We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two- and three-dimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.

bahl~microsoft, corn ymwang~microsoft, corn rogerwa~microsoft, corn The topology of a wireless multi-hop network can be con-trolled by varying the transmission power at each node. In this paper, we give a detailed analysis of a cone-based dis-tributed topology control algorithm. This algorithm, intro-duced in [16], does not assume that nodes have GPS in-formation available; rather it depends only on directional information. Roughly speaking, the basic idea of the algo-rithm is that a node u transmits with the minimum power P~,,a required to ensure that in every cone of degree a around u, there is some node that u can reach with power Pma- We show that taking a = 57r/6 is a necessary and sufficient con-dition to guarantee that network connectivity is preserved. More precisely, if there is a path from a to t when every node communicates at maximum power then, if a < _ 5~r/6, there is still a path in the smallest symmetric graph Ga con-taining all edges (u, v) such that u can communicate with v using power p~,a. On the other hand, if ~> 51r/6, connec-tivity is not necessarily preserved. We also propose a set of optimizations that further reduce power consumption and prove that they retain network connectivity. Dynamic re-configuration in the presence of failures and mobility is also discussed. Simulation results are presented to demonstrate the effectiveness of the algorithm and the optimizations. 1.

Several localized routing protocols [1] guarantee the delivery of the packets when the underlying network topology is the Delaunay triangulation of all wireless nodes. However, it is expensive to construct the Delaunay triangulation in a distributed manner. Given a set of wireless nodes, we more accurately model the network as a unit-disk graph UDG , in which a link in between two nodes exist only if the distance in between them is at most the maximum transmission range.

We consider online routing strategies for routing between the vertices of embedded planar straight line graphs. Our results include (1) two deterministic memoryless routing strategies, one that works for all Delaunay triangulations and the other that works for all regular triangulations, (2) a randomized memoryless strategy that works for all triangulations, (3) an O(1) memory strategy that works for all convex subdivisions, (4) an O(1) memory strategy that approximates the shortest path in Delaunay triangulations, and (5) theoretical and experimental results on the competitiveness of these strategies.

Sensor networks pose a number of challenging conceptual and optimization problems such as location, deployment, and tracking [1]. One of the fundamental problems in sensor networks is the calculation of the coverage. In [1], it is assumed that the sensor has the uniform sensing ability. In this paper, we give efficient distributed algorithms to optimally solve the best-coverage problem raised in [1]. Here, we consider the sensing model: the sensing ability diminishes as the distance increases. As energy conservation is a major concern in wireless (or sensor) networks, we also consider how to find an optimum bestcoverage -path with the least energy consumption. We also consider how to find an optimum best-coverage-path that travels a small distance. In addition, we justify the correctness of the method proposed in [1] that uses the Delaunay triangulation to solve the best coverage problem. Moreover, we show that the search space of the best coverage problem can be confined to the relative neighborhood graph, which can be constructed locally.

Let G = (V

We propose a novel localized algorithm that constructs a bounded degree and planar spanner for wireless ad hoc networks modeled by unit disk graph (UDG). Every node only has to know its 2-hop neighbors to find the edges in this new structure. Our method applies the Yao structure on the local Delaunay graph [21] in an ordering that are computed locally. This new structure has the following attractive properties: (1) it is a planar graph; (2) its node degree is bounded from above by a positive constant 19 + ⌈ 2π α ⌉; (3) it is a t-spanner (given any two nodes u and v, there is a path connecting them in the structure such that its length is no more than t ≤ max { π α,πsin 2 2 +1}·Cdel times of the shortest path in UDG); (4) it can be constructed locally and is easy to maintain when the nodes move around; (5) moreover, we show that the total communication cost is O(n), where n is the number of wireless nodes, and the computation cost of each node is at most O(d log d), where d is its 2-hop neighbors in the original unit disk graph. Here Cdel is the spanning ratio of the Delaunay triangulation, which is at most 4 √ 3 9 π. And the adjustable parameter α satisfies 0 <α<π/3. In addition, experiments are conducted to show this topology is efficient in practice, compared with other well-known topologies used in wireless ad hoc networks. Previously, only centralized method [5] of constructing bounded degree planar spanner is known, with degree bound 27 and spanning ratio t ≃ 10.02. The distributed implementation of their centralized method takes O(n 2) communications in the worst case. No localized methods were known previously for constructing bounded degree planar spanner.

This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.-R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, the exact and approximate post-office problem, and the problem of constructing spanners are discussed in detail. Contents 1 Introduction 1 2 The static closest pair problem 4 2.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Algorithms that are optimal in the algebraic computation tree model . 5 2.2.1 An algorithm based on the Voronoi diagram . . . . . . . . . . . 5 2.2.2 A divide-and-conquer algorithm . . . . . . . . . . . . . . . . . . 5 2.2.3 A plane sweep algorithm . . . . . . . . . . . . . . . . . . . . . . 6 2.3 A deterministic algorithm that uses indirect addressing . . . . . . . . . 7 2.3.1 The degraded grid . . . . . . . . . . . . . . . . . . ...

We propose a new geometric spanner for static wireless ad hoc networks, which can be constructed efficiently in a localized manner. It integrates the connected dominating set and the local Delaunay graph to form a backbone of the wireless network.

We can control the topology of a multi-hop wireless network by varying the transmission power at each node. The life-time of such networks depends on battery power at each node. This paper presents a distributed fault-tolerant topology control algorithm for minimum energy consumption in these networks. More precisely, we present algorithms which preserve the connectivity of a network upon failing of, at most, k nodes (k is constant) and simultaneously minimize the transmission power at each node to some extent. In addition, we present simulations to support the effectiveness of our algorithm. We also demonstrate some optimizations to further minimize the power at each node. Finally, we show how our algorithms can be extended to 3-dimensions.