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39
More algorithms for allpairs shortest paths in weighted graphs
 In Proceedings of 39th Annual ACM Symposium on Theory of Computing
, 2007
"... In the first part of the paper, we reexamine the allpairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general realweighted dense graphs. In the second part of the paper, we use fast matrix multipl ..."
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Cited by 54 (3 self)
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In the first part of the paper, we reexamine the allpairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general realweighted dense graphs. In the second part of the paper, we use fast matrix multiplication to obtain truly subcubic APSP algorithms for a large class of “geometrically weighted ” graphs, where the weight of an edge is a function of the coordinates of its vertices. For example, for graphs embedded in Euclidean space of a constant dimension d, we obtain a time bound near O(n 3−(3−ω)/(2d+4)), where ω < 2.376; in two dimensions, this is O(n 2.922). Our framework greatly extends the previously considered case of smallintegerweighted graphs, and incidentally also yields the first truly subcubic result (near O(n 3−(3−ω)/4) = O(n 2.844) time) for APSP in realvertexweighted graphs, as well as an improved result (near O(n (3+ω)/2) = O(n 2.688) time) for the allpairs lightest shortest path problem for smallintegerweighted graphs. 1
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 absolute spa ..."
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Cited by 54 (8 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.
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
Approximate Distance Oracles for Geometric Graphs
, 2002
"... Given a geometric tspanner graph G in E d with n points and m edges, with edge lengths that lie within a polynomial (in n) factor of each other. Then, after O(m+n log n) preprocessing, we present an approximation scheme to answer (1+") approximate shortest path queries in O(1) time. The data str ..."
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Cited by 34 (10 self)
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Given a geometric tspanner graph G in E d with n points and m edges, with edge lengths that lie within a polynomial (in n) factor of each other. Then, after O(m+n log n) preprocessing, we present an approximation scheme to answer (1+") approximate shortest path queries in O(1) time. The data structure uses O(n log n) space.
The BinCovering Technique for Thresholding Random Geometric Graph Properties
, 2005
"... We study the emerging phenomenon of ad hoc, sensorbased communication networks. The communication is modeled by the random geometric graph model G(n, r, ℓ) where n points randomly placed within [0, ℓ] d form the nodes, and any two nodes that correspond to points at most distance r away from each ot ..."
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Cited by 34 (3 self)
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We study the emerging phenomenon of ad hoc, sensorbased communication networks. The communication is modeled by the random geometric graph model G(n, r, ℓ) where n points randomly placed within [0, ℓ] d form the nodes, and any two nodes that correspond to points at most distance r away from each other are connected. We study fundamental properties of G(n, r, ℓ) of interest: connectivity, coverage, and routingstretch. Our main contribution is a simple analysis technique we call bincovering that we apply uniformly to get (asymptotically) tight thresholds for each of these properties. Typically, in the past, random geometric graph analyses involved sophisticated methods from continuum percolation theory; on contrast, our bincovering approach is discrete and very simple, yet it gives us tight threshold bounds. The technique also yields algorithmic benefits as illustrated by a simple local routing algorithm for finding paths with low stretch. Our specific results should also prove interesting to the networking community that has seen a recent increase in the study of random geometric graphs motivated by engineering ad hoc networks.
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
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.
Minimum dilation stars
 In Proc. ACM Symposium on Computational Geometry
, 2005
"... Abstract. The dilation of a Euclidean graph is defined as the ratio of distance in the graph divided by distance in IR d. In this paper we consider the problem of positioning the root of a star such that the dilation of the resulting star is minimal. We present a deterministic O(nlog n)time algorit ..."
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Cited by 15 (4 self)
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Abstract. The dilation of a Euclidean graph is defined as the ratio of distance in the graph divided by distance in IR d. In this paper we consider the problem of positioning the root of a star such that the dilation of the resulting star is minimal. We present a deterministic O(nlog n)time algorithm for evaluating the dilation of a given star; a randomized O(n log n) expectedtime algorithm for finding an optimal center in IR d; and for the case d = 2, a randomized O(nα(n) log 2 n) expectedtime algorithm for finding an optimal center among the input points. 1
Computing the Detour of Polygonal Curves
, 2002
"... Let P be a simple polygonal chain in E with n edges. The detour of P between two points, x and y, is the ratio between the length of P between x any y and their Euclidean distance. The detour ..."
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Cited by 12 (3 self)
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Let P be a simple polygonal chain in E with n edges. The detour of P between two points, x and y, is the ratio between the length of P between x any y and their Euclidean distance. The detour