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57
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" ..."
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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...
New Sparseness Results on Graph Spanners
, 1992
"... Let G = (V, E) be an nvertex connected graph with positive edge weights. A subgraph G ’ = (V, E’) is a tspanner of G if for all u, v E V, the weighted distance between u and v in G ’ is at most t times the weighted distance between u and v in G. We consider the problem of constructing sparse span ..."
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Cited by 86 (8 self)
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Let G = (V, E) be an nvertex connected graph with positive edge weights. A subgraph G ’ = (V, E’) is a tspanner of G if for all u, v E V, the weighted distance between u and v in G ’ is at most t times the weighted distance between u and v in G. We consider the problem of constructing sparse spanners. Sparseness of spanners is measured by two criteria, the size, defined as the number of edges in the spanner, and the weight, defined as the sum of the edge weights in the spanner. In this paper, we concentrate on constructing spanners of small weight. For an arbitrary positive edgeweighted graph G, for any t> 1, and any c>0, we show that a tspanner of G with weight O(n * ). wt(MST) can be constructed in polynomial time. We also show that (logz n)spanners of weight O(1). wt(MST) can be constructed. We then consider spanners for complete graphs induced by a set of points in ddimensional real normed space. The weight of an edge Zy is the norm of the ~y vector. We show that for these graphs, tspanners with total weight O(log n). wt(MST) can be constructed in polynomial time.
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 54 (13 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 46 (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).
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 39 (7 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.
Competitive Online Routing in Geometric Graphs
 Theoretical Computer Science
, 2001
"... We consider online routing algorithms for finding paths between the vertices of plane graphs. ..."
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Cited by 37 (4 self)
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We consider online routing algorithms for finding paths between the vertices of plane graphs.
The greedy triangulation approximates the minimum weight triangulation and can be computed in linear time in the average case
 Department of Computer Science, Lund University
, 1992
"... Abstract This paper settles the following two longstanding open problems: 1. What is the worstcase approximation ratio between the greedy and the minimum weight triangulation? 2. Is there a polynomial time algorithm that always produces a triangulation whose length is within a constant factor from ..."
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Cited by 31 (3 self)
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Abstract This paper settles the following two longstanding open problems: 1. What is the worstcase approximation ratio between the greedy and the minimum weight triangulation? 2. Is there a polynomial time algorithm that always produces a triangulation whose length is within a constant factor from the minimum? The answer to the first question is that the known \Omega (pn) lower bound is tight. The second question is answered in the affirmative by using a slight modification of an O(n log n) algorithm for the greedy triangulation. We also derive some other interesting results. For example, we show that a constantfactor approximation of the minimum weight convex partition can be obtained within the same time bounds. 1 Introduction Let S be any set of n points in the plane. A triangulation of S is a maximal straightline graph whose vertices are the points in S. Any triangulation of S partitions the convex hull of S into empty triangles. A triangulation that has received special attention is the minimum weight triangulation, in which the optimization criteria is to minimize the total edge length. This triangulation has some good properties [2] and is e.g. useful for numerical approximation of bivariate data [20].
Optimally sparse spanners in 3dimensional euclidean space
 In ACM Symposium on Computational Geometry
, 1993
"... Let V be a set of n points in 3dimensional Euclidean space. A subgraph of the complete Euclidean graph is a tspanner if for any u and v in V, the length of the shortest path from u to v in the spanner is at most t times d(u, v). We show that for any t> 1, a greedy algorithm produces a tspanner ..."
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Cited by 30 (1 self)
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Let V be a set of n points in 3dimensional Euclidean space. A subgraph of the complete Euclidean graph is a tspanner if for any u and v in V, the length of the shortest path from u to v in the spanner is at most t times d(u, v). We show that for any t> 1, a greedy algorithm produces a tspanner with O(n) edges, and total edge weight O(1). tot(it4ST), where MST is a minimum spanning tree of V. 1
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 algo ..."
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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...