问题
I'm currently taking an algorithm class, and we're covering Big O notations and such. Last time, we talked about how
O (n^2 + 3n + 5) = O(n^2)
And I was wondering, if the same rules apply to this:
O(n^2) + O(3n) + O(5) = O(n^2)
Also, do the following notations hold ?
O(n^2) + n
or
O(n^2) + Θ (3n+5)
The later n is outside of O, so I'm not sure what it should mean. And in the second notation, I'm adding O and Θ .
回答1:
At least for practical purposes, the Landau O(...) can be viewed as a function (hence the appeal of its notation). This function has properties for standard operations, for example:
O(f(x)) + O(g(x)) = O(f(x) + g(x))
O(f(x)) * O(g(x)) = O(f(x) * g(x))
O(k*f(x)) = O(f(x))
for well defined functions f(x) and g(x), and some constant k.
Thus, for your examples,
Yes: O(n^2) + O(3n) + O(5) = O(n^2)
and:O(n^2) + n = O(n^2) + O(n) = O(n^2),O(n^2) + Θ(3n+5) = O(n^2) + O(3n+5) = O(n^2)
回答2:
The notation:
O(n^2) + O(3n) + O(5) = O(n^2)
as well as, for example:
f(n,m) = n^2 + m^3 + O(n+m)
is abusing the equality symbol, as it violates the axiom of equality. To be more formally correct, you would need to define O(g(x)) as a set-valued function, the value of which is all functions that do not grow faster than g(x), and use set membership notation to indicate that a specific function is a member of the set.
Addition and multiplication is not defined for Landau's symbol (Big O).
回答3:
In complexity theory, the Landau symbols are used for sets of functions. Therefore O(*) does not represent a single function but an entire set. The + operator is not defined for sets, however, the following is commonly used when analyzing functions:
O(*) + g(n)
This usually represents a set of functions where g(n) is added to every function in O(*). The resulting set can be represented in big-O notation again.
O(*) + O(**)
This is similar. However, it behaves like a kind of cartesian product. Every function from O(**) is added to every function from O(*).
O(*) + Θ(*)
The same rules apply here. However, the result can usually not be expressed as Θ(**) because of the loosening by O(*). Expressing it as O(**) is still possible.
回答4:
Also the following notations hold
O(n^2) + n = O(n^2)
and
O(n^2) + Θ(3n+5) = O(n^2), Θ(n)
Hope it makes sense...
来源：https://stackoverflow.com/questions/30168800/can-you-do-addition-multiplication-with-big-o-notations