  In this paper, we introduce properties including groupoid comparison, pure
infiniteness and paradoxical comparison as well as a new algebraic tool called
groupoid semigroup for locally compact Hausdorff \'{e}tale groupoids. We show
these new tools help establishing pure infiniteness of reduced groupoid
$C^*$-algebras. As an application, we show a dichotomy of stably finiteness
against pure infiniteness for reduced groupoid $C^*$-algebras arising from
locally compact Hausdorff \'{e}tale minimal topological principal groupoids.
This generalizes the dichotomy obtained by B\"{o}nicke-Li and Rainone-Sims. We
also study the relation among our paradoxical comparison, $n$-filling property
and locally contracting property appeared in the literature for locally compact
Hausdorff \'{e}tale groupoids.
