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45
Design and Analysis of an MSTBased Topology Control Algorithm
, 2002
"... In this paper, we present a Minimum Spanning Tree (MST) based topology control algorithm, called Local Minimum Spanning Tree (LMST), for wireless multihop networks. In this algorithm, each node builds its local minimum spanning tree independently and only keeps ontree nodes that are onehop away a ..."
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Cited by 192 (4 self)
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In this paper, we present a Minimum Spanning Tree (MST) based topology control algorithm, called Local Minimum Spanning Tree (LMST), for wireless multihop networks. In this algorithm, each node builds its local minimum spanning tree independently and only keeps ontree nodes that are onehop away as its neighbors in the final topology. We analytically prove several important properties of LMST: (1) the topology derived under LMST preserves the network connectivity; (2) the node degree of any node in the resulting topology is bounded by 6; and (3) the topology can be transformed into one with bidirectional links (without impairing the network connectivity) after removal of all unidirectional links. These results are corroborated in the simulation study.
On the complexity of computing minimum energy consumption broadcast subgraphs
 in Symposium on Theoretical Aspects of Computer Science
, 2001
"... Abstract. We consider the problem of computing an optimal range assignment in a wireless network which allows a specified source station to perform a broadcast operation. In particular, we consider this problem as a special case of the following more general combinatorial optimization problem, calle ..."
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Cited by 96 (11 self)
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Abstract. We consider the problem of computing an optimal range assignment in a wireless network which allows a specified source station to perform a broadcast operation. In particular, we consider this problem as a special case of the following more general combinatorial optimization problem, called Minimum Energy Consumption Broadcast Subgraph (in short, MECBS): Given a weighted directed graph and a specified source node, find a minimum cost range assignment to the nodes, whose corresponding transmission graph contains a spanning tree rooted at the source node. We first prove that MECBS is not approximable within a constant factor (unless P=NP). We then consider the restriction of MECBS to wireless networks and we prove several positive and negative results, depending on the geometric space dimension and on the distancepower gradient. The main result is a polynomialtime approximation algorithm for the NPhard case in which both the dimension and the gradient are equal to 2: This algorithm can be generalized to the case in which the gradient is greater than or equal to the dimension. 1
Proximity Problems on Moving Points
 In Proc. 13th Annu. ACM Sympos. Comput. Geom
, 1997
"... A kinetic data structure for the maintenance of a multidimensional range search tree is introduced. This structure is used as a building block to obtain kinetic data structures for two classical geometric proximity problems in arbitrary dimensions: the first structure maintains the closest pair o ..."
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Cited by 49 (15 self)
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A kinetic data structure for the maintenance of a multidimensional range search tree is introduced. This structure is used as a building block to obtain kinetic data structures for two classical geometric proximity problems in arbitrary dimensions: the first structure maintains the closest pair of a set of continuously moving points, and is provably e#cient. The second structure maintains a spanning tree of the moving points whose cost remains within some prescribed factor of the minimum spanning tree. The method for maintaining the closest pair of points can be extended to the maintenance of closest pair of other distance functions which allows us to maintain the closest pair of a set of moving objects with similar sizes and of a set of points on a smooth manifold.
Closing the Gap: NearOptimal Steiner Trees in Polynomial Time
 IEEE Trans. ComputerAided Design
, 1994
"... The minimum rectilinear Steiner tree (MRST) problem arises in global routing and wiring estimation, as well as in many other areas. The MRST problem is known to be NPhard, and the best performing MRST heuristic to date is the Iterated 1Steiner (I1S) method recently proposed by Kahng and Robins. In ..."
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Cited by 42 (13 self)
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The minimum rectilinear Steiner tree (MRST) problem arises in global routing and wiring estimation, as well as in many other areas. The MRST problem is known to be NPhard, and the best performing MRST heuristic to date is the Iterated 1Steiner (I1S) method recently proposed by Kahng and Robins. In this paper we develop a straightforward, efficient implementation of I1S, achieving a speedup factor of three orders of magnitude over previous implementations. We also give a parallel implementation that achieves nearlinear speedup on multiple processors. Several performanceimproving enhancements enable us to obtain Steiner trees with average cost within 0.25% of optimal, and our methods produce optimal solutions in up to 90% of the cases for typical nets. We generalize I1S and its variants to three dimensions, as well as to the case where all the pins lie on k parallel planes, which arises in, e.g., multilayer routing. Motivated by the goal of reducing the running times of our algorith...
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.
Approximation Algorithms for DegreeConstrained MinimumCost NetworkDesign Problems
, 2001
"... We study networkdesign problems with two different design objectives: the total cost of the edges and nodes in the network and the maximum degree of any node in the network. A prototypical example is the degreeconstrained nodeweighted Steiner tree problem: We are given an undirected graph ..."
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Cited by 31 (2 self)
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We study networkdesign problems with two different design objectives: the total cost of the edges and nodes in the network and the maximum degree of any node in the network. A prototypical example is the degreeconstrained nodeweighted Steiner tree problem: We are given an undirected graph , with a nonnegative integral function that specifies an upper bound on the degree of each vertex in the Steiner tree to be constructed, nonnegative costs on the nodes, and a subset of nodes called terminals. The goal is to construct a Steiner containing all the terminals such that the degree of any node is at most the specified upper bound and the total cost of the nodes in is minimum. Our main result is a bicriteria approximation algorithm whose output is approximate in terms of both the degree and cost criteria  the degree of any node in the output Steiner tree is and the cost of the tree is times that of a minimumcost Steiner tree that obeys the degree bound for each node . Our result extends to the more general problem of constructing oneconnected networks such as generalized Steiner forests. We also consider the special case in which the edge costs obey the triangle inequality and present simple approximation algorithms with better performance guarantees.
Low Degree Spanning Trees Of Small Weight
, 1996
"... . Given n points in the plane, the degreeK spanning tree problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most K. This paper addresses the problem of computing lowweight degreeK spanning trees for K ? 2. It is shown that for an arbitrary collection of n ..."
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Cited by 30 (2 self)
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. Given n points in the plane, the degreeK spanning tree problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most K. This paper addresses the problem of computing lowweight degreeK spanning trees for K ? 2. It is shown that for an arbitrary collection of n points in the plane, there exists a spanning tree of degree three whose weight is at most 1.5 times the weight of a minimum spanning tree. It is shown that there exists a spanning tree of degree four whose weight is at most 1.25 times the weight of a minimum spanning tree. These results solve open problems posed by Papadimitriou and Vazirani. Moreover, if a minimum spanning tree is given as part of the input, the trees can be computed in O(n) time. The results are generalized to points in higher dimensions. It is shown that for any d 3, an arbitrary collection of points in ! d contains a spanning tree of degree three, whose weight is at most 5/3 times the weight of a minimum spanning tre...
On Embedding an OuterPlanar Graph in a Point Set
 CGTA: Computational Geometry: Theory and Applications
, 1997
"... Given an nvertex outerplanar graph G and a set P of n points in the plane, we present an O(n log n) time and O(n) space algorithm to compute a straightline embedding of G in P , improving upon the algorithm in [GMPP91, CU96] that requires O(n ) time. Our algorithm is nearoptimal as the ..."
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Cited by 27 (1 self)
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Given an nvertex outerplanar graph G and a set P of n points in the plane, we present an O(n log n) time and O(n) space algorithm to compute a straightline embedding of G in P , improving upon the algorithm in [GMPP91, CU96] that requires O(n ) time. Our algorithm is nearoptimal as there is an\Omega (n log n) lower bound for the problem [BMS95]. We present a simpler O(nd) time and O(n) space algorithm to compute a straightline embedding of G in P where log n d 2n is the length of the longest vertex disjoint path in the dual of G. Therefore, the time complexity of the simpler algorithm varies between O(n log n) and O(n ) depending on the value of d. More efficient algorithms are presented for certain restricted cases. If the dual of G is a path, then an optimal \Theta(n log n) time algorithm is presented. If the given point set is in convex position then we show that O(n) time suffices.
An Efficient Evolutionary Algorithm for the DegreeConstrained Minimum Spanning Tree Problem
, 2000
"... The representation of candidate solutions and the variation operators are fundamental design choices in an evolutionary algorithm (EA). This paper proposes a novel representation technique and suitable variation operators for the degreeconstrained minimum spanning tree problem. For a weighted, undi ..."
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Cited by 23 (5 self)
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The representation of candidate solutions and the variation operators are fundamental design choices in an evolutionary algorithm (EA). This paper proposes a novel representation technique and suitable variation operators for the degreeconstrained minimum spanning tree problem. For a weighted, undirected graph G(V, E), this problem seeks to identify the shortest spanning tree whose node degrees do not exceed an upper bound d 2. Within the EA, a candidate spanning tree is simply represented by its set of edges. Special initialization, crossover, and mutation operators are used to generate new, always feasible candidate solutions. In contrast to previous spanning tree representations, the proposed approach provides substantially higher locality and is nevertheless computationally efficient; an offspring is always created in O(V time. In addition, it is shown how problemdependent heuristics can be effectively incorporated into the initialization, crossover, and mutation operators without increasing the timecomplexity. Empirical results are presented for hard problem instances with up to 500 vertices. Usually, the new approach identifies solutions superior to those of several other optimization methods within few seconds. The basic ideas of this EA are also applicable to other network optimization tasks.
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 23 (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.