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58
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 126 (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.
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 93 (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...
A Theory of Network Localization
, 2004
"... In this paper we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigid ..."
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Cited by 62 (6 self)
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In this paper we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigidity theory to test the conditions for unique localizability and to construct uniquely localizable networks. We further study the computational complexity of network localization and investigate a subclass of grounded graphs where localization can be computed efficiently. We conclude with a discussion of localization in sensor networks where the sensors are placed randomly.
Unit Disk Graph Approximation
 In Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications (DIALM
, 2004
"... Finding a good embedding of a unit disk graph given by its connectivity information is a problem of practical importance in a variety of fields. In wireless ad hoc and sensor networks, such an embedding can be used to obtain virtual coordinates. In this paper, we prove a nonapproximability result f ..."
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Cited by 49 (9 self)
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Finding a good embedding of a unit disk graph given by its connectivity information is a problem of practical importance in a variety of fields. In wireless ad hoc and sensor networks, such an embedding can be used to obtain virtual coordinates. In this paper, we prove a nonapproximability result for the problem of embedding a given unit disk graph. Particularly, we show that if nonneighboring nodes are not allowed to be closer to each other than distance 1, then two neighbors can be as far apart as #, where # goes to 0 as n goes to infinity, unless P = NP . We further show that finding a realization of a dquasi unit disk graph with 1/ # 2 is NP hard.
Virtual Coordinates for Ad hoc and Sensor Networks
, 2004
"... In many applications of wireless ad hoc and sensor networks, positionawareness is of great importance. Often, as in the case of geometric routing, it is sufficient to have virtual coordinates, rather than real coordinates. In this paper, we address the problem of obtaining virtual coordinates based ..."
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Cited by 48 (9 self)
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In many applications of wireless ad hoc and sensor networks, positionawareness is of great importance. Often, as in the case of geometric routing, it is sufficient to have virtual coordinates, rather than real coordinates. In this paper, we address the problem of obtaining virtual coordinates based on connectivity information. In particular, we propose the first approximation algorithm for this problem and discuss implementational aspects.
Map Labeling and Its Generalizations
"... Map labeling is of fundamental importance in cartography and geographical information systems and is one of the areas targeted for research by the ACM Computational Geometry Impact Task Force. Previous work on map labeling has focused on the problem of placing maximal uniform, axisaligned, disjoint ..."
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Cited by 41 (5 self)
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Map labeling is of fundamental importance in cartography and geographical information systems and is one of the areas targeted for research by the ACM Computational Geometry Impact Task Force. Previous work on map labeling has focused on the problem of placing maximal uniform, axisaligned, disjoint rectangles on the plane so that each point feature to be labeled lies at the corner of one rectangle. Here, we consider a number of variants of the map labeling problem. We obtain three general types of results. First, we devise constantfactor polynomialtime approximation algorithms for labeling point features by rectangular labels, where the feature may lie anywhere on the boundary of its label region and where labeling rectangles may be placed in any orientation. These results generalize to the case of elliptical labels. Secondly, we consider the problem of labeling a map consisting of disjoint rectilinear line segments. We obtain constantfactor polynomialtime approximation algorithms for the general problem and an optimal algorithm for the special case where all segments are horizontal. Finally, we formulate a bicriteria version of the maplabeling problem and provide bicriteria polynomialtime approximation schemes for a number of such problems.
Fast Deterministic Distributed Maximal Independent Set Computation on GrowthBounded Graphs
 IN PROC. 19TH CONFERENCE ON DISTRIBUTED COMPUTING (DISC
, 2005
"... The distributed complexity of computing a maximal independent set in a graph is of both practical and theoretical importance. While there exists an elegant O(log n) time randomized algorithm for general graphs [20], no deterministic polylogarithmic algorithm is known. In this paper, we study the p ..."
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Cited by 40 (12 self)
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The distributed complexity of computing a maximal independent set in a graph is of both practical and theoretical importance. While there exists an elegant O(log n) time randomized algorithm for general graphs [20], no deterministic polylogarithmic algorithm is known. In this paper, we study the problem in graphs with bounded growth, an important family of graphs which includes the wellknown unit disk graph and many variants thereof. Particularly, we propose a deterministic algorithm that computes a maximal independent set in time O(log \Delta * log*n) in graphs with bounded growth, where n and \Delta denote the number of nodes and the maximal degree in G, respectively.
Computing 2Hop Neighborhoods in Ad Hoc Wireless Networks
 In ADHOCNOW’03
, 2003
"... We present efficient distributed algorithms for computing 2hop neighborhoods in Ad Hoc Wireless Networks. The knowledge of the 2hop neighborhood is assumed in many protocols and algorithms for routing, clustering, and distributed channel assignment, but no efficient distributed algorithms for comp ..."
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Cited by 36 (0 self)
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We present efficient distributed algorithms for computing 2hop neighborhoods in Ad Hoc Wireless Networks. The knowledge of the 2hop neighborhood is assumed in many protocols and algorithms for routing, clustering, and distributed channel assignment, but no efficient distributed algorithms for computing the 2hop neighborhoods were previously published. The problem is nontrivial,...
Neighborhoodbased topology recognition in sensor networks
 In ALGOSENSORS04
, 2004
"... Abstract. We consider a crucial aspect of selforganization of a sensor network consisting of a large set of simple sensor nodes with no location hardware and only very limited communication range. After having been distributed randomly in a given twodimensional region, the nodes are required to de ..."
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Cited by 36 (1 self)
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Abstract. We consider a crucial aspect of selforganization of a sensor network consisting of a large set of simple sensor nodes with no location hardware and only very limited communication range. After having been distributed randomly in a given twodimensional region, the nodes are required to develop a sense for the environment, based on a limited amount of local communication. We describe algorithmic approaches for determining the structure of boundary nodes of the region, and the topology of the region. We also develop methods for determining the outside boundary, the distance to the closest boundary for each point, the Voronoi diagram of the different boundaries, and the geometric thickness of the network. Our methods rely on a number of natural assumptions that are present in densely distributed sets of nodes, and make use of a combination of stochastics, topology, and geometry. Evaluation requires only a limited number of simple local computations. ACM classification: C.2.1 Network architecture and design; F.2.2 Nonnumerical algorithms and problems; G.3 Probability and statistics
On Coloring Unit Disk Graphs
 ALGORITHMICA
, 1994
"... In this paper the coloring problem for unit disk (UD) graphs is considered. UD graphs are the intersection graphs of equal sized disks in the plane. Colorings of UD graphs arise in the study of channel assignment problems in broadcast networks. Improving on a result of Clark, Colbourn and Johnson (1 ..."
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Cited by 36 (0 self)
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In this paper the coloring problem for unit disk (UD) graphs is considered. UD graphs are the intersection graphs of equal sized disks in the plane. Colorings of UD graphs arise in the study of channel assignment problems in broadcast networks. Improving on a result of Clark, Colbourn and Johnson (1990) it is shown that the coloring problem for UD graphs remains NPcomplete for any fixed number of colors k 3. Furthermore, a 3approximation algorithm for the problem is presented which is based on network flow and matching techniques, and it is pointed out how this technique can be applied to more general classes of disk graphs.