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30
Localized construction of bounded degree and planar spanner for wireless ad hoc networks
 In DIALMPOMC
, 2003
"... 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 2hop neighbors to find the edges in this new structure. Our method applies the Yao structure on the local Delauna ..."
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Cited by 69 (8 self)
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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 2hop 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 tspanner (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 2hop 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 wellknown 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.
Localized algorithms for energy efficient topology in wireless ad hoc networks
 In ACM MobiHoc’04
, 2004
"... Abstract. Topology control in wireless ad hoc networks is to select a subgraph of the communication graph (when all nodes use their maximum transmission range) with some properties for energy conservation. In this paper, we propose two novel localized topology control methods for homogeneous wireles ..."
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Cited by 41 (2 self)
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Abstract. Topology control in wireless ad hoc networks is to select a subgraph of the communication graph (when all nodes use their maximum transmission range) with some properties for energy conservation. In this paper, we propose two novel localized topology control methods for homogeneous wireless ad hoc networks. Our first method constructs a structure with the following attractive properties: power efficient, bounded node degree, and planar. Its power stretch factor is at most ρ = 1 1−(2 sin π k)β, and each node only has to maintain at most k + 5 neighbors where the integer k> 6 is an adjustable parameter, and β is a real constant between 2 and 5 depending on the wireless transmission environment. It can be constructed and maintained locally and dynamically. Moreover, by assuming that the node ID and its position can be represented in O(log n) bits each for a wireless network of n nodes, we show that the structure can be constructed using at most 24n messages, where each message is O(log n) bits. Our second method improves the degree bound to k, relaxes the theoretical power spanning ratio to ρ = √ 2 β 1−(2 √ 2 sin π, where k> 8 is an adjustable parameter, and keeps all other)β k properties. We show that the second structure can be constructed using at most 3n messages, where each message has size of O(log n) bits. We also experimentally evaluate the performance of these new energy efficient network topologies. The theoretical results are corroborated by the simulations: these structures are more efficient in practice, compared with other known structures used in wireless ad hoc networks and are easier to construct. In addition, the power assignment based on our new structures shows low energy cost and small interference at each wireless node.
Efficient Construction of Low Weight Bounded Degree Planar Spanner
 International Journal of Computational Geometry and Applications
, 2003
"... Given a set V of n points in a twodimensional plane, we give an O(n log n)time centralized algorithm that constructs a planar tspanner for V, for t <= +1} C del , such that the degree of each node is bounded from above by 19 + and the total edge length is proportional to the weight of the minimum ..."
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Cited by 17 (4 self)
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Given a set V of n points in a twodimensional plane, we give an O(n log n)time centralized algorithm that constructs a planar tspanner for V, for t <= +1} C del , such that the degree of each node is bounded from above by 19 + and the total edge length is proportional to the weight of the minimum spanning tree of V , where 0 < # < #/2 is an adjustable parameter...
The Geometric Dilation of Finite Point Sets
, 2006
"... Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The supremum of all these ratios is called the geometric dilation of G. Given a finite point set, we would ..."
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Cited by 17 (10 self)
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Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The supremum of all these ratios is called the geometric dilation of G. Given a finite point set, we would like to know the smallest possible dilation of any graph that contains the given points. In this paper we prove that a dilation of 1.678 is always sufficient, and that #/2 1.570 ... is sometimes necessary in order to accommodate a finite set of points.
Approximating geometric bottleneck shortest paths
 Computational Geometry: Theory and Applications
"... In a geometric bottleneck shortest path problem, we are given a set S of n points in the plane, and want to answer queries of the following type: Given two points p and q of S and a real number L, compute (or approximate) a shortest path between p and q in the subgraph of the complete graph on S con ..."
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Cited by 17 (9 self)
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In a geometric bottleneck shortest path problem, we are given a set S of n points in the plane, and want to answer queries of the following type: Given two points p and q of S and a real number L, compute (or approximate) a shortest path between p and q in the subgraph of the complete graph on S consisting of all edges whose lengths are less than or equal to L. We present efficient algorithms for answering several query problems of this type. Our solutions are based on Euclidean minimum spanning trees, spanners, and the Delaunay triangulation. A result of independent interest is the following. For any two points p and q of S, there is a path between p and q in the Delaunay triangulation, whose length is less than or equal to 2π/(3 cos(π/6)) times the Euclidean distance pq  between p and q, and all of whose edges have length at most pq.
A Unified EnergyEfficient Topology for Unicast and Broadcast
 In ACM MOBICOM
, 2005
"... We propose a novel communication efficient topology control algorithm for each wireless node to select communication neighbors and adjust its transmission power, such that all nodes together selfform a topology that is energy efficient simultaneously for both unicast and broadcast communications. W ..."
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Cited by 12 (3 self)
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We propose a novel communication efficient topology control algorithm for each wireless node to select communication neighbors and adjust its transmission power, such that all nodes together selfform a topology that is energy efficient simultaneously for both unicast and broadcast communications. We prove that the proposed topology is planar, which guarantees packet delivery if a certain localized routing method is used; it is power efficient for unicast – the energy needed to connect any pair of nodes is within a small constant factor of the minimum under a common power attenuation model; it is efficient for broadcast: the energy consumption for broadcasting data on top of it is asymptotically the best compared with structures constructed locally; it has a constant bounded logical degree, which will potentially reduce interference and signal contention. We further prove that the average physical degree of all nodes is bounded by a small constant. To the best of our knowledge, this is the first communicationefficient distributed algorithm to achieve all these properties. Previously, only a centralized algorithm was reported in [3]. Moreover, by assuming that the ID and position of every node can be represented in O(log n) bits for a wireless network of n nodes, our method uses at most 13n messages, where each message is of O(log n) bits. We also show that this structure can be efficiently updated for dynamical network environment. Our theoretical results are corroborated in the simulations.
Geometric dilation of closed planar curves: A new lower bound
, 2004
"... Given any simple closed curve C in the Euclidean plane, let w and D denote the minimal and the maximal caliper distances of C, correspondingly. We show that any such curve C has a geometric dilation of at least arcsin( D ) + ( w ) 1. ..."
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Cited by 10 (8 self)
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Given any simple closed curve C in the Euclidean plane, let w and D denote the minimal and the maximal caliper distances of C, correspondingly. We show that any such curve C has a geometric dilation of at least arcsin( D ) + ( w ) 1.
LowLight Trees, and Tight Lower Bounds for Euclidean Spanners
"... We show that for every npoint metric space M and positive integer k, there exists a spanning tree T with unweighted diameter O(k) and weight w(T) = O(k · n 1/k) · w(MST(M)), and a spanning tree T ′ with weight w(T ′ ) = O(k) · w(MST(M)) and unweighted diameter O(k · n 1/k). These trees also ach ..."
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Cited by 7 (7 self)
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We show that for every npoint metric space M and positive integer k, there exists a spanning tree T with unweighted diameter O(k) and weight w(T) = O(k · n 1/k) · w(MST(M)), and a spanning tree T ′ with weight w(T ′ ) = O(k) · w(MST(M)) and unweighted diameter O(k · n 1/k). These trees also achieve an optimal maximum degree. Furthermore, we demonstrate that these trees can be constructed efficiently. We prove that these tradeoffs between unweighted diameter and weight are tight up to constant factors in the entire range of parameters. Moreover, our lower bounds apply to a basic 1dimensional Euclidean space. Our lower bounds for the particular case of unweighted diameter O(log n) are of independent interest, settling a longstanding open problem in Computational Geometry. In STOC’95 Arya et al. devised a construction of Euclidean Spanners with unweighted diameter O(log n) and weight O(log n) · w(MST(M)). In SODA’05 Agarwal et al. showed that this result is tight up to a factor of O(log log n). We close this gap and show that the result of Arya et al. is tight up to constant factors. Finally, our upper bounds imply improved approximation algorithms for the minimum hhop spanning
Diamond Triangulations Contain Spanners of Bounded Degree
"... Given a triangulation G, whose vertex set V is a set of n points in the plane, and given a real number γ with 0 < γ < π, we design an O(n)time algorithm that constructs a connected spanning subgraph G ′ of G whose maximum degree is at most 14 + ⌈2π/γ⌉. If G is the Delaunay triangulation of V, and γ ..."
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Cited by 5 (2 self)
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Given a triangulation G, whose vertex set V is a set of n points in the plane, and given a real number γ with 0 < γ < π, we design an O(n)time algorithm that constructs a connected spanning subgraph G ′ of G whose maximum degree is at most 14 + ⌈2π/γ⌉. If G is the Delaunay triangulation of V, and γ = 2π/3, we show that G ′ is a tspanner of V (for some constant t) with maximum degree at most 17, thereby improving the previously best known degree bound of 23. If G is a triangulation satisfying the diamond property, then for a specific range of values of γ dependent on the angle of the diamonds, we show that G ′ is a tspanner of V (for some constant t) whose maximum degree is bounded by a constant dependent on γ. If G is the graph consisting of all Delaunay edges of length at most 1, and γ = π/3, we show that a modified version of the algorithm produces a plane subgraph G ′ of the unitdisk graph which is a tspanner (for some constant t) of the unitdisk graph of V, whose maximum degree is at most 20, thereby improving the previously best known degree bound of 25. 1
On the dilation of Delaunay triangulations of points in convex position
 In Proc. Canadian Conf. on Computational Geometry
, 2009
"... Let S be a finite set of points in the Euclidean plane, and let E be the complete graph whose pointset is S. Chew, in 1986, proved a lower bound of π/2 on the stretch factor of the Delaunay triangulation of S (with respect to E), and conjectured that this bound is tight. Dobkin, Friedman, and Supow ..."
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Cited by 3 (0 self)
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Let S be a finite set of points in the Euclidean plane, and let E be the complete graph whose pointset is S. Chew, in 1986, proved a lower bound of π/2 on the stretch factor of the Delaunay triangulation of S (with respect to E), and conjectured that this bound is tight. Dobkin, Friedman, and Supowit, in 1987, showed that the stretch factor of the Delaunay triangulation of S is at most π ( √ 5 + 1)/2 ≈ 5.084. This upper bound was later improved by Keil and Gutwin in 1989 to 2π/(3 cos(π/6)) ≈ 2.42. Since then (1989), Keil and Gutwin’s bound has stood as the best upper bound on the stretch factor of Delaunay triangulations, even though Chew’s conjecture is now widely believed to be true. Whether the stretch factor of Delaunay triangulations is π/2 or not remains a challenging and intriguing problem in computational geometry. Bose, in an openproblem session at CCCG 2007, suggested looking at the special case when the points in S are in convex position. In this paper we show that the stretch factor of the Delaunay triangulation of a pointset in convex position is at most ρ = 2.33. 1