Finding a shortest diagonal of a simple polygon in linear time

A diagonal of a planar, simple polygon P is an open line segment that connects two nonadjacent vertices and lies in the relative interior of P. We present a linear time algorithm for finding a shortest diagonal (in the L(2) norm) of a simple polygon, improving the previous best result by a factor of log n. Our result provides an interesting contrast to a known Omega(n log n) lower bound for finding a closest pair of vertices in a simple polygon-observe that a shortest diagonal is defined by a closest pair of vertices satisfying an additional visibility constraint.