Given a simple n-vertex polygon, the triangulation problem is to partition the interior of the polygon into n-2 triangles by adding n-3 nonintersecting diagonals. We propose an O(n log logn)-time algorithm for this problem, improving on the previously best bound of O (n log n) and showing that triangulation is not as hard as sorting. Improved algorithms for several other computational geometry problems, including testing whether a polygon is simple, follow from our result.

We have developed techniques which contribute to efficient algorithms for certain geometric optimiza-tion problems involving simple polygons: computing minimum separators, maximum inscribed triangles,

The goal of this paper is to show that the concept of the shortest path inside a polygonal region contributes to the design of efficient algorithms for certain geometric optimization problems involving simple polygons: computing optimum separators, maximum area or perimeter inscribed triangles, a minimum area circumscribed concave quadrilateral, or a maximum area contained triangle. The structure for our algorithms is as follows: a) decompose the initial problem into a low-degree polynomial number of optimization problems; b) solve each individual subproblem in constant time using standard methods of calculus, basic methods of numerical analysis, or linear programming. These same optimization techniques can be applied to splinegons (curved polygons). To do this, we first develop a decomposition technique for curved polygons which we substitute for triangulation in creating equally efficient curved versions of the algorithms for the shortest-path tree, ray-shooting and two-point shortes...

Important applications in science and engineering, such as modeling traffic flow, seismic waves, electromagnetics, and the simulation of mechanical stresses in materials, require the high-fidelity numerical solution of hyperbolic partial differential equations (PDEs) in space and time variables. Many interesting physical problems involve nonlinear and anisotropic behavior, and the PDEs modeling them exhibit discontinuities in their solutions. Spacetime discontinuous Galerkin (SDG) finite element methods are used to solve such PDEs arising from wave propagation phenomena. To support an accurate and efficient solution procedure using SDG methods and to exploit the flexibility of these methods, we give a meshing algorithm to construct an unstructured simplicial spacetime mesh over an arbitrary sim-plicial space domain. Our algorithm is the first spacetime meshing algorithm suitable for efficient solution of nonlinear phenomena in anisotropic media using novel discontinuous Galerkin finite element methods for implicit solutions di-rectly in spacetime. Given a triangulated d-dimensional Euclidean space domain M (a simplicial complex) and initial conditions of the underlying hyperbolic spacetime PDE, we construct an unstructured simplicial mesh of the (d + 1)-dimensional spacetime domain M × [0,∞). Our algorithm uses a near-optimal number of spacetime elements, each with bounded temporal aspect ratio for any finite prefix M × [0,T] of spacetime. Unlike Delaunay meshes, the facets of our mesh satisfy gradient constraints that allow interleaving the construction of the mesh by adding new space-

This paper presents a brief survey of some problems and solutions related to the triangulation of surfaces. A surface (a two dimensional manifold, in the context of this paper) can be represented as a three dimensional function on a planar disk. In that sense, the triangulation of the disk induces a triangulation of the surface. Hence the emphasis of this paper is on triangulation on a plane. Apart from the issues in triangulation, this survey talks about the known upper and lower bounds on various triangulation problems. It is intended as a broad compilation of known results rather than an intensive treatise, and the details of most algorithms are skipped. 1 Introduction This survey assumes familiarity with the fundamental concepts of computational geometry. We define the triangulation problem as follows: Input: i. A set S of points, fp i g, such that each p i lies on the surface ii. A set of conditions, fC i g Output: A set S 0 of triples f(p i 1 ; p i 2 ; p i 3 )g such that e...

The use of 3D models is gaining popularity since they are important for computer graphics applications. Recently, similarity retrieval techniques for 3D models have been investigated intensively for handling databases of 3D models systematically. The techniques extract shape descriptors from 3D models and use these descriptors for indices for comparing shape similarities. Various shape descriptors have been proposed for improving shape similarity search results. Most similarity search techniques focus on comparisons of each individual 3D model from databases. However, our similarity search technique can compare not only each individual 3D model, but also partial shape similarities. The partial shape matching technique extends the user’s query request by (1) finding similar parts of 3D models, (2) finding 3D models which contain similar parts, and (3) counting the number of similar portions automatically. We have implemented an experimental partial shape-matching search system for 3D model databases, and preliminary experiments show that the system successfully retrieves similar 3D model parts efficiently.

Energy efficiency has become increasingly critical in designing and operating wireless networks, especially for mobile ad hoc networks consisting of portable mobile wireless computing/communication devices powered by limited battery capacity. Since the energy required to transmit a given amount of data is a convex and monotonically increasing function of the transmission rate [5, 12], theoretically one can improve energy efficiency by transmitting data at lower rates. Unfortunately, low data rates result in longer transmission duration and larger communication delay at receiving end, which is usually undesirable. How to optimally schedule transmission process to both minimize the total power consumption and observe all time constraints (available times and transmission deadlines) is a challenging and interesting problem. In this paper, we propose a technique to solve the above rate scheduling problem by transforming it into finding the shortest path between two vertices of a two dimensional polygon, which yields an elegant analytical solution and easy-to-prove optimality. To the best of our knowledge, this is the first solution to the rate scheduling problem in its general form.

In this paper we investigate several variations of the following problem: Given a collection of pairwise disjoint polygons and their spatial positions in the plane, cover each with a polygonal hull such that (i) the hulls are pairwise disjoint, and

Given a set S of n points in the plane, each point having one of c colors, the bichromatic all-nearestneighbors problem is the task to find (in the set S) a closest point of different color for each of the n points in S. We consider a dynamic variant of this problem where the points are fixed but can change color. More precisely, we consider restricted problem instances, which allow us to improve over the time needed for solving the problem from scratch after each color change. In these variants we maintain, in O(n) time per color change, a data structure of size O(cn) or O(n), with which the closest neighbor of different color of any point in S can be found in time O(log n), and the restrictions allow us to bound the number of look ups that are necessary in each step.