Table 1.

Complexities of $\mathbf{pf}$ operators

Operator	Worst case	Expected case
${\mathbf{pf}}_{\text{sort}}$	$O(NlogN)$	$O(NlogN)$
${\mathbf{pf}}_{\text{isort}}$	$O(N^2)$	$O(NlogN)$
${\mathbf{pf}}_{\text{lex}}$	O(N)	O(N)
${\mathbf{pf}}_{\text{smooth}}$	$O(N^2)$	$O(NlogN)$
${\mathbf{pf}}_{\text{nosort}}$	$O(N^2)$	$O(NlogN)$