D. Eppstein. Spanning trees and spanners. In Handbook of Computational Geometry, pp. 425--461. Elsevier Science B.V., 1997. Home/Search Document Details and Download
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Sublinear-Time Approximation of Euclidean Minimum.. - Czumaj, Ergun.. (2003) (Correct)
....or k0 2 . Let b = b k0 and = k0 . Notice that b 2 and . By our arguments from Section 3, the active blocks at level k 0 can be found by querying the range query oracle O(b 2 log(n= times. Spanners and connected block components. For any t 1, a t spanner (see, e.g. [8, 15, 21]) for a set S of points in a Euclidean space is any Euclidean graph G with the vertex set S such that for every pair of points x; y 2 S there is a path in G between x and y of total length at most t jxyj. Let B be the set of centers of active blocks and let SPN be a (1 =4) spanner of B with ....
D. Eppstein. Spanning trees and spanners. In Handbook of Computational Geometry, pp. 425--461. Elsevier Science B.V., 1997.
Design and Analysis of Spatiotemporal Multicast Protocols.. - Huang, Lu, Roman (2003) (Correct)
....d(i, j) 4) We call #(i, j) the pairwise # compactness between nodes i and j. The # compactness of a geometric graph G(V, E) is defined as the smallest # compactness of all node pairs of the network: i,j#V #(i, j) 5) Note that # compactness has a close relation with the terms dilation [9], spanning ratio [3] and stretch factor [21] used in the graph and computational geometry community. Dilation is defined as the maximum ratio between Euclidean path distance and geometric distance, while # compactness is defined as minimum ratio between the geometric distance and the ....
D. Eppstein. Spanning trees and spanners. In In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425--461, Amsterdam, 1999. Elsevier Science.
Euclidean Bounded-Degree Spanning Tree Ratios - Timothy Chan School (2003) (3 citations) (Correct)
....factors for bounded degree spanning trees. Khuller, Raghavachari, and Young [10] took an in depth look into this question and managed to achieve factors 3 2 and 5 4 for K = 3 and K = 4 respectively in the plane. Since then, no improvements have been made, despite frequent references to their work [2, 3, 7, 8, 11, 16, 17]. We report the first progress in 8 years: in the Euclidean plane, there always exists degree 3 and degree 4 spanning trees with weights within factors 1.402 and 1.143 respectively of the MST weight. Such trees can be constructed in polynomial time. Immediately, we obtain a factor 1.402 and ....
D. Eppstein. Spanning trees and spanners. In J.-R. Sack and J. Urrutia (eds.), Handbook of Computational Geometry, Elsevier Science Publishers B.V. North-Holland, Amsterdam, pages 425--461, 2000.
Spatiotemporal Multicast in Sensor Networks - Huang, Lu, Roman (2003) (1 citation) (Correct)
....i.e. #(i, j) 1) We call #(i, j) the # compactness between nodes i and j. The # compactness of a geometric graph G(V, E) is defined as the smallest # compactness of all node pairs of the network: i,j#V #(i, j) 2) Note that # compactness has a close relation with the terms dilation [6], spanning ratio [3] and stretch factor [17] used in the graph and computational geometry community. Dilation is defined as the maximal ratio between graph and geometric distance, while # compactness is defined as minimum ratio between the geometric distance and the corresponding shortest ....
D. Eppstein. Spanning trees and spanners. In In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425--461, Amsterdam, 1999. Elsevier Science.
Constructing Plane Spanners of Bounded Degree and Low Weight - Bose, Gudmundsson, Smid (2002) (5 citations) (Correct)
.... spanner diameter (i.e. any two points are connected by a t spanner path consisting of only a small number of edges) low weight (i.e. the total length of all edges is proportional to the weight of a minimum spanning tree of S) and fault tolerance; see, e.g. 1, 11, 12, 14] and the surveys [10, 16]. All these algorithms compute t spanners for any given constant t 1. In this paper, we consider the construction of plane t spanners. Obviously, in order for a t spanner to be plane, t must be at least 2. It is known that the Delaunay triangulation is a t spanner for t = 2 = 3 cos( 6) ....
D. Eppstein. Spanning trees and spanners. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425-461. Elsevier Science Publishers, Amsterdam, 2000.
Geometric Spanner for Routing in Mobile Networks - Gao, Guibas, Hershberger.. (2001) (32 citations) (Correct)
....performs quite well when each individual node s visible range is large and nodes are randomly distributed, it does not perform well in more general cases. In particular, the GG and RNG are not good spanners: nodes that can be reached via a path with few hops might become far apart in the GG or RNG [10]. This fact limits the quality of paths even if we use globally optimum routing methods on these subgraphs. The stretch factor of paths in a graph captures this aspect of path quality. Roughly speaking, the stretch factor of a subgraph G 0 of a graph G measures the worst case ratio between the ....
....uses a log n approximation of the minimum connected dominating set. Wu et al. used a distributed algorithm to compute the connected dominating set; however, this could perform badly in the worst case (O(n) approximation) 23] Spanner graphs have been heavily studied in computational geometry [10]. The Delaunay triangulation has been shown to be a planar spanner [5, 8, 14] However, little is known about restricted spanner graphs, where only edges shorter than 1 are allowed. Also, little is known about how to maintain spanner graphs in a distributed manner when the points move. 3 Routing ....
D. Eppstein. Spanning trees and spanners. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425--461. Elsevier Science Publishers B.V. NorthHolland, Amsterdam, 2000.
Approximation Algorithms for the Bottleneck Stretch Factor.. - Narasimhan, Smid (2000) (Correct)
....built in subquadratic time. 1.2 Related work There has been substantial work on the problem of constructing a Euclidean graph on a given set of points whose stretch factor is bounded by a given constant t 1. A good overview of results in this direction can be found in the surveys by Eppstein [11] and Smid [15] The problem of approximating the stretch factor of any given Euclidean graph has been considered by the authors in [13] There, we prove the following result, which will be used in the current paper. Theorem 1 ( 13] Let S be a set of n points in R d , let G = S; E) be an ....
D. Eppstein. Spanning trees and spanners. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425-461. Elsevier Science, Amsterdam, 1999.
Approximation Algorithms for the Bottleneck Stretch Factor.. - Giri Narasimhan Michiel (2002) (Correct)
....be built in subquadratic time. 1.2 Related work There has been substantial work on the problem of constructing a Euclidean graph on a given set of points whose stretch factor is bounded by a given constant t 1. A good overview of results in this direction can be found in the surveys by Eppstein [11] and Smid [15] The problem of approximating the stretch factor of any given Euclidean graph has been considered by the authors in [13] There, we prove the following result, which will be used in the current paper. Theorem 1 ( 13] Let S be a set of n points in R d , let G = S, E) be an ....
D. Eppstein. Spanning trees and spanners. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425--461. Elsevier Science, Amsterdam, 1999.
Experiments with Computing Geometric Minimum Spanning Trees - Narasimhan, Zhu, Zachariasen (2000) (2 citations) (Correct)
....vs. n 2 ) and is thus better suited for higher dimensional geometric problems, and (b) it readily generalizes to L p and polyhedral metrics. 3. THE MST ALGORITHMS 3.1 The Basic Algorithm Let S be a given set of n points in d . Algorithm GeoMST (see Callahan and Kosaraju [1993] and Eppstein [1999]) shown in Figure 1 computes a GMST of S. In Section 3.3, we present GeoMST2, which improves on GeoMST. algorithm GeoMST(S) f(A 1 ; B 1 ) Am ; Bm )g : WSPD(S) E : for i : 1 to m do . fa i ; b i ; Dg : BCP(A i ; B i ; 1) E : E [ f(a i ; b i )g . endfor . G : S; ....
Eppstein, D. 1999. Spanning trees and spanners. In J.-R. Sack and J. Urrutia Eds., Handbook of Computational Geometry . Elsevier Science Publishers B.V. North-Holland.
Approximating The Stretch Factor Of Euclidean Graphs - Narasimhan, Smid (2000) (15 citations) (Correct)
....1 2 G. NARASIMHAN AND M. SMID C M Fig. 1.1. A section of the Scandinavian rail network; C denotes Copenhagen, and M denotes Malm o. Spanners have applications in network design, robotics, distributed algorithms, and many other areas, and have been the subject of considerable research [1, 4, 8, 11, 14, 18, 24]. More recently, spanners have received a lot of attention by researchers with the discovery of new applications for them in the design of approximation algorithms for geometric optimization problems such as the Euclidean traveling salesperson problem [3, 23] If the graph represents, say, a ....
D. Eppstein. Spanning trees and spanners. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425-461. Elsevier Science, Amsterdam, 1999.
Experiments with Computing Geometric Minimum Spanning Trees.. - Narasimhan, al. (1999) (2 citations) (Correct)
....vs. Theta(n 2 ) and is thus better suited for higher dimensional geometric problems, and (b) it easily generalizes to L p and polyhedral metrics. 3 The MST Algorithms 3.1 The Basic Algorithm Let S be a given set of n points in d . Algorithm GeoMST (see Callahan et al. 8] and Eppstein [12]) shown in Figure 1 computes a GMST of S. In Section 3.3, we present GeoMST2, which improves on GeoMST. algorithm GeoMST(S) f(A 1 ; B 1 ) Am ; Bm )g : WSPD(S) E : for i : 1 to m do . fa i ; b i ; Dg : BCP(A i ; B i ; 1) E : E [ f(a i ; b i )g . endfor . G : S; ....
D. Eppstein. Spanning trees and spanners. In Handbook of Comp. Geom. Elsevier, 1999.
On the Spanning Ratio of Gabriel Graphs and beta-Skeletons - Bose, Devroye, Evans.. (2001) (13 citations) (Correct)
.... variation analysis, geographic information systems, computational geometry, computational morphology, and computer vision use the underlying structure (also referred to as the skeleton or internal shape) of a set of data points revealed by means of a proximity graph (see for example [16] 13] [7], 9] A proximity graph attempts to exhibit the relation between points in a point set. Two points are joined by an edge if they are deemed close by some proximity measure. It is the measure that determines the type of graph that results. Many different measures of proximity have been defined, ....
....by L(u; v) while the direct Euclidean distance is D(u; v) The spanning ratio of the graph is defined by S def = max (u;v) L(u; v) D(u; v) where the maximum is over all Gamma n 2 Delta pairs of data points. Graphs with small spanning ratios are important in some applications (see [7] for a survey on spanners) The history for the Delaunay triangulation is interesting. First, Chew [2, 3] showed that in the worst case, S =2. Subsequently, Dobkin et al. 5] showed that the Delaunay triangulation was a ( 1 p 5) 2) 5:08 spanner. Finally, Keil and Gutwin [10, 11] improve this ....
D. Eppstein. Spanning Trees and Spanners, Tech. Report 96-16, Dept. of Comp. Sci, University of California, Irvine, 1996.
Exploiting Domain Geometry in Analogical Route Planning - Haigh, Shewchuk, Veloso (1997) (3 citations) (Correct)
....by a type of sparse graph known as a low dilation graph; a graph that has the property that the shortest path between any two vertices in the graph is at most a constant factor longer than a straight line. A variety of low dilation graphs that could be used for this purpose are surveyed by Eppstein (1996). For our implementation, we have chosen a planar graph called a Delaunay triangulation (Delaunay 1934; Fortune 1992) Delaunay triangulations have several desirable properties. ffl They provide a structure that makes it possible to quickly determine edge weights in the graph. ffl Local ....
Eppstein, D. (1996). Spanning trees and spanners. Technical Report 96-16, Department of Information and Computer Science, University of California at Irvine.
Geometric Shortest Paths and Network Optimization - Mitchell (1998) (39 citations) (Correct)
....also give Omega Gamma n log n) lower bounds on computing various types of t spanners, which are graphs that, for every pair of points, contain a path whose length is at most t times the interpoint distance (Euclidean, geodesic, etc. see the survey on spanners in this handbook by Eppstein [150], as well as [79, 106, 351] Two Point Queries Two point queries in a polygonal domain are much more challenging than the case of simple polygons, where optimal algorithms are known. One approach, observed by Chen, Daescu, and Klenk [94] is to proceed as follows. Using O(n 2 ) space, we can ....
....this time bound, the approximate length is reported; in additional time proportional to the number of vertices, a path can be reported that achieves the length bound. These results have been improved recently by Arikati et al. 21] who give a family of results, based on planar spanners (see [150]) with tradeoffs among the approximation factor and the preprocessing time, storage space, and query time. One such result obtains a (3 p 2 ffl) approximation using O(n 3=2 = log 1=2 n) time to build a data structure of size O(n log n) after which queries are performed in time O(log n) ....
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D. Eppstein. Spanning trees and spanners. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, page ?? Elsevier Science Publishers B.V. North-Holland, Amsterdam, 1998.
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D. Eppstein. Spanning trees and spanners. In Handbook of Computational Geometry, pp. 425--461. Elsevier Science B.V., 1997.
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D. Eppstein. Spanning trees and spanners. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, chapter 9, pages 425--461. Elsevier Science B.V., 1997.
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D. Eppstein. Spanning trees and spanners. In Handbook of Computational Geometry, J.-R. Sack and J. Urrutia, editors, pp. 425-461. Elsevier, 1999.
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D. Eppstein. Spanning trees and spanners. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425--461. Elsevier Science Publishers B.V. North-Holland, Amsterdam, 2000.
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D. Eppstein. Spanning trees and spanners. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425--461. Elsevier, Amsterdam, 1999.
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D. Eppstein. Spanning trees and spanners. In Handbook of Computational Geometry, J.-R. Sack and J. Urrutia, editors, pp. 425-461. Elsevier, 1999.
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Eppstein, D. (1999). Spanning trees and spanners. In Sack, J.-R. and Urrutia, J., editors, Handbook of Computational Geometry, pages 425-461. Elsevier Science Publishers B.V., North-Holland,Amsterdam.
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D. Eppstein. Spanning trees and spanners. In In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425--461, Amsterdam, 1999. Elsevier Science.
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D. Eppstein. Spanning trees and spanners. In In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425--461, Amsterdam, 1999. Elsevier Science.
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D. Eppstein. Spanning trees and spanners. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425-461. Elsevier Science Publishers B.V. North-Holland, Amsterdam, 2000.
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David Eppstein. Spanning trees and spanners. In In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425--461, Amsterdam, 1999. Elsevier Science.
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D. Eppstein. Spanning trees and spanners. In J.-R. Sack and J. Urrutia (eds.), Handbook of Computational Geometry, Elsevier Science Publishers B.V. North-Holland, Amsterdam, pages 425-461, 2000.
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D. Eppstein. Spanning trees and spanners. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425-461. Elsevier, 1999.
Approximating the Weight of the Euclidean Minimum.. - Czumaj, Ergün.. (Correct)
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D. Eppstein. Spanning trees and spanners. In Handbook of Computational Geometry, pp. 425--461. Elsevier Science B.V., 1997.
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D. Eppstein. Spanning trees and spanners. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425-461. Elsevier Science, Amsterdam, 2000.
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D. Eppstein. Spanning trees and spanners. In J. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425--461. Elsevier, Amsterdam, 2000.
Approximating Geometric Bottleneck Shortest Paths - Bose, Maheshwari.. (Correct)
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D. Eppstein. Spanning trees and spanners. In Handbook of Computational Geometry, pages 425-461. Elsevier, 2000.
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D. Eppstein. Spanning trees and spanners. In J. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425-461. Elsevier, Amsterdam, 2000.
Sublinear-Time Approximation of Euclidean Minimum.. - Czumaj, Ergün.. (2003) (Correct)
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D. Eppstein. Spanning trees and spanners. In Handbook of Computational Geometry, pp. 425--461. Elsevier Science B.V., 1997.
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D. Eppstein. Spanning trees and spanners. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 425--461. Elsevier Science Publishers B.V. North-Holland, Amsterdam, 2000.
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David Eppstein: "Spanning Trees and Spanners", Technical Report 96-16 http://www.ics.uci.edu/eppstein/, Dept. Information and Computer Science, University of California.
Partitioned Neighborhood Spanners of Minimal Outdegree - Fischer, Lukovszki, Ziegler (1999) (5 citations) (Correct)
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David Eppstein: "Spanning Trees and Spanners", Technical Report 96-16 http://www.ics.uci.edu/eppstein/, Dept. Information and Computer Science, University of California.
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