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Simple heuristics for unit disk graphs
 NETWORKS
, 1995
"... Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NPhard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring ..."
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Cited by 130 (6 self)
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Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NPhard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an online coloring heuristic which achieves a competitive ratio of 6 for unit disk graphs. Our heuristics do not need a geometric representation of unit disk graphs. Geometric representations are used only in establishing the performance guarantees of the heuristics. Several of our approximation algorithms can be extended to intersection graphs of circles of arbitrary radii in the plane, intersection graphs of regular polygons, and to intersection graphs of higher dimensional regular objects.
On Reducing Broadcast Redundancy in Ad Hoc Wireless Networks
, 2003
"... Unlike in a wired network, a packet transmitted by a node in an ad hoc wireless network can reach all neighbors. Therefore, the total number of transmissions (forward nodes) is generally used as the cost criterion for broadcasting. The problem of finding the minimum number of forward nodes is NPcomp ..."
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Cited by 123 (23 self)
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Unlike in a wired network, a packet transmitted by a node in an ad hoc wireless network can reach all neighbors. Therefore, the total number of transmissions (forward nodes) is generally used as the cost criterion for broadcasting. The problem of finding the minimum number of forward nodes is NPcomplete. Among various approximation approaches, dominant pruning [7] utilizes 2hop neighborhood information to reduce redundant transmissions. In this paper, we analyze some deficiencies of the dominant pruning algorithm and propose two better approximation algorithms: total dominant pruning and partial dominant pruning. Both algorithms utilize 2hop neighborhood information more effectively to reduce redundant transmissions. Simulation results of applying these two algorithms show performance improvements compared with the original dominant pruning. In addition, two termination criteria are discussed and compared through simulation.
NCApproximation Schemes for NP and PSPACEHard Problems for Geometric Graphs
, 1997
"... We present NC approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance tradeoff as the best known approximation schemes for planar gr ..."
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Cited by 103 (1 self)
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We present NC approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance tradeoff as the best known approximation schemes for planar graphs. We also define the concept of precision unit disk graphs and show that for such graphs the approximation schemes have a better time versus performance tradeoff than the approximation schemes for arbitrary unit disk graphs. Moreover, compared to unit disk graphs, we show that for precision unit disk graphs, many more graph problems have efficient approximation schemes. Our NC approximation schemes can also be extended to obtain efficient NC approximation schemes for several PSPACEhard problems on unit disk graphs specified using a restricted version of the hierarchical specification language of Bentley, Ottmann and Widmayer. The approximation schemes for hierarchically specified un...
Unit Disk Graph Recognition is NPHard
 Computational Geometry. Theory and Applications
, 1993
"... Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have s ..."
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Cited by 94 (2 self)
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Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have sphericity at most 3, is NPhard. We show how this reduction can be extended to 3 dimensions, thereby showing that unit sphere graph recognition, or determining if a graph has sphericity 3 or less, is also NPhard. We conjecture that Ksphericity is NPhard for all fixed K greater than 1. 1 Introduction A unit disk graph is the intersection graph of a set of unit diameter closed disks in the plane. That is, each vertex corresponds to a disk in the plane, and two vertices are adjacent in the graph if the corresponding disks intersect. The set of disks is said to realize the graph. Of course, the unit of distance is not critical, since the disks realize the same graph even if the coordina...
A Clustering Scheme for Hierarchical Control in Multihop Wireless Networks
, 2001
"... In this paper we present a clustering scheme to create a hierarchical control structure for multihop wireless networks. A cluster is defined as a subset of vertices, whose induced graph is connected. In addition, a cluster is required to obey certain constraints that are useful for management and s ..."
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Cited by 94 (0 self)
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In this paper we present a clustering scheme to create a hierarchical control structure for multihop wireless networks. A cluster is defined as a subset of vertices, whose induced graph is connected. In addition, a cluster is required to obey certain constraints that are useful for management and scalability of the hierarchy. All these constraints cannot be met simultaneously for general graphs, but we show how such a clustering can be obtained for wireless network topologies. Finally, we present an efficient distributed implementation of our clustering algorithm for a set of wireless nodes to create the set of desired clusters. KeywordsClustering, Adhoc networks, Wireless networks, Sensor networks, Hierarchy I. INTRODUCTION R APID advances in hardware design have greatly reduced cost, size and the power requirements of network elements. As a consequence, it is now possible to envision networks comprising of a large number of such small devices. In the Smart Dust project at UC...
Distributed Heuristics for Connected Dominating Sets in Wireless Ad Hoc Networks
 Journal of Communications and Networks
, 2002
"... A connected dominating set (CDS) for a graph is a subset of , such that each node in is adjacent to some node in , and induces a connected subgraph. CDSs have been proposed as a virtual backbone for routing in wireless ad hoc networks. However, it is NPhard to find a minimum connecte ..."
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Cited by 79 (5 self)
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A connected dominating set (CDS) for a graph is a subset of , such that each node in is adjacent to some node in , and induces a connected subgraph. CDSs have been proposed as a virtual backbone for routing in wireless ad hoc networks. However, it is NPhard to find a minimum connected dominating set (MCDS). An approximation algorithm for MCDS in general graphs has been proposed in the literature with performance guarantee of where is the maximal nodal degree [1]. This algorithm has been implemented in distributed manner in wireless networks [2][4]. This distributed implementation suffers from high time and message complexity, and the performance ratio remains . Another distributed algorithm has been developed in [5], with performance ratio of . Both algorithms require twohop neighborhood knowledge and a message length of . On the other hand, wireless ad hoc networks have a unique geometric nature, which can be modeled as a unitdisk graph (UDG), and thus admits heuristics with better performance guarantee. In this paper we propose two destributed heuristics with constant performance ratios. The time and message complexity for any of these algorithms is , and "!$# , respectively. Both of these algorithms require only singlehop neighborhood knowledge, and a message length of &%' .
PolynomialTime Approximation Schemes for Geometric Graphs
, 2001
"... A disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane. For the case that the disk representation is given, we present polynomialtime approximation schemes (PTASs) for the maximum weight independent set problem (selecting disjoint disks of maximum total weigh ..."
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Cited by 78 (4 self)
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A disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane. For the case that the disk representation is given, we present polynomialtime approximation schemes (PTASs) for the maximum weight independent set problem (selecting disjoint disks of maximum total weight) and for the minimum weight vertex cover problem in disk graphs. These are the first known PTASs for NPhard optimization problems on disk graphs. They are based on a novel recursive subdivision of the plane that allows applying a shifting strategy on different levels simultaneously, so that a dynamic programming approach becomes feasible. The PTASs for disk graphs represent a common generalization of previous results for planar graphs and unit disk graphs. They can be extended to intersections graphs of other "disklike" geometric objects (such as squares or regular polygons), also in higher dimensions.
On the Computational Complexity of Sensor Network Localization
 In Proceedings of First International Workshop on Algorithmic Aspects of Wireless Sensor Networks
, 2004
"... Determining the positions of the sensor nodes in a network is essential to many network functionalities such as routing, coverage and tracking, and event detection. The localization problem for sensor networks is to reconstruct the positions of all of the sensors in a network, given the distances ..."
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Cited by 73 (4 self)
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Determining the positions of the sensor nodes in a network is essential to many network functionalities such as routing, coverage and tracking, and event detection. The localization problem for sensor networks is to reconstruct the positions of all of the sensors in a network, given the distances between all pairs of sensors that are within some radius r of each other. In the past few years, many algorithms for solving the localization problem were proposed, without knowing the computational complexity of the problem. In this paper, we show that no polynomialtime algorithm can solve this problem in the worst case, even for sets of distance pairs for which a unique solution exists, unless RP = NP. We also discuss the consequences of our result and present open problems.
Minimizing broadcast latency and redundancy in ad hoc networks
 In Proc. of the Fourth ACM Int. Symposium on Mobile Ad Hoc Networking and Computing (MOBIHOC'03
, 2003
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PolynomialTime Approximation Scheme for Minimum Connected Dominating Set in Ad Hoc Wireless Networks
 Networks
"... A connected dominating set in a graph is a subset of vertices such that every vertex is either in the subset or adjacent to a vertex in the subset and the subgraph induced by the subset is connected. The minimum connected dominating set is such a vertex subset with minimum cardinality. An applicatio ..."
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Cited by 59 (10 self)
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A connected dominating set in a graph is a subset of vertices such that every vertex is either in the subset or adjacent to a vertex in the subset and the subgraph induced by the subset is connected. The minimum connected dominating set is such a vertex subset with minimum cardinality. An application in ad hoc wireless networks requires the study of the minimum connected dominating set in unitdisk graphs. In this paper, we design (1+1=s)approximation for the minimum connected dominating set in unitdisk graphs, running in time n O((slogs) 2 )