Asymptotic Notation: Big-Oh, Big-Omega, Big-Theta, Small-Oh, Small-Omega

Big-Oh (upper bond / worst-case)

If there exists positive constants $c$ and $n_0$, s.t. $f(n)\le c\cdot g(n)$ for all $n\ge n_0$, we can denote this by $f(n)=O(g(n))$.

Big-Omega (lower bond / best-case)

If there exists positive constants $c$ and $n_0$, s.t. $f(n)\ge c\cdot g(n)$ for all $n\ge n_0$, we can denote this by $f(n)=\Omega (g(n))$.

Big-Theta(tight bond / average-case)

If $f(n)=O(g(n))$ and $f(n)=\Omega (g(n))$, we define $f(n)=\Theta (g(n))$.
$f(n)=\Theta (g(n))$ if and only if there exists positive constants $c_1$, $c_2$ and $n_0$, s.t. $c_1\cdot g(n)\le f(n)\le c_2\cdot g(n)$ for all $n\ge n_0$.

Little-Oh

If there exists positive constants $c$ and $n_0$, s.t. $f(n)<c\cdot g(n)$ for all $n\ge n_0$, we can denote this by $f(n)=o(g(n))$.

Little-Omega

If there exists positive constants $c$ and $n_0$, s.t. $f(n)>c\cdot g(n)$ for all $n\ge n_0$, we can denote this by $f(n)=\omega (g(n))$.

Others

• $\log n<\sqrt{n}<n<n^2<n^3<2^n$
• Special classes of algorithms:
• • Logarithmic: $O(\log n)$
• • Linear: $O(n)$
• • Quadratic: $O(n^2)$
• • Polynomial: $O(n^k)$, $k\ge 1$
• • Exponential: $O(a^n)$, $a>1$