  We define a measure of "complexity" of a braid which is natural with respect
to both an algebraic and a geometric point of view. Algebraically, we modify
the standard notion of the length of a braid by introducing generators
$\Delta\_{ij}$, which are Garside-like half-twists involving strings $i$
through $j$, and by counting powered generators $\Delta\_{ij}^k$ as
$\log(|k|+1)$ instead of simply $|k|$. The geometrical complexity is some
natural measure of the amount of distortion of the $n$ times punctured disk
caused by a homeomorphism. Our main result is that the two notions of
complexity are comparable. This gives rise to a new combinatorial model for the
Teichmueller space of an $n+1$ times punctured sphere. We also show how to
recover a braid from its curve diagram in polynomial time. The key r\^ole in
the proofs is played by a technique introduced by Agol, Hass, and Thurston.
