We describe a randomized algorithm for computing the trapezoidal decomposition
of a simple polygon. Its expected running time is linear in the size of the polygon.
By a well-known and simple linear time reduction, this implies a linear time
algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle's (1991) celebrated optimal deterministic algorithm. The new algorithm can be viewed as a combination of Chazelle's algorithm and of
simple non-optimal randomized algorithms due to Clarkson et al. (1991)
and to Seidel (1991). As in Chazelle's algorithm, it is indispensable to include
a bottom-up preprocessing phase, in addition to the actual top-down construction.
An essential new idea is the use of random sampling on subchains of the initial
polygonal chain, rather than on individual edges as is normally done.