Exceptional collections on toric varieties and birational contractions

We show the existence of a full strong exceptional collection on every
smooth toric Fano $3$-fold.
This result is already known by Bernardi and Tirabassi under assuming
Bondal's conjecture,
which states that the Frobenius push-forward of the structure sheaf
$\mathcal O_X$ generates the derived category $D^b(X)$ for all
smooth projective toric varieties $X$.
In the proof, we show Bondal's conjecture for smooth toric Fano $3$-folds and
also improve the proof due to Bernardi and Tirabassi by using the birational
geometry.