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113
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 35 (5 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
The FreezeTag Problem: How to Wake Up a Swarm of Robots
 In Proc. 13th ACMSIAM Sympos. Discrete Algorithms
, 2002
"... An optimization problem that naturally arises in the study of "swarm robotics" is to wake up a set of "asleep" robots, starting with only one "awake" robot. One robot can only awaken another when they are in the same location. As soon as a robot is awake, it assists in waking up other robots. The go ..."
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Cited by 33 (6 self)
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An optimization problem that naturally arises in the study of "swarm robotics" is to wake up a set of "asleep" robots, starting with only one "awake" robot. One robot can only awaken another when they are in the same location. As soon as a robot is awake, it assists in waking up other robots. The goal is to compute an optimal awakening schedule such that all robots are awake by time t , for the smallest possible value of t . We consider both scenarios on graphs and in geometric environments. In the graph setting, robots sleep at vertices and there is a length function on the edges. An awake robot can travel from vertex to vertex along edges, and the length of an edge determines the time it takes to travel from one vertex to the other. While this problem bears some resemblance to problems from various areas in combinatorial optimization such as routing, broadcasting, scheduling and covering, its algorithmic characteristics are surprisingly different. We prove that the problem is NPhard, even for the special case of star graphs. We also establish hardness of approximation, showing that it is NPhard to obtain an approximation factor better than 5/3, even for graphs of bounded degree. These lower bounds are complemented with several algorithmic results. We present a simple online algorithm that is O(log)competitive for graphs with maximum degree . Other results include algorithms that require substantially more sophistication and development of new techniques: (1) The natural greedy strategy on star graphs has a worstcase performance of 7/3, which is tight. (2) There exists a PTAS for star graphs. (3) For the problem Dept. of Appl. Math. and Statistics, SUNY Stony Brook, NY 117943600, festie, jsbmg@ams.sunysb.edu. y Dept. of Computer Science, SUNY St...
On Local Algorithms for Topology Control and Routing in Ad Hoc Networks
 In Proc. SPAA
, 2003
"... An ad hoc network is a collection of wireless mobile hosts forming a temporary network without the aid of any fixed infrastructure. Indeed, an important task of an ad hoc network is to determine an appropriate topology over which highlevel routing protocols are implemented. Furthermore, since the u ..."
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Cited by 28 (1 self)
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An ad hoc network is a collection of wireless mobile hosts forming a temporary network without the aid of any fixed infrastructure. Indeed, an important task of an ad hoc network is to determine an appropriate topology over which highlevel routing protocols are implemented. Furthermore, since the underlying topology may change with time, we need to design routing algorithms that effectively react to dynamically changing network conditions. This paper studies algorithms...
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 24 (0 self)
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We introduce Tukey and Tukey scagnostics and develop graphtheoretic methods for implementing their procedure on large datasets.
Computing the maximum detour and spanning ratio of planar chains, trees and cycles
 In Proc. 19th Internat. Symp. Theor. Aspects of C.Sc., LNCS 2285:250–261
, 2002
"... Let G = (V, E) be an embedded connected graph with n vertices and m edges. Specifically, the vertex set V consists of points in R 2, and E consists ..."
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Cited by 21 (1 self)
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Let G = (V, E) be an embedded connected graph with n vertices and m edges. Specifically, the vertex set V consists of points in R 2, and E consists
Many distances in planar graphs
 In SODA ’06: Proc. 17th Symp. Discrete algorithms
, 2006
"... We show how to compute in O(n 4/3 log 1/3 n+n 2/3 k 2/3 logn) time the distance between k given pairs of vertices of a planar graph G with n vertices. This improves previous results whenever (n/logn) 5/6 ≤ k ≤ n 2 /log 6 n. As an application, we speed up previous algorithms for computing the dilatio ..."
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Cited by 18 (5 self)
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We show how to compute in O(n 4/3 log 1/3 n+n 2/3 k 2/3 logn) time the distance between k given pairs of vertices of a planar graph G with n vertices. This improves previous results whenever (n/logn) 5/6 ≤ k ≤ n 2 /log 6 n. As an application, we speed up previous algorithms for computing the dilation of geometric planar graphs. 1
On the spanning ratio of gabriel graphs and βskeletons
 SIAM Journal on Discrete Mathematics
, 2002
"... Abstract. The spanning ratio of a graph defined on n points in the Euclidean plane is the maximum ratio, over all pairs of data points (u, v), of the minimum graph distance between u and v divided by the Euclidean distance between u and v. A connected graph is said to be a Sspanner if the spanning ..."
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Cited by 18 (8 self)
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Abstract. The spanning ratio of a graph defined on n points in the Euclidean plane is the maximum ratio, over all pairs of data points (u, v), of the minimum graph distance between u and v divided by the Euclidean distance between u and v. A connected graph is said to be a Sspanner if the spanning ratio does not exceed S. For example, for any S, there exists a point set whose minimum spanning tree is not a Sspanner. At the other end of the spectrum, a Delaunay triangulation is guaranteed to be a 2.42spanner[11]. For proximity graphs between these two extremes, such as Gabriel graphs[8], relative neighborhood graphs[16] and βskeletons[12] with β ∈ [0, 2] some interesting questions arise. We show that the spanning ratio for Gabriel graphs (which are βskeletons with β = 1) is Θ ( √ n) in the worst case. For all βskeletons with β ∈ [0, 1], we prove that the spanning ratio is at most O(nγ) where γ = (1 − log2 (1 + p 1 − β2))/2. For all βskeletons with β ∈ [1, 2), we prove that there exist point sets whose spanning ratio is at least ` 1 2 − o(1) ´ √ n. For relative neighborhood graphs[16] (skeletons with β = 2), we show that there exist point sets where the spanning ratio is Ω(n). For points drawn independently from the uniform distribution on the unit square, we show that the spanning ratio of the (random) Gabriel graph and all βskeletons with β ∈ [1, 2] tends to ∞ in probability as p log n / log log n.
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.
Estimating the weight of metric minimum spanning trees in sublineartime
 in Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC
"... In this paper we present a sublinear time (1+ ɛ)approximation randomized algorithm to estimate the weight of the minimum spanning tree of an npoint metric space. The running time of the algorithm is Õ(n/ɛO(1)). Since the full description of an npoint metric space is of size Θ(n 2),the complexity ..."
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Cited by 16 (5 self)
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In this paper we present a sublinear time (1+ ɛ)approximation randomized algorithm to estimate the weight of the minimum spanning tree of an npoint metric space. The running time of the algorithm is Õ(n/ɛO(1)). Since the full description of an npoint metric space is of size Θ(n 2),the complexity of our algorithm is sublinear with respect to the input size. Our algorithm is almost optimal as it is not possible to approximate in o(n) time the weight of the minimum spanning tree to within any factor. Furthermore,it has been previously shown that no o(n 2) algorithm exists that returns a spanning tree whose weight is within a constant times the optimum.