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37
The geometry of graphs and some of its algorithmic applications
 Combinatorica
, 1995
"... In this paper we explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graphtheoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that r ..."
Abstract

Cited by 457 (19 self)
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In this paper we explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graphtheoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect the metric of the (possibly weighted) graph. Given a graph G we map its vertices to a normed space in an attempt to (i) Keep down the dimension of the host space and (ii) Guarantee a small distortion, i.e., make sure that distances between vertices in G closely match the distances between their geometric images. In this paper we develop efficient algorithms for embedding graphs lowdimensionally with a small distortion. Further algorithmic applications include: 0 A simple, unified approach to a number of problems on multicommodity flows, including the LeightonRae Theorem [29] and some of its extensions. 0 For graphs embeddable in lowdimensional spaces with a small distortion, we can find lowdiameter decompositions (in the sense of [4] and [34]). The parameters of the decomposition depend only on the dimension and the distortion and not on the size of the graph. 0 In graphs embedded this way, small balanced separators can be found efficiently. Faithful lowdimensional representations of statistical data allow for meaningful and efficient clustering, which is one of the most basic tasks in patternrecognition. For the (mostly heuristic) methods used
Geographic routing without location information
 In Proc. of ACM MOBICOM
, 2003
"... For many years, scalable routing for wireless communication systems was a compelling but elusive goal. Recently, several routing algorithms that exploit geographic information (e.g., GPSR) have been proposed to achieve this goal. These algorithms refer to nodes by their location, not address, and us ..."
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Cited by 306 (10 self)
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For many years, scalable routing for wireless communication systems was a compelling but elusive goal. Recently, several routing algorithms that exploit geographic information (e.g., GPSR) have been proposed to achieve this goal. These algorithms refer to nodes by their location, not address, and use those coordinates to route greedily, when possible, towards the destination. However, there are many situations where location information is not available at the nodes, and so geographic methods cannot be used. In this paper we define a scalable coordinatebased routing algorithm that does not rely on location information, and thus can be used in a wide variety of ad hoc and sensornet environments. 1.
Social Potential Fields: A Distributed Behavioral Control for Autonomous Robots
, 1999
"... A Very Large Scale Robotic (VLSR) system may consist of from hundreds to perhaps tens of thousands or more autonomous robots. The costs of robots are going down, and the robots are getting more compact, more capable, and more flexible. Hence, in the near future, we expect to see many industrial and ..."
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Cited by 124 (1 self)
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A Very Large Scale Robotic (VLSR) system may consist of from hundreds to perhaps tens of thousands or more autonomous robots. The costs of robots are going down, and the robots are getting more compact, more capable, and more flexible. Hence, in the near future, we expect to see many industrial and military applications of VLSR systems in tasks such as assembling, transporting, hazardous inspection, patrolling, guarding and attacking. In this paper, we propose a new approach for distributed autonomous control of VLSR systems. We define simple artificial force laws between pairs of robots or robot groups. The force laws are inversepower force laws, incorporating both attraction and repulsion. The force laws can be distinct and to some degree they reflect the 'social relations' among robots. Therefore we call our method social potential fields. An individual robot's motion is controlled by the resultant artificial force imposed by other robots and other components of the system. The approach is distributed in that the force calculations and motion control can be done in an asynchronous and distributed manner. We also extend the social potential fields model to use spring laws as force laws. This paper presents the first and a preliminary study on applying potential fields to distributed autonomous multirobot control. We describe the generic framework of our social potential fields method. We show with computer simulations that the method can yield interesting and useful behaviors among robots, and we give examples of possible industrial and military applications. We also identify theoretical problems for future studies. 1999 Published by Elsevier Science B.V. All rights reserved.
Semidefinite programs and combinatorial optimization (Lecture notes)
, 1995
"... this paper, we are only concerned about the last question, which can be answered using semidefinite programming. For a survey of other aspects of such geometric representations, see [64]. ..."
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Cited by 30 (1 self)
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this paper, we are only concerned about the last question, which can be answered using semidefinite programming. For a survey of other aspects of such geometric representations, see [64].
ThreeDimensional Orthogonal Graph Drawing
, 2000
"... vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . ..."
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Cited by 27 (10 self)
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vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1
Discrete OneForms on Meshes and Applications to 3D Mesh Parameterization
 Journal of CAGD
, 2006
"... We describe how some simple properties of discrete oneforms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated "springembedding" theorem for planar graphs, which is widely used for parameterizing mesh ..."
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Cited by 25 (1 self)
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We describe how some simple properties of discrete oneforms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated "springembedding" theorem for planar graphs, which is widely used for parameterizing meshes with the topology of a disk as a planar embedding with a convex boundary. Our second result generalizes the first, dealing with the case where the mesh contains multiple boundaries, which are free to be nonconvex in the embedding. We characterize when it is still possible to achieve an embedding, despite these boundaries being nonconvex. The third result is an analogous embedding theorem for meshes with genus 1 (topologically equivalent to the torus). Applications of these results to the parameterization of meshes with disk and toroidal topologies are demonstrated. Extensions to higher genus meshes are discussed.
Greedy routing with guaranteed delivery using ricci flows
 In Proc. of the 8th International Symposium on Information Processing in Sensor Networks (IPSN’09
, 2009
"... Greedy forwarding with geographical locations in a wireless sensor network may fail at a local minimum. In this paper we propose to use conformal mapping to compute a new embedding of the sensor nodes in the plane such that greedy forwarding with the virtual coordinates guarantees delivery. In parti ..."
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Cited by 22 (15 self)
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Greedy forwarding with geographical locations in a wireless sensor network may fail at a local minimum. In this paper we propose to use conformal mapping to compute a new embedding of the sensor nodes in the plane such that greedy forwarding with the virtual coordinates guarantees delivery. In particular, we extract a planar triangulation of the sensor network with nontriangular faces as holes, by either using the nodes ’ location or using a landmarkbased scheme without node location. The conformal map is computed with Ricci flow such that all the nontriangular faces are mapped to perfect circles. Thus greedy forwarding will never get stuck at an intermediate node. The computation of the conformal map and the virtual coordinates is performed at a preprocessing phase and can be implemented by local gossipstyle computation. The method applies to both unit disk graph models and quasiunit disk graph models. Simulation results are presented for these scenarios.
Greedy Drawings of Triangulations
, 2007
"... Greedy Routing is a class of routing algorithms in which the packets are forwarded in a manner that reduces the distance to the destination at every step. In an attempt to provide theoretical guarantees for a class of greedy routing algorithms, Papadimitriou and Ratajczak [PR05] came up with the fol ..."
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Cited by 18 (1 self)
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Greedy Routing is a class of routing algorithms in which the packets are forwarded in a manner that reduces the distance to the destination at every step. In an attempt to provide theoretical guarantees for a class of greedy routing algorithms, Papadimitriou and Ratajczak [PR05] came up with the following conjecture: Any 3connected planar graph can be drawn in the plane such that for every pair of vertices s and t a distance decreasing path can be found. A path s = v1,v2,...,vk = t in a drawing is said to be distance decreasing if �vi − t � < �vi−1 − t�, 2 ≤ i ≤ k where �... � denotes the Euclidean distance. We settle this conjecture in the affirmative for the case of triangulations. A partitioning of the edges of a triangulation G into 3 trees, called the realizer of G, was first developed by Walter Schnyder who also gave a drawing algorithm based on this. We generalize Schnyder’s algorithm to obtain a whole class of drawings of any given triangulation G. We show, using the KnasterKuratowskiMazurkiewicz Theorem, that some drawing of G belonging to this class is greedy. 1 1
MultiDimensional Orthogonal Graph Drawing with Small Boxes
 Proc. 7th International Symp. on Graph Drawing (GD '99
, 1999
"... In this paper we investigate the general position model for the drawing of arbitrary degree graphs in the Ddimensional (D >= 2) orthogonal grid. In this model no two vertices lie in the same grid hyperplane. ..."
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Cited by 13 (5 self)
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In this paper we investigate the general position model for the drawing of arbitrary degree graphs in the Ddimensional (D >= 2) orthogonal grid. In this model no two vertices lie in the same grid hyperplane.