Energy conservation is a critical issue in designing wireless ad hoc networks, as the nodes are powered by batteries only. Given a set of wireless network nodes, the directed weighted transmission graph Gt has an edge uv if and only if node v is in the transmission range of node u and the weight of uv is typically defined as II,,vll + c for a constant 2 <_ t ~ < 5 and c> O. The minimum power topology Gm is the smallest subgraph of Gt that contains the shortest paths between all pairs of nodes, i.e., the union of all shortest paths. In this paper, we described a distributed position-based networking protocol to construct an enclosure graph G~, which is an approximation of Gin. The time complexity of each node u is O(min(dG ~ (u)dG ~ (u), dG ~ (u) log dG ~ (u))), where dc(u) is the degree of node u in a graph G. The space required at each node to compute the minimum power topology is O(dG ~ (u)). This improves the previous result that computes Gm in O(dG, (u) a) time using O(dGt(U) 2) spaces. We also show that the average degree dG,(u) is usually a constant, which is at most 6. Our result is first developed for stationary network and then extended to mobile networks. I.

A sliver is a tetrahedron whose four vertices lie close to a plane and whose orthogonal projection to that plane is a convex quadrilateral with no short edge. Slivers are notoriously common in 3-dimensional Delaunay triangulations even for well-spaced point sets. We show that if the Delaunay triangulation has the ratio property introduced in [15] then there is an assignment of weights so the weighted Delaunay triangulation contains no slivers. We also give an algorithm to compute such a weight assignment.

Delaunay meshes with bounded circumradius to shortest edge length ratio have been proposed in the past for quality meshing. The only poor quality tetrahedra called slivers that can occur in such a mesh can be eliminated by the sliver exudation method. This method has been shown to work for periodic point sets, but not with boundaries. Recently a randomized point-placement strategy has been proposed to remove slivers while conforming to a given boundary. In this paper we present a deterministic algorithm for generating a weighted Delaunay mesh which respects the input boundary and has no poor quality tetrahedron including slivers. This success is achieved by combining the weight pumping method for sliver exudation and the Delaunay refinement method for boundary conformation. We show that an incremental weight pumping can be mixed seamlessly with vertex insertions in our weighted Delaunay refinement paradigm. 1

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A triangular mesh in 3D is a decomposition of a given geometric domain into tetrahedra. The mesh is wellshaped if the aspect ratio of every of its tetrahedra is bounded from aboveby a constant. It is Delaunayifthe interior of the circum-sphere of each of its tetrahedra does not contain any other mesh vertices. Generating a well-shaped Delaunay mesh for any 3D domain has been a long term outstanding problem. In this paper, wepresent an efficient 3D Delaunay meshing algorithm that mathematically guarantees the well-shape quality of the mesh, if the domain does not have acute angles. The main ingredient of our algorithm is a novel refinement technique which systematically forbids the formation of slivers, a family of bad elements that none of the previous known algorithms can cleanly remove, especially near the domain boundary --- needless to say, that our algorithm ensure that there is no sliver near the boundary of the domain.

Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearest-neighbor searching, clustering, finite-element mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms in these application domains begin by constructing the Delaunay triangulation or Voronoi diagram of a set of points in R³. Since three-dimensional Delaunay triangulations can have complexity Ω(n²) in the worst case, these algorithms have worst-case running time \Omega (n2). However, this behavior is almost never observed in practice except for highly-contrived inputs. For all practical purposes, three-dimensional Delaunay triangulations appear to have linear complexity. This frustrating

a conformal Delaunay mesh in arbitrary dimension with guaranteed mesh size and quality. Our algorithm runs in output-sensitive time O(nlog(L/s) + m), with constants depending only on dimension and on prescribed element shape quality bounds. For a large class of inputs, including integer coordinates, this matches the optimal time bound of Θ(n log n + m). Our new technique uses interleaving: we maintain a sparse mesh as we mix the recovery of input features with the addition of Steiner vertices for quality improvement. 1

. A key step in the finite element method is to generate a high quality mesh that is as small as possible for an input domain. Several meshing methods and heuristics have been developed and implemented. Methods based on advancing front, Delaunay triangulations, and quadtrees/octrees are among the most popular ones. Advancing front uses simple data structures and is efficient. Unfortunately, in general, it does not provide any guarantee on the size and quality of the mesh it produces. On the other hand, the circle-packing based Delaunay methods generate a well-shaped mesh whose size is within a constant factor of the optimal. In this paper, we present a new meshing algorithm, the biting method, which combines the strengths of advancing front and circle packing. We prove that it generates a high quality 2D mesh, and the size of the mesh is within a constant factor of the optimal. Keywords. unstructured mesh generation, advancing front, paving, circle packing, biting. 1 Introduction A...

We present a Delaunay refinement algorithm for meshing a piecewise smooth complex in three dimensions. The algorithm protects edges with weighted points to avoid the difficulty posed by small angles between adjacent input elements. These weights are chosen to mimic the local feature size and to satisfy a Lipschitz-like property. A Delaunay refinement algorithm using the weighted Voronoi diagram is shown to terminate with the recovery of the topology of the input. Guaranteed bounds on the aspect ratios, normal variation and dihedral angles are also provided. To this end, we present new concepts and results including a new definition of local feature size and a proof for a generalized topological ball property. 1

A bounded aspect-ratio coarsening sequence of an unstructured mesh M 0 is a sequence of meshes M 1 ; : : : ; M k such that: ffl M i is a bounded aspect-ratio mesh, and ffl M i is an approximation of M i\Gamma1 that has fewer elements, where a mesh is called a bounded aspect-ratio mesh if all its elements are of bounded aspectratio. The sequence is node-nested if the set of the nodes of M i is a subset of that of M i\Gamma1 . The problem of constructing good quality coarsening sequences is a key step for hierarchical and multi-level numerical calculations. In this paper, we give an algorithm for finding a bounded aspect-ratio, node--nested, coarsening sequence that is of optimal size: that is, the number of meshes in the sequence, as well as the number of elements in each mesh, are within a constant factor of the smallest possible. 1 Introduction Numerical methods such as the finite element, finite difference, and finite volume methods apply the following basic steps to sol...