Abstract
We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include intersections of lattices with convex sets, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.
Highlights
Triangulations, partitions of the convex hull of a point set into triangles having the given points as vertices and meeting edge-to-edge, form a fundamental object of study in computational geometry, of importance due to applications including computer graphics and finite element mesh generation
Even for convex point sets, it remains unknown whether flip distances may be computed in polynomial time, in this case tight linear bounds are known on how large the flip distance can be as a function of the number of points [20, 30]
We show that the flip graph of any point set with no empty pentagon is a partial cube, a graph that can be embedded isometrically into a hypercube [8]
Summary
Triangulations, partitions of the convex hull of a point set into triangles having the given points as vertices and meeting edge-to-edge, form a fundamental object of study in computational geometry, of importance due to applications including computer graphics and finite element mesh generation. If a set of n points is in convex position (its points form the vertices of a convex polygon, with one polygon edge chosen arbitrarily as root), its triangulations are in one-to-one correspondence, by a form of planar duality, to the binary trees with n − 1 leaves [30]; see Figure 2 for an example According to this correspondence, a flip of a triangulation corresponds to a binary tree rotation, a standard operation in the theory of data structures that interchanges the heights of two adjacent tree nodes while preserving the left-to-right traversal ordering of the tree nodes [30]. For some sets of points that are not in convex position, there exist non-regular triangulations In such cases, the vertices and edges of the secondary polytope do not represent the entire flip graph. The analogous result for points with no empty quadrilateral is an immediate consequence of our Theorem 1
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