Are O(n+m) and O(n) notations equivalent if m

Question

I am reading about the Rabin-Karp algorithm on Wikipedia and the time complexity mentioned in there is O(n+m). Now, from my understanding, m is necessarily between 0 and n, so in the best case the complexity is O(n) and in the worst case it is also O(2n)=O(n), so why isn't it just O(n)?

Answer 1

Basically, Robin-Karp expresses it's asymptotic notation as O(m+n) as a means to express the fact that it takes linear time relative to m+n and not just n. Essentially, the variables, m and n, have to mean something whenever you use asymptotic notation. For the case of the Robin-Karp algorithm, n represents the length of the text, and m represents the combined length of both the text and the pattern. Note that O(2n) means the same thing as O(n), because O(2n) is still a linear function of just n. However, in the case of Robin-Karp, m+n isn't really a function of just n. Rather, it's a function of both m and n, which are two independent variables. As such, O(m+n) doesn't mean the same thing as O(n) in the same way that O(2n) equates to O(n).

I hope that makes sense. :-P

Answer 2

m and n measure different dimensions of the input data. Text of length n and patterns of length m is not the same as text of length 2n and patterns of length 0.

O(m+n) tells us that the complexity is proportional to both the length of the text and the length of the patterns.

Answer 3

There are some scenarios where saying complexity in form of O(n+m) is suitable than just saying O(max(m,n)).

Scenario:

Consider BFS(Breadth First Search) or DFS(Depth First Search) as Scenario. It will be more intuitive and will convey more information to say that the complexity is O(E+V) than max{E,V}. The former is in Sync with actual algorithmic description.