Comparing functions asymptotically

Question

I am given two functions: F₁(n)=2n+20 and F₂(n)=n+1. I have to show which one is better.

Our lecturer solved a similar problem. Given F₁(n)=n² and F₂(n)=2n+20, he did:

F₂(n)/F₁(n)=(2/n)+(20/n²)

and he said it would be always less than 22, hence 2n+20 is better.

My doubt is how do I solve these types of comparison between functions. I have gone through all questions previously asked here and didn't quite understand it.

Also if given (an²+bn+c)=O(n²), please help in choosing constants c₁, c₂, n₀ such that

c₁g(n) ≤ f(n) ≤ c₂g(n).

Sorry my mistake Between n^2 and 2n+20 The latter function performs better — user9440643, Mar 04 '18 at 07:35
That depends on your definition of *good*. Obviously, `2n+20` is always greater than `n+1` for positive `n`. — Nico Schertler, Mar 04 '18 at 07:45
Yes @nico but can you prove that by some mathematical means? I would appreciate it — user9440643, Mar 04 '18 at 07:49
Well, the difference `(2n + 20) - (n + 1) = n + 19`. And this difference is obviously positive for all `n > -19`. Therefore, `2n + 20` is greater than `n + 1` for all `n > -19`. — Nico Schertler, Mar 04 '18 at 07:51

score 0 · Answer 1 · answered Mar 04 '18 at 08:46

As stated in one of the comment, what a better complexity is first needs to be defined.

It is likely your lecturer means a complexity C(n) better than K(n) if and only if C(n) << K(n) (or using Landau notations C(n) = o(K(n))) as n -> +Inf.

If you want to compare two complexities F1(n) and F2(n), one method is to check if the quotient F2(n)/F1(n) converges and if so, to which value. With F2(n) ~ n and F1(n) ~ n^2, F2(n)/F1(n) converges towards 0 and therefore F1(n) dominates F2(n) towards infinity. You would say F1(n) is better given the language convention adopted earlier.

Note that "it (F2(n)/F1(n)) would be always less than 22 and told 2n+20 is better" is not a correct approach to asymptotic analysis in case of domination relationships (o(.) or equivalently <<)(I doubt your lecturer said that for the F1(n) and F2(n) defined above.).

In your last question - could you please define f(n) and g(n) ?

Comparing functions asymptotically

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