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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 156 (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
Spanning Trees and Spanners
, 1996
"... We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" ..."
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Cited by 148 (2 self)
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We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. For instance, one may wish to connect components of a VLSI circuit by networks of wires, in a way that uses little surface area on the chip, draws little power, and propagates signals quickly. Similar problems come up in other applications such as telecommunications, road network design, and medical imaging [1]. One network design problem, the Traveling Salesman problem, is sufficiently important to have whole books devoted to it [79]. Problems involving some form of geometric minimum or maximum spanning tree also arise in the solution of other geometric problems such as clustering [12], mesh generation [56], and robot motion planning [93]. One can vary the network design problem in many w...
Euclidean spanners: short, thin, and lanky
 IN: 27TH ACM SYMPOSIUM ON THEORY OF COMPUTING
, 1995
"... Euclidean spanners are important data structures in geometric algorithm design, because they provide a means of approximating the complete Euclidean graph with only O(n) edges, so that the shortest path length between each pair of points is not more than a constant factor longer than the Euclidean d ..."
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Cited by 112 (22 self)
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Euclidean spanners are important data structures in geometric algorithm design, because they provide a means of approximating the complete Euclidean graph with only O(n) edges, so that the shortest path length between each pair of points is not more than a constant factor longer than the Euclidean distance between the points. In many applications of spanners, it is important that the spanner possess a number of additional properties: low tot al edge weight, bounded degree, and low diameter. Existing research on spanners has considered one property or the other. We show that it is possible to build spanners in optimal O(n log n) time and O(n) space that achieve optimal or near optimal tradeoffs between all combinations of these
New Sparseness Results on Graph Spanners
, 1992
"... Let G = (V, E) be an nvertex connected graph with positive edge weights. A subgraph G ’ = (V, E’) is a tspanner of G if for all u, v E V, the weighted distance between u and v in G ’ is at most t times the weighted distance between u and v in G. We consider the problem of constructing sparse span ..."
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Cited by 95 (8 self)
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Let G = (V, E) be an nvertex connected graph with positive edge weights. A subgraph G ’ = (V, E’) is a tspanner of G if for all u, v E V, the weighted distance between u and v in G ’ is at most t times the weighted distance between u and v in G. We consider the problem of constructing sparse spanners. Sparseness of spanners is measured by two criteria, the size, defined as the number of edges in the spanner, and the weight, defined as the sum of the edge weights in the spanner. In this paper, we concentrate on constructing spanners of small weight. For an arbitrary positive edgeweighted graph G, for any t> 1, and any c>0, we show that a tspanner of G with weight O(n * ). wt(MST) can be constructed in polynomial time. We also show that (logz n)spanners of weight O(1). wt(MST) can be constructed. We then consider spanners for complete graphs induced by a set of points in ddimensional real normed space. The weight of an edge Zy is the norm of the ~y vector. We show that for these graphs, tspanners with total weight O(log n). wt(MST) can be constructed in polynomial time.
ClosestPoint Problems in Computational Geometry
, 1997
"... 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, th ..."
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Cited by 73 (14 self)
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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 postoffice 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 divideandconquer 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 . . . . . . . . . . . . . . . . . . ...
Faster Algorithms for Some Geometric Graph Problems in Higher Dimensions
, 1993
"... We show how to apply the wellseparated pair decomposition of a pointset P in ! d to significantly improve known time bounds on several geometric graph problems. We first present an algorithm to find an approximate Euclidean minimum spanning tree of P whose weight is at most 1 + ffl times the exa ..."
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Cited by 72 (2 self)
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We show how to apply the wellseparated pair decomposition of a pointset P in ! d to significantly improve known time bounds on several geometric graph problems. We first present an algorithm to find an approximate Euclidean minimum spanning tree of P whose weight is at most 1 + ffl times the exact minimum. We achieve a time complexity of O(n log n + (ffl \Gammad=2 log 1 ffl )n), improving the best known bound of O(ffl \Gammad n log n). We then show how to construct a graph with O(ffl \Gammad+1 n) edges in which the shortest path between any pair of points is within 1 + ffl of the Euclidean distance. Our time complexity is O(n log n+(ffl \Gammad log 1 ffl )n), a significant improvement over the best previous algorithm that produces a graph of this size. Finally, we show how to compute the exact Euclidean minimum spanning tree in time O(T d (n; n) log n), where T d (m; n) is the time to find the bichromatic closest pair between m red points and n blue points. The previo...
Randomized and Deterministic Algorithms for Geometric Spanners of Small Diameter
, 1994
"... Let S be a set of n points in IR d and let t ? 1 be a real number. A tspanner for S is a directed graph having the points of S as its vertices, such that for any pair p and q of points there is a path from p to q of length at most t times the Euclidean distance between p and q. Such a path is c ..."
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Cited by 39 (7 self)
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Let S be a set of n points in IR d and let t ? 1 be a real number. A tspanner for S is a directed graph having the points of S as its vertices, such that for any pair p and q of points there is a path from p to q of length at most t times the Euclidean distance between p and q. Such a path is called a tspanner path. The spanner diameter of such a spanner is defined as the smallest integer D such that for any pair p and q of points there is a tspanner path from p to q containing at most D edges. Randomized and deterministic algorithms are given for constructing tspanners consisting of O(n) edges and having O(logn) diameter. Also, it is shown how to maintain the randomized tspanner under random insertions and deletions. Previously, no results were known for spanners with low spanner diameter and for maintaining spanners under insertions and deletions. 1 Introduction Given a set S of n points in IR d and a real number t ? 1, a tspanner for S is a directed graph on S such th...
Connections between thetagraphs, Delaunay triangulations, and orthogonal surfaces
 In Proceedings of the 36th International Conference on Graph Theoretic Concepts in Computer Science (WG 2010
, 2010
"... Abstract. Θkgraphs are geometric graphs that appear in the context of graph navigation. The shortestpath metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TDDelaunay graphs, a.k.a. triangulard ..."
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Cited by 32 (2 self)
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Abstract. Θkgraphs are geometric graphs that appear in the context of graph navigation. The shortestpath metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TDDelaunay graphs, a.k.a. triangulardistance Delaunay triangulations, introduced by Chew, have been shown to be plane 2spanners of the 2D Euclidean complete graph, i.e., the distance in the TDDelaunay graph between any two points is no more than twice the distance in the plane. Orthogonal surfaces are geometric objects defined from independent sets of points of the Euclidean space. Orthogonal surfaces are well studied in combinatorics (orders, integer programming) and in algebra. From orthogonal surfaces, geometric graphs, called geodesic embeddings can be built. In this paper, we introduce a specific subgraph of the Θ6graph defined in the 2D Euclidean space, namely the halfΘ6graph, composed of the evencone edges of the Θ6graph. Our main contribution is to show that these graphs are exactly the TDDelaunay graphs, and are strongly connected to the geodesic embeddings of orthogonal surfaces of coplanar points in the 3D Euclidean space. Using these new bridges between these three fields, we establish: – Every Θ6graph is the union of two spanning TDDelaunay graphs. In particular, Θ6graphs are 2spanners of the Euclidean graph, and the bound of 2 on the stretch factor is the best possible. It was not known that Θ6graphs are tspanners for some constant t, and Θ7graphs were only known to be tspanners for t ≈ 7.562. – Every plane triangulation is TDDelaunay realizable, i.e., every combinatorial plane graph for which all its interior faces are triangles is the TDDelaunay graph of some point set in the plane. Such realizability property does not hold for classical Delaunay triangulations.
Distributed Maintenance of Resource Efficient Wireless Network Topologies (Ext. Abstract
 In EUROPAR’02
"... Abstract. Multiple hop routing in mobile ad hoc networks can minimize energy consumption and increase data throughput. Yet, the problem of radio interferences remains. However if the routes are restricted to a basic network based on local neighborhoods, these interferences can be reduced such that s ..."
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Cited by 31 (6 self)
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Abstract. Multiple hop routing in mobile ad hoc networks can minimize energy consumption and increase data throughput. Yet, the problem of radio interferences remains. However if the routes are restricted to a basic network based on local neighborhoods, these interferences can be reduced such that standard routing algorithms can be applied. We compare different network topologies for these basic networks with respect to degree, spannerproperties, radio interferences, energy, and congestion, i.e. the Yaograph (aka.graph) and some also known related models, which will be called the SymmYgraph (aka. YSgraph), the SparsYgraph (aka.YYgraph) and the BoundYgraph. Further, we present a promising network topology called the HLgraph (based on Hierarchical Layers). Further, we compare the ability of these topologies to handle dynamic changes of the network when radio stations appear and disappear. For this we measure the number of involved radio stations and present distributed algorithms for repairing the network structure. 1