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20
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
FaultTolerant Geometric Spanners
 DISCRETE & COMPUTATIONAL GEOMETRY
, 2004
"... We present two new results about vertex and edge faulttolerant spanners in Euclidean spaces. We describe the first construction of vertex and edge faulttolerant spanners having optimal bounds for maximum degree and total cost. We present a greedy algorithm that for any t> 1 and any nonnegative ..."
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Cited by 24 (1 self)
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We present two new results about vertex and edge faulttolerant spanners in Euclidean spaces. We describe the first construction of vertex and edge faulttolerant spanners having optimal bounds for maximum degree and total cost. We present a greedy algorithm that for any t> 1 and any nonnegative integer k, constructs a kfaulttolerant tspanner in which every vertex is of degree O(k) and whose total cost is O(k2) times the cost of the minimum spanning tree; these bounds are asymptotically optimal. Our next contribution is an efficient algorithm for constructing good faulttolerant spanners. We present a new, sufficient condition for a graph to be a kfaulttolerant spanner. Using this condition, we design an efficient algorithm that finds faulttolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost.
Efficient construction of low weight bounded degree planar spanner
 In Proceedings of the 9th International Computing and Combinatorics Conference (COCOON
, 2003
"... Abstract. 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 maxf 2; sin 2 +1gCdel, such that the degree of each node is bounded from above by 19 + d 2 e, and the total edge length is proportional ..."
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Cited by 23 (4 self)
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Abstract. 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 maxf 2; sin 2 +1gCdel, such that the degree of each node is bounded from above by 19 + d 2 e, and the total edge length is proportional to the weight of the minimum spanning tree of V, where 0 < < =2 is an adjustable parameter. Here Cdel is the spanning ratio of the Delaunay triangulation, which is at most 4
Wellseparated pair decomposition for the unitdisk graph metric and its applications
 SIAM Journal on Computing
, 2003
"... Abstract. We extend the classic notion of wellseparated pair decomposition [10] to the unitdisk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We show that for the unitdisk graph metric of n points in the plane and for any constant c ≥ 1, there ex ..."
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Cited by 10 (2 self)
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Abstract. We extend the classic notion of wellseparated pair decomposition [10] to the unitdisk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We show that for the unitdisk graph metric of n points in the plane and for any constant c ≥ 1, there exists a cwellseparated pair decomposition with O(n log n) pairs, and the decomposition can be computed in O(n log n) time. We also show that for the unitball graph metric in k dimensions where k ≥ 3, there exists a cwellseparated pair decomposition with O(n 2−2/k) pairs, and the bound is tight in the worst case. We present the application of the wellseparated pair decomposition in obtaining efficient algorithms for approximating the diameter, closest pair, nearest neighbor, center, median, and stretch factor, all under the unitdisk graph metric. Keywords Well separated pair decomposition, Unitdisk graph, Approximation algorithm
Experimental Study of Geometric tSpanners
, 2009
"... The construction of tspanners of a given point set has received a lot of attention, especially from a theoretical perspective. In this article, we experimentally study the performance and quality of the most common construction algorithms for points in the Euclidean plane. We implemented the most w ..."
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Cited by 10 (2 self)
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The construction of tspanners of a given point set has received a lot of attention, especially from a theoretical perspective. In this article, we experimentally study the performance and quality of the most common construction algorithms for points in the Euclidean plane. We implemented the most wellknown tspanner algorithms and tested them on a number of different point sets. The experiments are discussed and compared to the theoretical results, and in several cases, we suggest modifications that are implemented and evaluated. The measures of quality that we consider are the number of edges, the weight, the maximum degree, the spanner diameter, and the number of crossings. This is the first time an extensive comparison has been made between the running times of construction algorithms of tspanners and the quality of the generated spanners.
Fast pruning of geometric spanners
 In Proc. 22nd International Symposium on Theoretical Aspects of Computer Science
, 2005
"... Abstract. Let S be a set of points in R d. Given a geometric spanner graph, G = (S, E), with constant dilation t, and a positive constant ε, we show how to construct a (1 + ε)spanner of G with O(S) edges in time O(E  + S  log S). Previous algorithms require a preliminary step in which the e ..."
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Cited by 9 (5 self)
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Abstract. Let S be a set of points in R d. Given a geometric spanner graph, G = (S, E), with constant dilation t, and a positive constant ε, we show how to construct a (1 + ε)spanner of G with O(S) edges in time O(E  + S  log S). Previous algorithms require a preliminary step in which the edges are sorted in nondecreasing order of their lengths and, thus, have running time Ω(E  log S). We obtain our result by designing a new algorithm that finds the pair in a wellseparated pair decomposition separating two given query points. Previously, it was known how to answer such a query in O(log S) time. We show how a sequence of such queries can be answered in O(1) amortized time per query. 1
Near Optimal Multicriteria Spanner Constructions in Wireless AdHoc Networks
, 2010
"... In this paper we study asymmetric power assignments which induce a low energy kstrongly connected communication graph with spanner properties. We address two spanner models: energy and distance. The former serves as an indicator for the energy consumed in a message propagation between two nodes, wh ..."
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Cited by 7 (3 self)
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In this paper we study asymmetric power assignments which induce a low energy kstrongly connected communication graph with spanner properties. We address two spanner models: energy and distance. The former serves as an indicator for the energy consumed in a message propagation between two nodes, while the latter reflects the geographic properties of routing in the induced communication graph. We consider a random wireless adhoc network with V = n nodes distributed uniformly and independently in a unit square. For k ∈ {1, 2} we propose several power assignments which obtain a good bicriteria approximation on the total cost and stretch factor under the two models. For k> 2 we analyze a power assignment developed in [1], and derive some interesting bounds on the stretch factor for both models as well. We also describe how to compute all the power assignments distributively, and provide simulation results. To the best of our knowledge, these are the first provable theoretical bounds for low cost spanners in wireless adhoc networks.
Finding the best shortcut in a geometric network
 ACM Symp. Comput. Geom
, 2005
"... Given a Euclidean graph G in R d with n vertices and m edges we consider the problem of adding a shortcut such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a Euclidean graph runs in O(mn + n 2 log n) time, resulting ..."
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Cited by 7 (2 self)
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Given a Euclidean graph G in R d with n vertices and m edges we consider the problem of adding a shortcut such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a Euclidean graph runs in O(mn + n 2 log n) time, resulting in a trivial O(mn 3 + n 4 log n) time algorithm for computing the optimal shortcut. First, we show that a simple modification yields the optimal solution in O(n 4) time using O(n 2) space. To reduce the running times we consider several approximation algorithms. Our main result is a (2 + ε)approximation algorithm with running time O(nm + n 2 (log n +1/ε 3d)) using O(n 2)space.
Wireless sensor networks and computational geometry
 In Proceedings of the Conference on Parallel and Distributed Computing and Systems
, 2003
"... Wireless Sensor Networks Due to its potential applications in various situations such as battleeld, emergency relief, environment monitoring, and so on, wireless sensor networks [50, 75, 118, 130] have recently emerged as a premier research topic. Sensor networks consist of a set of sensor nodes wh ..."
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Cited by 5 (0 self)
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Wireless Sensor Networks Due to its potential applications in various situations such as battleeld, emergency relief, environment monitoring, and so on, wireless sensor networks [50, 75, 118, 130] have recently emerged as a premier research topic. Sensor networks consist of a set of sensor nodes which are spread over a geographical area. These nodes are able to perform processing as well as sensing and are additionally capable of communi