The Edge-Flipping Distance of Triangulations

An edge-flipping operation in a triangulation T of a set of points in the plane is a local restructuring that changes T into a triangulation that differs from T in exactly one edge. The edge-flipping distance between two triangulations of the same set of points is the minimum number of edge-flipping operations needed to convert one into the other. In the context of computing the rotation distance of binary trees Sleator, Tarjan, and Thurston show an upper bound of 2n - 10 on the maximum edge-flipping distance between triangulations of convex polygons with n nodes, n > 12. Using volumetric arguments in hyperbolic 3-space they prove that the bound is tight. In this paper we establish an upper bound on the edge-flipping distance between triangulations of a general finite set of points in the plane by showing that no more edge-flipping operations than the number of intersections between the edges of two triangulations are needed to transform these triangulations into another, and we present an algorithm that computes such a sequence of edge-flipping operations.