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58
Approximating Layout Problems on Random Geometric Graphs
 Journal of Algorithms
, 2001
"... In this paper, we study the approximability of several layout problems on a family of random geometric graphs. Vertices of random geometric graphs are randomly distributed on the unit square and are connected by edges whenever they are closer than some given parameter. The layout problems that we co ..."
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Cited by 22 (10 self)
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In this paper, we study the approximability of several layout problems on a family of random geometric graphs. Vertices of random geometric graphs are randomly distributed on the unit square and are connected by edges whenever they are closer than some given parameter. The layout problems that we consider are: Bandwidth, Minimum Linear Arrangement, Minimum Cut Width, Minimum Sum Cut, Vertex Separation and Edge Bisection. We first prove that some of these problems remain NPcomplete even for geometric graphs. Afterwards, we compute lower bounds that hold, almost surely, for random geometric graphs. Then, we present two heuristics that, almost surely, turn to be constant approximation algorithms for our layout problems on random geometric graphs. In fact, for the Bandwidth and Vertex Separation problems, these heuristics are asymptotically optimal. Finally, we use the theoretical results in order to empirically compare these and other wellknown heuristics. # This research was partially ...
Simple Heuristics and PTASs for Intersection Graphs in Wireless Ad Hoc Networks
 in Wireless Ad Hoc Networks, in DialM’02
, 2002
"... In wireless ad hoc networks, each wireless device has a transmission range, which is usually modeled as a disk centered at this node. A wireless node can send message directly to all nodes lying inside this disk. We present several intersection graphs to model the wireless networks. Then we present ..."
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Cited by 18 (8 self)
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In wireless ad hoc networks, each wireless device has a transmission range, which is usually modeled as a disk centered at this node. A wireless node can send message directly to all nodes lying inside this disk. We present several intersection graphs to model the wireless networks. Then we present some simple heuristics and/or PTASs to approximate the maximum independent set, the minimum vertex cover and the minimum graph coloring in these graph models.
Greedy Routing with Bounded Stretch
"... Abstract—Greedy routing is a novel routing paradigm where messages are always forwarded to the neighbor that is closest to the destination. Our main result is a polynomialtime algorithm that embeds combinatorial unit disk graphs (CUDGs – a CUDG is a UDG without any geometric information) into O(log ..."
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Cited by 15 (0 self)
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Abstract—Greedy routing is a novel routing paradigm where messages are always forwarded to the neighbor that is closest to the destination. Our main result is a polynomialtime algorithm that embeds combinatorial unit disk graphs (CUDGs – a CUDG is a UDG without any geometric information) into O(log 2 n)dimensional space, permitting greedy routing with constant stretch. To the best of our knowledge, this is the first greedy embedding with stretch guarantees for this class of networks. Our main technical contribution involves extracting, in polynomial time, a constant number of isometric and balanced tree separators from a given CUDG. We do this by extending the celebrated LiptonTarjan separator theorem for planar graphs to CUDGs. Our techniques extend to other classes of graphs; for example, for general graphs, we obtain an O(log n)stretch greedy embedding into O(log 2 n)dimensional space. The greedy embeddings constructed by our algorithm can also be viewed as a constantstretch compact routing scheme in which each node is assigned an O(log 3 n)bit label. To the best of our knowledge, this result yields the best known stretchspace tradeoff for compact routing on CUDGs. Extensive simulations on random wireless networks indicate that the average routing overhead is about 10%; only few routes have a stretch above 1.5. I.
Distributed Localization Using Noisy Distance and Angle Information
 MOBIHOC'06
, 2006
"... Localization is an important and extensively studied problem in adhoc wireless sensor networks. Given the connectivity graph of the sensor nodes, along with additional local information (e.g. distances, angles, orientations etc.), the goal is to reconstruct the global geometry of the network. In th ..."
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Cited by 14 (3 self)
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Localization is an important and extensively studied problem in adhoc wireless sensor networks. Given the connectivity graph of the sensor nodes, along with additional local information (e.g. distances, angles, orientations etc.), the goal is to reconstruct the global geometry of the network. In this paper, we study the problem of localization with noisy distance and angle information. With no noise at all, the localization problem with both angle (with orientation) and distance information is trivial. However, in the presence of even a small amount of noise, we prove that the localization problem is NPhard. Localization with accurate distance information and relative angle information is also hard. These hardness results motivate our study of approximation schemes. We relax the nonconvex constraints to approximating convex constraints and propose linear programs (LP) for two formulations of the resulting localization problem, which we call the weak deployment and strong deployment problems. These two formulations give upper and lower bounds on the location uncertainty respectively: No sensor is located outside its weak deployment region, and each sensor can be anywhere in its strong deployment region without violating the approximate distance and angle constraints. Though LPbased algorithms are usually solved by centralized methods, we propose distributed, iterative methods, which are provably convergent to the centralized algorithm solutions. We give simulation results for the distributed algorithms, evaluating the convergence rate, dependence on measurement noises, and robustness to link dynamics.
Representing Graphs by Disks and Balls (a survey of recognitioncomplexity results)
"... . Practical applications, like radio frequency assignments, led to the denition of disk intersection graphs in the plane, called shortly disk graphs. If the disks in the representation are not allowed to overlap, we speak about disk contact graphs (coin graphs). In this paper we survey recogniti ..."
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Cited by 12 (1 self)
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. Practical applications, like radio frequency assignments, led to the denition of disk intersection graphs in the plane, called shortly disk graphs. If the disks in the representation are not allowed to overlap, we speak about disk contact graphs (coin graphs). In this paper we survey recognitioncomplexity results for disk intersection and contact graphs in the plane. In particular, we refer a classical result by Koebe about disk contact representations, and works of Breu and Kirkpatrick about boundedratio disk representations. We prove that the recognition of diskintersection graphs (in the unbounded ratio case) is NPhard. This result is proved in a more general setting of noncrossing arcconnected sets. We also show some partial results concerning recognition of ball intersection and contact graphs in higher dimensions. In particular, we prove that the recognition of unitball contact graphs is NPhard in dimensions 3; 4, and 8 (24). 1 Introduction 1.1 Intersect...
Good quality virtual realization of unit ball graphs
 of Lecture Notes in Computer Science
, 2007
"... The quality of an embedding Φ: V ↦ → R 2 of a graph G = (V, E) into the Euclidean plane is the ratio of max{u,v}∈E Φ(u) − Φ(v)2 to min{u,v}�∈E Φ(u) − Φ(v)2. Given a graph G = (V, E), that is known to be a unit ball graph in fixed dimensional Euclidean space R d, we seek algorithms to compu ..."
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Cited by 10 (2 self)
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The quality of an embedding Φ: V ↦ → R 2 of a graph G = (V, E) into the Euclidean plane is the ratio of max{u,v}∈E Φ(u) − Φ(v)2 to min{u,v}�∈E Φ(u) − Φ(v)2. Given a graph G = (V, E), that is known to be a unit ball graph in fixed dimensional Euclidean space R d, we seek algorithms to compute an embedding Φ: V ↦ → R 2 of best (smallest) quality. Note that G comes with no associated geometric information and in this setting, related problems such as recognizing if G is a unit disk graph (UDG), are NPhard. While any connected unit disk graph (UDG) has a 2dimensional embedding with quality between 1/2 and 1, as far as we know, Vempala’s random projection approach (FOCS 1998) provides the best quality bound of O(log 3 n · √ log log n) for this problem. This paper presents a simple, combinatorial algorithm for computing a O(log 2.5 n)quality 2dimensional embedding of a given graph, that is known to be a UBG in fixed dimensional Euclidean space R d. If the embedding is allowed to reside in higher dimensional space, we obtain improved results: a quality2 embedding in R O(d log d). The first step of our algorithm constructs a “growthrestricted approximation ” of the given UBG. While such a construction is trivial if the UBG comes with a geometric representation, we are not aware of any other algorithm that can perform this step without geometric information. Construction of a growthrestricted approximation permits us to bypass the standard and costly technique of solving a linear program with exponentially many “spreading constraints. ” As a side effect of our construction, we get a constantfactor approximation to the minimum clique cover problem for UBGs, described without geometry. The second step of our algorithm combines the probabilistic decomposition of growthrestricted graphs due to Lee and Krauthgamer (STOC 2003) with Rao’s embedding algorithm for planar graphs (SoCG 1999) to obtain a (k, O ( √ log n))volume respecting embedding of growthrestricted graphs. Our problem is a version of the well known localization problem in wireless sensor networks, in which network nodes are required to compute virtual 2dimensional Euclidean coordinates given little or (as in our case) no geometric information.
A ptas for the minimum dominating set problem in unit disk graphs
 In Proc.3rd Workshop on Approximation and Online (WAOA
, 2005
"... A PTAS for the minimum dominating set problem in unit disk graphs ..."
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Cited by 10 (1 self)
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A PTAS for the minimum dominating set problem in unit disk graphs
Surrounding nodes in coordinatefree networks
 In Workshop in Algorithmic Foundations of Robotics
, 2006
"... Summary. Consider a network of nodes in the plane whose locations are unknown but which establish communication links based on proximity. We solve the following problems: given a node in the network, (1) determine if a given cycle surrounds the node; and (2) find some cycle that surrounds the node. ..."
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Cited by 9 (1 self)
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Summary. Consider a network of nodes in the plane whose locations are unknown but which establish communication links based on proximity. We solve the following problems: given a node in the network, (1) determine if a given cycle surrounds the node; and (2) find some cycle that surrounds the node. The only localization capabilities assumed are unique IDs with binary proximity measure, and, in some cases, cyclic orientation of neighbors. We give complete algorithms for finding and verifying surrounding cycles when cyclic orientation data is available. We also provide an efficient but noncomplete algorithm in the case where angular data is not available. 1
Sensor networks continue to puzzle: Selected open problems
 In Proc. 9th Internat. Conf. Distributed Computing and Networking (ICDCN
, 2008
"... Abstract. While several important problems in the field of sensor networks have already been tackled, there is still a wide range of challenging, open problems that merit further attention. We present five theoretical problems that we believe to be essential to understanding sensor networks. The goa ..."
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Cited by 9 (0 self)
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Abstract. While several important problems in the field of sensor networks have already been tackled, there is still a wide range of challenging, open problems that merit further attention. We present five theoretical problems that we believe to be essential to understanding sensor networks. The goal of this work is both to summarize the current state of research and, by calling attention to these fundamental problems, to spark interest in the networking community to attend to these and related problems in sensor networks.
Random Scaled Sector Graphs
 Dept
, 2002
"... In this paper, we introduce a new model of random graph, that we call random sector graph. This model aims to provide a tool for studying communication problems in networks of sensors using laser communication such as the ones addressed in the Smart Dust project. Current technology allows steerin ..."
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Cited by 8 (1 self)
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In this paper, we introduce a new model of random graph, that we call random sector graph. This model aims to provide a tool for studying communication problems in networks of sensors using laser communication such as the ones addressed in the Smart Dust project. Current technology allows steering the laser cannon along a contigous sector, providing undirectional communication. Thus, random sector graphs are a generalization of random geometric graphs, in which this restricted communication is taken into account. We provide tight estimations of the maximum and minimum degree and show that random sector graphs are connected for an adequate selection of the sector radius.