Results 1  10
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16
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 139 (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...
Relay Placement for Higher Order Connectivity in Wireless Sensor Networks
"... Sensors typically use wireless transmitters to communicate with each other. However, sensors may be located in a way that they cannot even form a connected network (e.g, due to failures of some sensors, or loss of battery power). In this paper we consider the problem of adding the smallest number o ..."
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Cited by 22 (1 self)
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Sensors typically use wireless transmitters to communicate with each other. However, sensors may be located in a way that they cannot even form a connected network (e.g, due to failures of some sensors, or loss of battery power). In this paper we consider the problem of adding the smallest number of additional (relay) nodes so that the induced communication graph is 2connected 1. The problem is NPhard. In this paper we develop O(1)approximation algorithms that find close to optimal solutions in time O((kn) 2) for achieving kedge connectivity of n nodes. The worst case approximation guarantee is 10, but the algorithm produces solutions that are far better than this bound suggests. We also consider extensions to higher dimensions, and the scheme that we develop for points in the plane, yields a bound of 2dMST where dMST is the maximum degree of a minimumdegree Minimum Spanning Tree in d dimensions using Euclidean metrics. In addition, our methods extend with the same approximation guarantees to a generalization when the locations of relays are required to avoid certain polygonal regions (obstacles). We also prove that if the sensors are uniformly and identically distributed in a unit square, the expected number of relay nodes required goes to zero as the number of sensors goes to infinity.
A NetworkFlow Technique for Finding LowWeight BoundedDegree Spanning Trees
 JOURNAL OF ALGORITHMS
, 1996
"... Given a graph with edge weights satisfying the triangle inequality, and a degree bound for each vertex, the problem of computing a low weight spanning tree such that the degree of each vertex is at most its specified bound is considered. In particular, modifying a given spanning tree T using ad ..."
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Cited by 20 (1 self)
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Given a graph with edge weights satisfying the triangle inequality, and a degree bound for each vertex, the problem of computing a low weight spanning tree such that the degree of each vertex is at most its specified bound is considered. In particular, modifying a given spanning tree T using adoptions to meet the degree constraints is considered. A novel networkflow based algorithm for finding a good sequence of adoptions is introduced. The method yields a better performance guarantee than any previously obtained. Equally importantly, it takes this approach to the limit in the following sense: if any performance guarantee that is solely a function of the topology and edge weights of a given tree holds for any algorithm at all, then it also holds for our algorithm. The performance guarantee is the following. If the degree constraint d(v) for each v is at least 2, the algorithm is guaranteed to find a tree whose weight is at most the weight of the given tree times 2 \Gamma min n d(v)\Gamma2 deg T (v)\Gamma2 : deg T (v) ? 2 o ; where deg T (v) is the initial degree of v. Examples are provided in which no lighter tree meeting the degree constraint exists. Lineartime algorithms are provided with the same worstcase performance guarantee. Choosing T to be a minimum spanning tree yields approximation algorithms for the general problem on geometric graphs with distances induced by various Lp norms. Finally, examples of Euclidean graphs are provided in which the ratio of the lengths of an optimal Traveling Salesman path and a minimum spanning tree can be arbitrarily close to 2.
A New Heuristic for Rectilinear Steiner Trees
 In Proc. IEEE Int. Conf. on CAD
"... The minimum rectilinear Steiner tree (RST) problem is one of the fundamental problems in the field of electronic design automation. The problem is NPhard, and much work has been devoted to designing good heuristics and approximation algorithms; to date, the champion in solution quality among RST he ..."
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Cited by 20 (2 self)
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The minimum rectilinear Steiner tree (RST) problem is one of the fundamental problems in the field of electronic design automation. The problem is NPhard, and much work has been devoted to designing good heuristics and approximation algorithms; to date, the champion in solution quality among RST heuristics is the Batched Iterated 1Steiner (BI1S) heuristic of Kahng and Robins. In a recent development, exact RST algorithms have witnessed spectacular progress: The new release of the GeoSteiner code of Warme, Winter, and Zachariasen has average running time comparable to that of the fastest available BI1S implementation, due to Robins. We are thus faced with the paradoxical situation that an exact algorithm for an NPhard problem is competitive in speed with a stateoftheart heuristic for the problem. The main contribution of this paper is a new RST heuristic, which has at its core a recent 3=2 approximation algorithm of Rajagopalan and Vazirani for the metric Steiner tree problem on ...
Euclidean BoundedDegree Spanning Tree Ratios
, 2003
"... Let K be the worstcase (supremum) ratio of the weight of the minimum degreeK spanning tree to the weight of the minimum spanning tree, over all finite point sets in the Euclidean plane. It is known that ..."
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Cited by 16 (0 self)
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Let K be the worstcase (supremum) ratio of the weight of the minimum degreeK spanning tree to the weight of the minimum spanning tree, over all finite point sets in the Euclidean plane. It is known that
Efficient minimum spanning tree construction without Delaunay triangulation
 UNI 4.0 Security Addendum, ATM Forum BTDSIGSEC
, 2001
"... Minimum spanning tree problem is a very important problem in VLSI CAD. Given n points in a plane, a minimum spanning tree is a set of edges which connects all the points and has a minimum total length. A naive approach enumerates edges on all pairs of points and takes at least R(n2) time. More eff ..."
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Cited by 7 (0 self)
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Minimum spanning tree problem is a very important problem in VLSI CAD. Given n points in a plane, a minimum spanning tree is a set of edges which connects all the points and has a minimum total length. A naive approach enumerates edges on all pairs of points and takes at least R(n2) time. More efficient approaches find a minimum spanning tree only among edges in the Delaunay triangulation of the points. However, Delaunay triangulation is not well defined in rectilinear distance. In this paper, we first establish a framework for minimum spanning tree construction which is based on a general concept of spanning graphs. A spanning graph is a natural definition and not necessarily a Delaunay triangulation. Based on this framework, we then design an O(n log n) sweepline algorithm to construct a rectilinear minimum spanning tree without using Delaunay triangulation. 1
A note on the MST heuristic for bounded edgelength Steiner Trees with minimum number of Steiner Points
 Information Processing Letters, 75:165– 167
, 1999
"... We give a tight analysis of the MST heuristic recently introduced by G.H. Lin and G. Xue for approximating the Steiner tree with minimum number of Steiner points and bounded edgelengths. The approximation factor of the heuristic is shown to be one less than the MST number of the underlying space, ..."
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Cited by 7 (2 self)
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We give a tight analysis of the MST heuristic recently introduced by G.H. Lin and G. Xue for approximating the Steiner tree with minimum number of Steiner points and bounded edgelengths. The approximation factor of the heuristic is shown to be one less than the MST number of the underlying space, defined as the maximum possible degree of a minimumdegree MST spanning points from the space. In particular, on instances drawn from the Euclidean (resp. rectilinear) plane, the MST heuristic is shown to have tight approximation factors of 4, respectively 3. Keywords: Approximation algorithms, Steiner trees, MST heuristic, fixed wireless network design, VLSI CAD. 1 Introduction The classical Steiner tree problem is that of finding a shortest tree spanning a given set of terminal points. The tree may use additional points besides the terminals, these points are commonly referred to as Steiner points. In the Minimum number of Steiner Points Tree (MSPT) problem [7,5] one also seeks a tree ...
Approximation Algorithms for the Capacitated Minimum Spanning Tree Problem and its Variants in Network Design
, 2004
"... Given an undirected graph G = (V, E) with nonnegative costs on its edges, a root node r V with demand v D wishing to route w(v) units of flow (weight) to r, and a positive number k, the Capacitated Minimum Steiner Tree (CMStT) problem asks for a minimum Steiner tree, rooted at r, spannin ..."
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Cited by 6 (4 self)
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Given an undirected graph G = (V, E) with nonnegative costs on its edges, a root node r V with demand v D wishing to route w(v) units of flow (weight) to r, and a positive number k, the Capacitated Minimum Steiner Tree (CMStT) problem asks for a minimum Steiner tree, rooted at r, spanning the vertices in D in which the sum of the vertex weights in every subtree hanging o# r is at most k. When D = V , this problem is known as the Capacitated Minimum Spanning Tree (CMST) problem. Both CMStT and CMST problems are NPhard. In this paper, we present approximation algorithms for these problems and several of their variants in network design. Our main results are the following.
DegreeBounded Minimum Spanning Trees
, 2004
"... Given n points in the Euclidean plane, the degree MST problem asks for a spanning tree of minimum weight in which the degree of each node is at most . ..."
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Cited by 3 (0 self)
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Given n points in the Euclidean plane, the degree MST problem asks for a spanning tree of minimum weight in which the degree of each node is at most .
Small degree outbranchings
, 2001
"... Using a suitable orientation, we give a short proof of a result of Czumaj and Strothmann [3]: Every 2edgeconnected graph G contains a spanning tree T with the property that dT (v) ≤ dG(v)+3 for every vertex v. 2 Trying to find an analogue of this result in the directed case, we prove that every 2 ..."
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Cited by 3 (0 self)
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Using a suitable orientation, we give a short proof of a result of Czumaj and Strothmann [3]: Every 2edgeconnected graph G contains a spanning tree T with the property that dT (v) ≤ dG(v)+3 for every vertex v. 2 Trying to find an analogue of this result in the directed case, we prove that every 2arcstrong digraph D has an outbranching B such that d + d+