Asymptotic comparison of functions

One-dimensional case

Let [math]\displaystyle{ f:\mathbb{R}\rightarrow\mathbb{R}^+_0 }[/math] be a function. The following sets (a.k.a. classes) of functions are defined for [math]\displaystyle{ f }[/math]:

1. [math]\displaystyle{ \mathcal{O}(f) }[/math] consists of all functions [math]\displaystyle{ g:\mathbb{R}\rightarrow\mathbb{R}^+_0 }[/math] such that there are [math]\displaystyle{ N_g,\,c_g\in\mathbb{R} }[/math] that fulfill [math]\displaystyle{ g(n)\leq c_g\cdot f(n) }[/math] for all [math]\displaystyle{ n\geq N_g }[/math], [math]\displaystyle{ n\in\mathbb{R} }[/math];
2. [math]\displaystyle{ \Omega(f) }[/math] consists of all functions [math]\displaystyle{ g:\mathbb{R}\rightarrow\mathbb{R}^+_0 }[/math] such that there are [math]\displaystyle{ N_g,\,c_g\in\mathbb{R} }[/math] that fulfill [math]\displaystyle{ g(n)\geq\frac{1}{c_g}\cdot f(n) }[/math] for all [math]\displaystyle{ n\geq N_g }[/math], [math]\displaystyle{ n\in\mathbb{R} }[/math];
3. [math]\displaystyle{ \Theta(f):=\mathcal{O}(f)\cap\Omega(f) }[/math];
4. [math]\displaystyle{ o(f):=\mathcal{O}(f)\setminus\Theta(f) }[/math].
5. [math]\displaystyle{ \omega(f):=\Omega(f)\setminus\Theta(f) }[/math].

Remark: This notation is usually called the big O notation or asymptotic notation and is also known as the Landau symbols or Landau-Bachmann symbols.

Mathematical rules for asymptotic comparison

Let [math]\displaystyle{ f,g,h:\mathbb{R}\rightarrow\mathbb{R}^+_0 }[/math] be three functions.

1. Anti-reflexivity: If [math]\displaystyle{ f\in\mathcal{O}(g) }[/math], it is [math]\displaystyle{ g\in\Omega(f) }[/math], and vice versa.
2. Transitivity: If [math]\displaystyle{ f\in\oplus(g) }[/math] and [math]\displaystyle{ g\in\oplus(h) }[/math] then [math]\displaystyle{ f\in\oplus(h) }[/math], where "[math]\displaystyle{ \oplus }[/math]" is anyone of "[math]\displaystyle{ \mathcal{O} }[/math]", "[math]\displaystyle{ \Omega }[/math]", "[math]\displaystyle{ \Theta }[/math]", "[math]\displaystyle{ o }[/math]", and "[math]\displaystyle{ \omega }[/math]".
3. It is [math]\displaystyle{ \mathcal{O}(f)\cup\mathcal{O}(g)\subseteq\mathcal{O}(f+g) }[/math].
4. If [math]\displaystyle{ f\in\mathcal{O}(g) }[/math], it is [math]\displaystyle{ \oplus(f+g)=\oplus(g) }[/math], where "[math]\displaystyle{ \oplus }[/math]" is anyone of "[math]\displaystyle{ \mathcal{O} }[/math]", "[math]\displaystyle{ \Omega }[/math]", and "[math]\displaystyle{ \Theta }[/math]".
5. It is [math]\displaystyle{ f\in\mathcal{O}(g) }[/math] if, and only if, the limit superior of the series [math]\displaystyle{ f(n)/g(n) }[/math] for [math]\displaystyle{ n\rightarrow+\infty }[/math] is finite.
6. It is [math]\displaystyle{ f\in o(g) }[/math] if, and only if, this limit superior is zero. Note that, due to nonnegativity, this is equivalent to the statement that [math]\displaystyle{ \lim_{n\rightarrow+\infty}f(n)/g(n) }[/math] exists and equals zero.
7. For [math]\displaystyle{ a,b\in\mathbb{R} }[/math], [math]\displaystyle{ a,b\gt 1 }[/math], it is [math]\displaystyle{ \oplus(\log_a(n))=\oplus(\log_b(n)) }[/math], where "[math]\displaystyle{ \oplus }[/math]" is anyone of "[math]\displaystyle{ \mathcal{O} }[/math]", "[math]\displaystyle{ \Omega }[/math]", "[math]\displaystyle{ \Theta }[/math]", "[math]\displaystyle{ o }[/math]", and "[math]\displaystyle{ \omega }[/math]" (follows immediately from the basic rule [math]\displaystyle{ \log_a(n)/\log_b(n)=\log_a(b)= }[/math] const). In particular, the base of a logarithm function may be omitted: [math]\displaystyle{ \oplus(\log(n))=\oplus(\log_a(n)) }[/math].
8. For all [math]\displaystyle{ k,\ell\in\mathbb{R} }[/math], [math]\displaystyle{ k\lt \ell }[/math], it is [math]\displaystyle{ n^k\in o(n^\ell) }[/math].
9. For all [math]\displaystyle{ k\in\mathbb{R}^+ }[/math], it is [math]\displaystyle{ \log^k(n)\in o(n) }[/math].
10. For all [math]\displaystyle{ k,a\in\mathbb{R} }[/math], [math]\displaystyle{ a\gt 1 }[/math], it is [math]\displaystyle{ n^k\in o(a^n) }[/math].
11. For all [math]\displaystyle{ a,b\in\mathbb{R} }[/math], [math]\displaystyle{ 1\lt a\lt b }[/math], it is [math]\displaystyle{ a^n\in o(b^n) }[/math].
12. For all [math]\displaystyle{ a\in\mathbb{R} }[/math], [math]\displaystyle{ a\gt 1 }[/math], it is [math]\displaystyle{ a^n\in o(n!) }[/math].

Comparison with specific functions

A function [math]\displaystyle{ f:\mathbb{R}\rightarrow\mathbb{R}^+_0 }[/math] is said to be

1. linear if [math]\displaystyle{ f\in\Theta(n) }[/math];
2. quadratic if [math]\displaystyle{ f\in\Theta(n^2) }[/math];
3. cubic if [math]\displaystyle{ f\in\Theta(n^3) }[/math];
4. logarithmic if [math]\displaystyle{ f\in\Theta(\log(n)) }[/math];
5. "n-log-n" if [math]\displaystyle{ f\in\Theta(n\cdot\log(n)) }[/math];
6. polynomial if there is a polynomial [math]\displaystyle{ p:\mathbb{R}\rightarrow\mathbb{R} }[/math] such that [math]\displaystyle{ f\in\mathcal{O}(p) }[/math];
7. subexponential if [math]\displaystyle{ f\in o(a^n) }[/math] for every [math]\displaystyle{ a\in\mathbb{R} }[/math], [math]\displaystyle{ a\gt 1 }[/math];
8. exponential if there are [math]\displaystyle{ a,b\in\mathbb{R} }[/math], [math]\displaystyle{ a\gt b\gt 1 }[/math], such that [math]\displaystyle{ f\in\mathcal{O}(a^n) }[/math] and [math]\displaystyle{ f\in\Omega(b^n) }[/math];
9. factorial if [math]\displaystyle{ f\in\Theta(n!) }[/math].

Remark: Note that the notion of polynomial is based on an "[math]\displaystyle{ \mathcal{O} }[/math]", not on a "[math]\displaystyle{ \Theta }[/math]". In fact, in this context, "polynomial" is usually used short for "polynomially bounded from above".

Multidimensional case

Let [math]\displaystyle{ k\in\mathbb{R} }[/math] and let [math]\displaystyle{ f:\mathbb{R}^k\rightarrow\mathbb{R}^+_0 }[/math]. The following sets (a.k.a. classes) are defined for [math]\displaystyle{ f }[/math]:

1. [math]\displaystyle{ \mathcal{O}(f) }[/math] consists of all functions [math]\displaystyle{ g:\mathbb{R}^k\rightarrow\mathbb{R}^+_0 }[/math] such that there are [math]\displaystyle{ N_g,\,c_g\in\mathbb{R} }[/math] that fulfill [math]\displaystyle{ g(n_1,\ldots,n_k)\leq c_g\cdot f(n_1,\ldots,n_k) }[/math] for all [math]\displaystyle{ n_1,\ldots,n_k\in\mathbb{R} }[/math] such that [math]\displaystyle{ n_1,\ldots,n_k\geq N_g }[/math];
2. [math]\displaystyle{ \Omega(f) }[/math] consists of all functions [math]\displaystyle{ g:\mathbb{R}^k\rightarrow\mathbb{R}^+_0 }[/math] such that there are [math]\displaystyle{ N_g,\,c_g\in\mathbb{R} }[/math] that fulfill [math]\displaystyle{ g(n_1,\ldots,n_k)\geq c_g\cdot f(n_1,\ldots,n_k) }[/math] for all [math]\displaystyle{ n_1,\ldots,n_k\in\mathbb{R} }[/math] such that [math]\displaystyle{ n_1,\ldots,n_k\geq N_g }[/math];
3. [math]\displaystyle{ \Theta(f):=\mathcal{O}(f)\cap\Omega(f) }[/math];
4. [math]\displaystyle{ o(f):=\mathcal{O}\setminus\Theta(f) }[/math].
5. [math]\displaystyle{ \omega(f):=\Omega(f)\setminus\Theta(f) }[/math].