渐近上界与紧界

Question

我在 CLRS (Introduction to Algorithms) 中遇到过这样一句话：“区分渐近上界和渐近紧界是算法文献中的标准”虽然我理解文本想要传达的本质，但会更好地理解如果我得到一个说明差异的例子。

Answer 1

O-notation gives us asymptotic upper bound.
Consider a function f(x), We can define a function g(x), such that f(x) = O(g(x)).
Here g(x) is the asymptotic upper bound of f(x), meaning for all values of x >= c, f(x) grows at the same rate or slower than g(x) as x increases.
Another thing to be noticed is that if h(x) is the asymptotic upper bound of g(x) then it could easily be concluded that h(x) is also the asymptotic upper bound of f(x). After all, if f(x) can only grow at an equal or smaller rate than g(x), it is bound to grow at an equal of smaller rate than h(x) as g(x) cannot grow at any faster than h(x).

Eg. if f(x) = 10x + 2, g(x) = 12x + 1 and h(x) = 2x^2.

We can safely say that f(x) = O(g(x)), g(x) = O(h(x)) and f(x) = O(h(x)).
Here g(x) is said to be asymptotic tight upper bound and h(x) is said to be the asymptotic upper bound of f(x).