CSP (NOIP) preliminary analysis of the time complexity of sorting

Original link: https://www.luogu.org/discuss/show/155153

given

$T(n)=\begin{cases}1,&n=1\\a T(\dfrac{n}{b})+f(n),&\text{otherwise}\end{cases}$

Calculate $T(n)$。

make $c_ {crit} = \ log_ba$：

• If the $f(n)=O(n^c)$and $c<c_{crit}$There $T(n)=\Theta(n^{c_{crit}})$
• If the $\exists k$ such that $f(n)=n^{c_{crit}}\log^kn$ , there are: $T(n)=\begin{cases}\Theta(n^{c_{crit}}\log^{k+1}n),&k>-1\\\Theta(n^{c_{crit}}\log\log n),&k=-1\\\Theta(n^{c_{crit}}),&k<-1\end{cases}$
• If the $f(n)=\Omega(n^c)$and $c>c_{crit}$There $T(n)=\Theta(f(n))$