算法学习之“Omega“、“Theta”和“Little Oh”

一、An asymptotical lower bound - Omega

The big oh notation introduced in the preceding section is an asymptotic upper bound. In this section, we introduce a similar notation for characterizing the asymptotic behavior of a function, but in this case it is a lower bound.

Definition (Omega)        Consider a function  f( n) which is non-negative for all integers  . We say that `` f( n) is omega  g( n),'' which we write  , if there exists an integer   and a constant  c>0 such that for all integers  ,  .

The definition of omega is almost identical to that of big oh. The only difference is in the comparison--for big oh it is ; for omega, it is . All of the same conventions and caveats apply to omega as they do to big oh.

“Omega Notattion”与“Big Oh”向对应，唯一不同的是比较符的方向，同时，Omega也拥有与Big Oh相同的性质（和、积及传递性）。

二、“Theta” 和 “Little Oh”

This section presents two less commonly used forms of asymptotic notation. They are:

• A notation, , to describe a function which is both O(g(n)) and , for the same g(n). (Definition ).
• A notation, , to describe a function which is O(g(n)) but not , for the same g(n). (Definition ).

Definition (Theta)        Consider a function  f( n) which is non-negative for all integers  . We say that `` f( n) is theta  g( n),'' which we write  , if and only if  f( n) is  O( g( n))  and f( n) is .

Recall that we showed in Section  that a polynomial in n, say , is . We also showed in Section  that a such a polynomial is . Therefore, according to Definition , we will write .

Definition (Little Oh)        Consider a function  f( n) which is non-negative for all integers  . We say that `` f( n) is little oh  g( n),'' which we write  f( n)= o( g( n)), if and only if  f( n) is  O( g( n)) but f( n) is  not  .

Little oh notation represents a kind of loose asymptotic bound  in the sense that if we are given that f(n)=o(g(n)), then we know that g(n) is an asymptotic upper bound since f(n)=O(g(n)), but g(n) is not an asymptotic lower bound since f(n)=O(g(n)) and  implies that .

For example, consider the function f(n)=n+1. Clearly, . Clearly too, , since not matter what c we choose, for large enough n, . Thus, we may write .

从上面的内容可知，Theta notation是Big Oh与Omega的交集，即同时满足Big Oh和Omega。