渐近符号max（m，n）= O（m + n）

Question

I have studied Introduction to Algorithms by CLRS in great details,but one thing is not clear yet.

Why is max(m,n)=O(m,n)?

Please explain,it would be great help!

Answer 1

max(m, n) = O(m+n) simply means that, asymptotically speaking, max(m, n) doesn't grow more quickly than m+n . Since max(m, n) < m + n for all m, n , this must be true. Note that max(m, n) is equal either to m or n , either of which is guaranteed to be less than m + n (as long as m and n are nonnegative, which can be assumed).

Answer 2

Strictly speaking G(n) ∈ O(F(n)) means means that G(n) belongs to the infinite set of functions that are asymptotically bound under or equal to some C * F(n).

Big Oh Cheat Sheet

• Big Oh - Bound under or equal to
• Little Oh - Bound under and not equal to
• Theta - Equal to, not under or over
• Little Omega - Bound over and not equal to
• Big Omega - Bound over and or equal to

Misconception

Expressing that something = O(f(n)) is mathematically incorrect although even most professors make this mistake, it should be something ∈ O(f(n))

So it is true that Max(M, N) ∈ O(M + N) because Max(M, N) is asymptotically bound under or equal to M + N.

So it is true that 1 ∈ O(log n) ∈ O(n) ∈ O(n^2) ∈ O(n^2) ∈ O(n!).

This took me some time to get my head around but it's very easy once you do. It's critical to fully grasp this once you get into more advanced topics in algorithms and data structures.