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31
Spanning Trees and Spanners
, 1996
"... We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. ..."
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Cited by 143 (2 self)
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We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. For instance, one may wish to connect components of a VLSI circuit by networks of wires, in a way that uses little surface area on the chip, draws little power, and propagates signals quickly. Similar problems come up in other applications such as telecommunications, road network design, and medical imaging [1]. One network design problem, the Traveling Salesman problem, is sufficiently important to have whole books devoted to it [79]. Problems involving some form of geometric minimum or maximum spanning tree also arise in the solution of other geometric problems such as clustering [12], mesh generation [56], and robot motion planning [93]. One can vary the network design problem in many w...
ClosestPoint Problems in Computational Geometry
, 1997
"... This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, th ..."
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Cited by 65 (14 self)
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This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, the exact and approximate postoffice problem, and the problem of constructing spanners are discussed in detail. Contents 1 Introduction 1 2 The static closest pair problem 4 2.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Algorithms that are optimal in the algebraic computation tree model . 5 2.2.1 An algorithm based on the Voronoi diagram . . . . . . . . . . . 5 2.2.2 A divideandconquer algorithm . . . . . . . . . . . . . . . . . . 5 2.2.3 A plane sweep algorithm . . . . . . . . . . . . . . . . . . . . . . 6 2.3 A deterministic algorithm that uses indirect addressing . . . . . . . . . 7 2.3.1 The degraded grid . . . . . . . . . . . . . . . . . . ...
On the Hardness of Approximating Spanners
 Algorithmica
, 1999
"... A k\Gammaspanner of a connected graph G = (V; E) is a subgraph G 0 consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G 0 is larger than the distance in G by no more than a factor of k. This paper concerns ..."
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Cited by 55 (16 self)
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A k\Gammaspanner of a connected graph G = (V; E) is a subgraph G 0 consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G 0 is larger than the distance in G by no more than a factor of k. This paper concerns the hardness of finding spanners with a number of edges close to the optimum. It is proved that for every fixed k, approximating the spanner problem is at least as hard as approximating the set cover problem We also consider a weighted version of the spanner problem, and prove an essential difference between the approximability of the case k = 2, and the case k 5. Department of Computer Science, The Open University, 16 Klauzner st., Ramat Aviv, Israel, guyk@shaked.openu.ac.il. 1 Introduction The concept of graph spanners has been studied in several recent papers in the context of communication networks, distributed computing, robotics and computational geometry [ADDJ90, C94, CK94,...
Generating Sparse 2spanners
, 1993
"... A kspanner of a connected graph G = (V; E) is a subgraph G 0 consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G 0 is larger than that distance in G by no more than a factor of k. This note concerns the prob ..."
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Cited by 41 (6 self)
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A kspanner of a connected graph G = (V; E) is a subgraph G 0 consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G 0 is larger than that distance in G by no more than a factor of k. This note concerns the problem of finding the sparsest 2spanner in a given graph, and presents an approximation algorithm for this problem with approximation ratio log(E/V).
Balancing Minimum Spanning Trees and ShortestPath Trees
, 2002
"... We give a simple algorithm to find a spanning tree that simultaneously approximates a shortestpath tree and a minimum spanning tree. The algorithm provides a continuous tradeoff: given the two trees and a fl? 0, the algorithm returns a spanning tree in which the distance between any vertex and the ..."
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Cited by 39 (1 self)
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We give a simple algorithm to find a spanning tree that simultaneously approximates a shortestpath tree and a minimum spanning tree. The algorithm provides a continuous tradeoff: given the two trees and a fl? 0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortestpath tree is at most 1 + p 2fl times the shortestpath distance, and yet the total weight of the tree is at most 1 + p 2=fl times the weight of a minimum spanning tree. Our algorithm runs in linear time and obtains the bestpossible tradeoff. It can be implemented on a CREW PRAM to run in logarithmic time using one processor per vertex.
Constructing Plane Spanners of Bounded Degree and Low Weight
 in Proceedings of European Symposium of Algorithms
, 2002
"... Given a set S of n points in the plane, we give an O(n log n)time algorithm that constructs a plane tspanner for S, for t 10:02, such that the degree of each point of S is bounded from above by 27, and the total edge length is proportional to the weight of a minimum spanning tree of S. These c ..."
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Cited by 36 (6 self)
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Given a set S of n points in the plane, we give an O(n log n)time algorithm that constructs a plane tspanner for S, for t 10:02, such that the degree of each point of S is bounded from above by 27, and the total edge length is proportional to the weight of a minimum spanning tree of S. These constants are all worst case constants that are artifacts of our proofs. In practice, we believe them to be much smaller. Previously, no algorithms were known for constructing plane tspanners of bounded degree.
Constructing Competitive Tours From Local Information
 Theoretical Computer Science
, 1994
"... We consider the problem of a searcher exploring an initially unknown weighted planar graph G. When the searcher visits a vertex v, it learns of each edge incident to v. The searcher's goal is to visit each vertex of G, incurring as little cost as possible. We present a constant competitive algorithm ..."
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Cited by 27 (2 self)
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We consider the problem of a searcher exploring an initially unknown weighted planar graph G. When the searcher visits a vertex v, it learns of each edge incident to v. The searcher's goal is to visit each vertex of G, incurring as little cost as possible. We present a constant competitive algorithm for this problem. 1 Introduction In this paper we consider the following situation. A salesperson is assigned to visit all the towns in some rural state that he/she knows nothing about. Of course, the salesperson wishes to accomplish this with as little time spent traveling as possible. The salesperson, however, is not given the benefit of having a map. Hence, when the salesperson visits a town, the only information that he/she may be able to glean about other cities is from the road signs on the roads leaving that town. Each road sign gives the name and the distance to the next city down that road. As the salesperson visits towns, new information may reveal shorter routes and may cause th...
Constructing degree3 spanners with other sparseness properties
 International Journal of Foundations of Computer Science
, 1996
"... Let V be any set of n points in kdimensional Euclidean space. A subgraph of the complete Euclidean graph is a tspanner if for any ~ and ~ in V, the length of the shortest path from u to v in the spanner is at most t times d(~, ~). We show that for any 5> 1, there exists a polynomialtime construct ..."
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Cited by 19 (0 self)
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Let V be any set of n points in kdimensional Euclidean space. A subgraph of the complete Euclidean graph is a tspanner if for any ~ and ~ in V, the length of the shortest path from u to v in the spanner is at most t times d(~, ~). We show that for any 5> 1, there exists a polynomialtime constructihle tspanner (where ~ is a constant that depends only on 5 and k) with the following properties. Its maximum degree is 3, it has at most n 9 6 edges, and its total edge weight is comparable to the minimum spanning tree of V (for/ ~ < 3 its weight is O(1). wt(MgT), and for k> 3 its weight is O(log n). wt(MST)). 1
Improved Approximation Schemes for Geometrical Graphs Via Spanners and Banyans
 In 30th ACM Symposium on Theory of Computing (STOC'98
, 1998
"... We give deterministic and randomized algorithms to find a Euclidean traveling salesman tour (TST) of length within (1 + 1=s) times optimal. They run in O(N log N) time and O(N) space for constant dimension and s. These time and space bounds are optimal in an algebraic computation tree model. We can ..."
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Cited by 17 (2 self)
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We give deterministic and randomized algorithms to find a Euclidean traveling salesman tour (TST) of length within (1 + 1=s) times optimal. They run in O(N log N) time and O(N) space for constant dimension and s. These time and space bounds are optimal in an algebraic computation tree model. We can also find a (1 + 1=s) times optimal length 2matching (M2M), edge cover (EC), minimum spanning tree (MST), Steiner minimal tree (SMT), rectilinear ditto (RSMT), and related graphs in the same time bound. This improves recent algorithms of Arora, which had used N(log N) O(s d\Gamma1 ) time in fixed dimension d to produce a (1 + 1=s) times optimal TST (or SMT, RSMT) with success probability 1=2. To verify success, however, Arora could only use a deterministic version of his algorithm that took a factor of N d more time. The increase in running time for our deterministic version depends only on s. Arora's approach can also be extended to produce other (1 + ffl)approximate geometrical grap...