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145
Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to ..."
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Cited by 187 (15 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
Geometric Spanner for Routing in Mobile Networks
, 2001
"... Abstract—We propose a new routing graph, the restricted Delaunay graph (RDG), for mobile ad hoc networks. Combined with a node clustering algorithm, the RDG can be used as an underlying graph for geographic routing protocols. This graph has the following attractive properties: 1) it is planar; 2) be ..."
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Cited by 183 (18 self)
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Abstract—We propose a new routing graph, the restricted Delaunay graph (RDG), for mobile ad hoc networks. Combined with a node clustering algorithm, the RDG can be used as an underlying graph for geographic routing protocols. This graph has the following attractive properties: 1) it is planar; 2) between any two graph nodes there exists a path whose length, whether measured in terms of topological or Euclidean distance, is only a constant times the minimum length possible; and 3) the graph can be maintained efficiently in a distributed manner when the nodes move around. Furthermore, each node only needs constant time to make routing decisions. We show by simulation that the RDG outperforms previously proposed routing graphs in the context of the Greedy perimeter stateless routing (GPSR) protocol. Finally, we investigate theoretical bounds on the quality of paths discovered using GPSR. Index Terms—Geographical routing, spanners, wireless ad hoc networks. I.
Topology Control and Routing in Ad hoc Networks: A Survey
 SIGACT News
, 2002
"... this article, we review some of the characteristic features of ad hoc networks, formulate problems and survey research work done in the area. We focus on two basic problem domains: topology control, the problem of computing and maintaining a connected topology among the network nodes, and routing. T ..."
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Cited by 164 (0 self)
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this article, we review some of the characteristic features of ad hoc networks, formulate problems and survey research work done in the area. We focus on two basic problem domains: topology control, the problem of computing and maintaining a connected topology among the network nodes, and routing. This article is not intended to be a comprehensive survey on ad hoc networking. The choice of the problems discussed in this article are somewhat biased by the research interests of the author
Exact and Approximate Distances in Graphs  a survey
 In ESA
, 2001
"... We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems. ..."
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Cited by 68 (0 self)
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We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems.
Spatiotemporal Multicast in Sensor Networks
 IN SENSYS
, 2003
"... Sensor networks often involve the monitoring of mobile phenomena, which can be facilitated by a spatiotemporal multicast protocol we call "mobicast". Mobicast is a novel spatiotemporal multicast protocol featuring a delivery zone that evolves over time. Mobicast can in theory provide ab ..."
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Cited by 65 (7 self)
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Sensor networks often involve the monitoring of mobile phenomena, which can be facilitated by a spatiotemporal multicast protocol we call "mobicast". Mobicast is a novel spatiotemporal multicast protocol featuring a delivery zone that evolves over time. Mobicast can in theory provide absolute spatiotemporal delivery guarantees by limiting communication to a mobile forwarding zone whose size is determined by the global worstcase value associated with a compactness metric defined over the geometry of the network.In this work, we first studied the compactness properties of sensor networks with uniform distribution. The results of this study motivate three approaches for improving the e#ciency of spatiotemporal multicast in such networks. First, one may achieve high savings on communication overhead by slightly relaxing spatiotemporal delivery guarantees. Second, spatiotemporal multicast may exploit local compactness values for higher e#ciency for networks with non uniform spatial distribution of compactness. Third, for random uniformly distributed sensor network deployment, one may choose a deployment density to best support spatiotemporal communication. We also explored all these directions via mobicast simulation and results are presented in this paper.
Graphtheoretic scagnostics
 In Proc. 2005 IEEE Symp. on Information Visualization (INFOVIS
, 2005
"... We introduce Tukey and Tukey scagnostics and develop graphtheoretic methods for implementing their procedure on large datasets. ..."
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Cited by 54 (1 self)
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We introduce Tukey and Tukey scagnostics and develop graphtheoretic methods for implementing their procedure on large datasets.
On Greedy Geographic Routing Algorithms in SensingCovered Networks
 IN PROCEEDINGS OF MOBIHOC ’04
, 2004
"... Greedy geographic routing is attractive in wireless sensor networks due to its efficiency and scalability. However, greedy geographic routing may incur long routing paths or even fail due to routing voids on random network topologies. We study greedy geographic routing in an important class of wire ..."
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Cited by 51 (3 self)
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Greedy geographic routing is attractive in wireless sensor networks due to its efficiency and scalability. However, greedy geographic routing may incur long routing paths or even fail due to routing voids on random network topologies. We study greedy geographic routing in an important class of wireless sensor networks that provide sensing coverage over a geographic area (e.g., surveillance or object tracking systems). Our geometric analysis and simulation results demonstrate that existing greedy geographic routing algorithms can successfully find short routing paths based on local states in sensingcovered networks. In particular, we derive theoretical upper bounds on the network dilation of sensingcovered networks under greedy geographic routing algorithms. Furthermore, we propose a new greedy geographic routing algorithm called Bounded Voronoi Greedy Forwarding (BVGF) that allows sensingcovered networks to achieve an asymptotic network dilation lower than 4.62 as long as the communication range is at least twice the sensing range. Our results show that simple greedy geographic routing is an effective routing scheme in many sensingcovered networks.
Deformable spanners and applications
 In Proc. of the 20th ACM Symposium on Computational Geometry (SoCG’04
, 2004
"... For a set S of points in R d,ansspanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)spanner with O(n/ε d) edges, where ε is a spe ..."
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Cited by 49 (6 self)
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For a set S of points in R d,ansspanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)spanner with O(n/ε d) edges, where ε is a specified parameter. The key property of this spanner is that it can be efficiently maintained under dynamic insertion or deletion of points, as well as under continuous motion of the points in both the kinetic data structures setting and in the more realistic blackbox displacement model we introduce. Our deformable spanner succinctly encodes all proximity information in a deforming point cloud, giving us efficient kinetic algorithms for problems such as the closest pair, the near neighbors of all points, approximate nearest neighbor search (aka approximate Voronoi diagram), wellseparated pair decomposition, and approximate kcenters. 1
WellSpaced Points for Numerical Methods
, 1997
"... mesh generation, mesh coarsening, multigrid Abstract A numerical method for the solution of a partial differential equation (PDE) requires the following steps: (1) discretizing the domain (mesh generation); (2) using an approximation method and the mesh to transform the problem into a linear system; ..."
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Cited by 49 (2 self)
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mesh generation, mesh coarsening, multigrid Abstract A numerical method for the solution of a partial differential equation (PDE) requires the following steps: (1) discretizing the domain (mesh generation); (2) using an approximation method and the mesh to transform the problem into a linear system; (3) solving the linear system. The approximation error and convergence of the numerical method depend on the geometric quality of the mesh, which in turn depends on the size and shape of its elements. For example, the shape quality of a triangular mesh is measured by its element's aspect ratio. In this work, we shift the focus to the geometric properties of the nodes, rather than the elements, of well shaped meshes. We introduce the concept of wellspaced points and their spacing functions, and show that these enable the development of simple and efficient algorithms for the different stages of the numerical solution of PDEs. We first apply wellspaced point sets and their accompanying technology to mesh coarsening, a crucial step in the multigrid solution of a PDE. A good aspectratio coarsening sequence of an unstructured mesh M0 is a sequence of good aspectratio meshes M1; : : : ; Mk such that Mi is an approximation of Mi\Gamma 1 containing fewer nodes and elements. We present a new approach to coarsening that guarantees the sequence is also of optimal size and width up to a constant factor the first coarsening method that provides these guarantees. We also present experimental results, based on an implementation of our approach, that substantiate the theoretical claims.
Approximating the stretch factor of Euclidean paths, cycles and trees
 SIAM J. Comput
, 1999
"... Given a set S of n points in R d , and a graph G having the points of S as its vertices, the stretch factor t of G is dened as the maximal value jpqj G =jpqj, where p; q 2 S, p 6= q, jpqj G is the length of a shortest path in G between p and q, and jpqj is the Euclidean distance between p and ..."
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Cited by 45 (7 self)
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Given a set S of n points in R d , and a graph G having the points of S as its vertices, the stretch factor t of G is dened as the maximal value jpqj G =jpqj, where p; q 2 S, p 6= q, jpqj G is the length of a shortest path in G between p and q, and jpqj is the Euclidean distance between p and q. We consider the problem of designing algorithms that, for an arbitrary constant > 0, compute an approximation to this stretch factor, i.e., a value t such that t t (1 + )t. We give eÆcient solutions for the cases when G is a path, cycle, or tree. The main idea used in all the algorithms is to use wellseparated pair decompositions to speed up the computations. 1 Introduction Let S be a set of n points in R d , where d 1 is a small constant, and let G be an undirected connected graph having the points of S as its vertices. The length of any edge (p; q) of G is dened as the Euclidean distance jpqj between the two vertices p and q. The length of a path in G is dened a...