A simple and relatively efficient triangulation of then-cube (original) (raw)

Asymptotically Efficient Triangulations of the d -Cube

Discrete and Computational Geometry, 2003

Let P and Q be polytopes, the first of "low" dimension and the second of "high" dimension. We show how to triangulate the product P × Q efficiently (i.e., with few simplices) starting with a given triangulation of Q. Our method has a computational part, where we need to compute an efficient triangulation of P × m , for a (small) natural number m of our choice. m denotes the m-simplex. Our procedure can be applied to obtain (asymptotically) efficient triangulations of the cube I n : We decompose I n = I k × I n−k , for a small k. Then we recursively assume we have obtained an efficient triangulation of the second factor and use our method to triangulate the product. The outcome is that using k = 3 and m = 2, we can triangulate I n with O(0.816 n n!) simplices, instead of the O(0.840 n n!) achievable before.

Properties of n-dimensional triangulations

Computer Aided Geometric Design, 1986

This paper establishes a number of mathematical results relevant to the problem of constructing a triangulation, i.e., a simplical tessellation of the convex hull of an arbitrary finite set of points in n-space.

More Efficient Generation of Plane Triangulations

Lecture Notes in Computer Science, 2004

In this paper we give an algorithm to generate all biconnected plane triangulations having exactly n vertices including exactly r vertices on the outer face. The algorithm uses O(n) space in total and generates such triangulations without duplications in O(rn) time per triangulation, while the previous best algorithm generates such triangulations in O(r 2 n) time per triangulation.

Efficient Generation of Plane Triangulations without Repetitions

Lecture Notes in Computer Science, 2001

A "based" plane triangulation is a plane triangulation with one designated edge on the outer face. In this paper we give a simple algorithm to generate all biconnected based plane triangulations with at most n vertices. The algorithm uses O(n) space and generates such triangulations in O(1) time per triangulation without duplications. The algorithm does not output entire triangulations but the difference from the previous triangulation. By modifying the algorithm we can generate all biconnected based plane triangulation having exactly n vertices including exactly r vertices on the outer face in O(1) time per triangulation without duplications, while the previous best algorithm generates such triangulations in O(n 2) time per triangulation. Also we can generate without duplications all biconnected (non-based) plane triangulations having exactly n vertices including exactly r vertices on the outer face in O(r 2 n) time per triangulation, and all maximal planar graphs having exactly n vertices in O(n 3) time per graph.

Efficient generation of triconnected plane triangulations

Computational Geometry, 2004

A "rooted" plane triangulation is a plane triangulation with one designated vertex on the outer face. A simple algorithm to generate all triconnected rooted plane triangulations with at most n vertices is presented. The algorithm uses O(n) space and generates such triangulations in O(1) time per triangulation without duplications. The algorithm does not output entire triangulations but the difference from the previous triangulation. By modifying the algorithm all triconnected rooted plane triangulations having exactly n vertices including exactly r vertices on the outer face in O(r) time per triangulation can be generated without duplicates, while the previous best algorithm generates such triangulations in O(n 2) time per triangulation. All triconnected (non-rooted) plane triangulations having exactly n vertices including exactly r vertices on the outer face can also be generated without duplicates in O(r 2 n) time per triangulation, and all maximal planar graphs can be generated in O(n 3) time per graph.

Polynomial-Size Nonobtuse Triangulation

1992

We describe methods for triangulating polygonal regions of the plane so that no triangle has a large angle. Our main result is that a polygon with n sides can be triangulated with O(n 2) nonobtuse triangles. We also show that any triangulation (without Steiner points) of a simple polygon has a refinement with O(n 4) nonobtuse triangles. Finally we show that a triangulation whose dual is a path has a refinement with only O(n 2) nonobtuse triangles.

Combinatories and Triangulations

The problem searching for an optimal triangulation with required properties (in a plane) is solved in this paper. Existing approaches are shortly introduced here and, specially, this paper is dedicated to the brute force methods. Several new brute force methods that solve the problem from different points of view are described here. Although they have NP time complexity, we accelerate the time needed for computation maximally to get results of as large sets of points as possible. Note that our goal is to design the method that can be used for arbitrary criterion without another prerequisite. Therefore, it can serve as a generator of optimal triangulations. For example, those results can be used in verification of developed heuristic methods or in other problems where accurate results are needed and no methods for required criterion have been developed yet.

Polygon triangulation inO(n log logn) time with simple data structures

Discrete & Computational Geometry, 1992

We give a new O(n log log n)-time deterministic algorithm for triangulating simple n-vertex polygons, which avoids the use of complicated data structures. In addition, for polygons whose vertices have integer coordinates of polynomially bounded size, the algorithm can be modified to run in O(n log* n) time. The major new techniques employed are the efficient location of horizontal visibility edges that partition the interior of the polygon into regions of approximately equal size, and a linear-time algorithm for obtaining the horizontal visibility partition of a subchain of a polygonal chain, from the horizontal visibility partition of the entire chain. The latter technique has other interesting applications, including a linear-time algorithm to convert a Steiner triangulation of a polygon into a true triangulation.