upper (O notation) and lower bounds:
- f(x)=Strike Strike (g(x))Yes and only iff(x)=O(g(x))andf(x)=Ω Ω (g(x)){displaystyle f(x)=Theta (g(x))){mbox{ si y solo si }f(x)=O(g(x)){mbox{ y }}f(x)=Omega (g(x)}}
Examples
- Function x2 can be lowered by function x. To prove it, just note that for all value x≥1 is fulfilled x≤x2. So... x2 = Ω(x) (However, x It doesn't serve as a top boot for x2).
- Function x2+200x It's lowered by x2. You mean when x tends to infinite the value of 200x can be despised with respect to x2. Besides he's never gonna touch zero.