在图中的每个节点上运行 DFS 的时间复杂度是多少？

Question

I have to write a function that performs DFS on each node in a directed graph. I believe the run time would be O(n^2+mn) but I am not positive about this. Would this be considered polynomial time?

Answer 1

Yes, it would be considered polynomial:

Everything thing that has a time complexity of O( n^k ), n \\in \\N is polynomial.

You have O( n^2 + mn ), m can be written as m = n u, u \\in \\R; Now you can insert if: n^2 + u n n = n^2 + u n^2 = ( 1 + u ) * n^2. ( 1 + u ) is a constant value, so it is negligible.

Therefore, you have the form of a polynomial time: O( n^2 ) (k=2).