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 edgeflipping 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.