Adapting (Pseudo)-Triangulations with a Near-Linear Number of Edge Flips

Abstract

In geometric data processing, structures that partition the geometric input, as well as connectivity structures for geometric objects, play an important role. A versatile tool in this context are triangular meshes, often called triangulations; see e.g., the survey articles [6, 12, 5]. A triangulation of a finite set S of points in the plane is a maximal planar straight-line graph that uses all and only the points in S as its vertices. Each face in a triangulation is a triangle spanned by S. In the last few years, a relaxation of triangulations, called pseudo-triangulations (or geodesic triangulations), has received considerable attention. Here, faces bounded by three concave chains, rather than by three line segments, are allowed. The scope of applications of pseudo-triangulations as a geometric data stucture ranges from ray shooting [10, 14] and visibility [25, 26] to kinetic collision detection [1, 21, 22], rigidity [32, 29, 15], and guarding [31]. Still, only very recently, results on the combinatorial properties of pseudo-triangulations have been obtained. These include bounds on the minimal vertex and face degree [20] and on the number of possible pseudo-triangulations [27, 3]. The usefulness of (pseudo-)triangulations partially stems from the fact that these structures can be modified by constant-size combinatorial changes, commonly called flip operations. Flip operations allow for an adaption to local requirements, or even for generating globally optimal structures [6, 12]. A classical result states that any two triangulations of a given planar point set can be made to coincide by applying a quadratic number of edge flips; see e.g. [16, 19]. A similar result has been proved recently for the class of minimum pseudo-triangulations [8, 29].