Why do we ignore co-efficients in Big O notation?

Question

While searching for answers relating to \"Big O\" notation, I have seen many SO answers such as this, this, or this, but still I have not clearly understood some points.

Answer 1

An other thing is that, what I have understood, the complexity of 2N^3 + 99N^2 + 500 will be O(N^3). So how do we ignore/remove 99N^2 portion even? Will it not make difference when let's say N is one miilion?

That's right, in that case the 99N^2 term is far overshadowed by the 2N^3 term. The point where they cross is at N=49.5, much less than one million.

But you bring up a good point. Asymptotic computational complexity analysis is in fact often criticized for ignoring constant factors that can make a huge difference in real-world applications. However, big-O is still a useful tool for capturing the efficiency of an algorithm in a few syllables. It's often the case that an n^2 algorithm will be faster in real life than an n^3 algorithm for nontrivial n, and it's almost always the case that a log(n) algorithm will be much faster than an n^2 algorithm.

In addition to being a handy yardstick for approximating practical efficiency, it's also an important tool for the theoretical analysis of algorithm complexity. Many useful properties arise from the composability of polynomials - this makes sense because nested looping is fundamental to computation, and those correspond to polynomial numbers of steps. Using asymptotic complexity analysis, you can prove a rich set of relationships between different categories of algorithms, and that teaches us things about exactly how efficiently certain problems can be solved.