O with tilde
\(\quad\ f(n)=\tilde{O}(g(n))\)
\(\Leftrightarrow f(n)=O(g(n)\cdot{\rm log}^c n)\)
\(\Leftrightarrow \exists\) constant \(c^{'}\) s.t. \(f(n)\leq c^{'}\cdot g(n)\cdot{\rm log}^c n\) for large enough \(n\).
\(\quad\ f(n)=\tilde{O}(g(n))\)
\(\Leftrightarrow f(n)=O(g(n)\cdot{\rm log}^c n)\)
\(\Leftrightarrow \exists\) constant \(c^{'}\) s.t. \(f(n)\leq c^{'}\cdot g(n)\cdot{\rm log}^c n\) for large enough \(n\).
