Computing Simple Paths on Points in Simple Polygons

Given a simple polygon P with m vertices, a set X = {x(1), x(2), ..., x(n)} of n points within P, a start point s is an element of X, and an end point t is an element of X, we give an O(n(3) log m + mn) tune, O(n(3) + m) space algorithm to find O(n(2)) simple polygonal paths from s to t, that have their vertices among the points in X and stay inside P, or report that no such path exists.