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16
Some results on greedy embeddings in metric spaces
 In Proc. of the 49th IEEE Annual Symposium on Foundations of Computer Science
, 2008
"... Geographic Routing is a family of routing algorithms that uses geographic point locations as addresses for the purposes of routing. Such routing algorithms have proven to be both simple to implement and heuristically effective when applied to wireless sensor networks. Greedy Routing is a natural abs ..."
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Cited by 23 (0 self)
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Geographic Routing is a family of routing algorithms that uses geographic point locations as addresses for the purposes of routing. Such routing algorithms have proven to be both simple to implement and heuristically effective when applied to wireless sensor networks. Greedy Routing is a natural abstraction of this model in which nodes are assigned virtual coordinates in a metric space, and these coordinates are used to perform pointtopoint routing. Here we resolve a conjecture of Papadimitriou and Ratajczak that every 3connected planar graph admits a greedy embedding into the Euclidean plane. This immediately implies that all 3connected graphs that exclude K3,3 as a minor admit a greedy embedding into the Euclidean plane. Additionally, we provide the first nontrivial examples of graphs that admit no such embedding. These structural results provide efficiently verifiable certificates that a graph admits a greedy embedding or that a graph admits no greedy embedding into the Euclidean plane.
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.
Succinct greedy graph drawing in the hyperbolic plane
 In Proc. 16th Int. Symp. Graph Drawing
, 2008
"... Abstract. We describe an efficient method for drawing any nvertex simple graph G in the hyperbolic plane. Our algorithm produces greedy drawings, which support greedy geometric routing, so that a message M between any pair of vertices may be routed geometrically, simply by having each vertex that r ..."
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Cited by 16 (4 self)
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Abstract. We describe an efficient method for drawing any nvertex simple graph G in the hyperbolic plane. Our algorithm produces greedy drawings, which support greedy geometric routing, so that a message M between any pair of vertices may be routed geometrically, simply by having each vertex that receives M pass it along to any neighbor that is closer in the hyperbolic metric to the message’s eventual destination. More importantly, for networking applications, our algorithm produces succinct drawings, in that each of the vertex positions in one of our embeddings can be represented using O(log n) bits and the calculation of which neighbor to send a message to may be performed efficiently using these representations. These properties are useful, for example, for routing in sensor networks, where storage and bandwidth are limited. 1
On the efficiency of a local iterative algorithm to compute delaunay realizations
 In Workshop on Experimental Algorithms (WEA
, 2008
"... Abstract. Greedy routing protocols for wireless sensor networks (WSNs) are fast and efficient but in general cannot guarantee message delivery. Hence researchers are interested in the problem of embedding WSNs in low dimensional space (e.g., R 2) in a way that guarantees message delivery with greedy ..."
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Cited by 8 (1 self)
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Abstract. Greedy routing protocols for wireless sensor networks (WSNs) are fast and efficient but in general cannot guarantee message delivery. Hence researchers are interested in the problem of embedding WSNs in low dimensional space (e.g., R 2) in a way that guarantees message delivery with greedy routing. It is well known that Delaunay triangulations are such embeddings. We present the algorithm FindAngles, which is a fast, simple, local distributed algorithm that computes a Delaunay triangulation from any given combinatorial graph that is Delaunay realizable. Our algorithm is based on a characterization of Delaunay realizability due to Hiroshima et al. (IEICE 2000). When compared to the PowerDiagram algorithm of Chen et al. (SoCG 2007) that embeds triangulations in the plane so as to permit successful greedy routing, our algorithm requires on average 1/6 th the number of iterations. FindAngles also scales linearly to larger networks and has a much faster distributed implementation than PowerDiagram, requiring just a single round of communication in each iteration. The PowerDiagram algorithm was proposed as an improvement on another algorithm due to Thurston (unpublished, 1988). Our experiments show that on average the PowerDiagram algorithm uses about 19 % fewer iterations than the Thurston algorithm, whereas our algorithm uses about 89 % fewer iterations. Experimentally, FindAngles exhibits well behaved convergence. Theoretically, we prove that with certain initial conditions the error term decreases monotonically. Taken together, these suggest our algorithm may have polynomial time convergence for certain classes of graphs. We note that our algorithm runs only on Delaunay realizable triangulations. This is not a significant concern because Hiroshima et al. (IEICE 2000) indicate that most combinatorial triangulations are indeed Delaunay realizable, which we have also observed experimentally: out of 5000 randomly generated combinatorial triangulations on 100 vertices, only one was not Delaunay realizable.
Drawing graphs in the plane with a prescribed outer face and polynomial area
 Proc. 18th Int. Symp. on Graph Drawing (GD 2010
"... We study the classic graph drawing problem of drawing a planar graph using straightline edges with a prescribed convex polygon as the outer face. Unlike previous algorithms for this problem, which may produce drawings with exponential area, our method produces drawings with polynomial area. In addi ..."
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Cited by 3 (1 self)
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We study the classic graph drawing problem of drawing a planar graph using straightline edges with a prescribed convex polygon as the outer face. Unlike previous algorithms for this problem, which may produce drawings with exponential area, our method produces drawings with polynomial area. In addition, we allow for collinear points on the boundary, provided such vertices do not create overlapping edges. Thus, we solve an open problem of Duncan et al., which, when combined with their work, implies that we can produce a planar straightline drawing of a combinatoriallyembedded genusg graph with the graph’s canonical polygonal schema drawn as a convex polygonal external face. Submitted:
Succinct greedy geometric routing in the euclideanplane
 In Y.Dong, D.Z.Du, andO.Ibarra,editors, International Symposium on Algorithms and Computation (ISAAC ’09), LNCS
, 2009
"... In greedy geometric routing, messages are passed in a network embedded in a metric space according to the greedy strategy of always forwarding messages to nodes that are closer to the destination. We show that greedy geometric routing schemes exist for the Euclidean metric in R 2, for 3connected pl ..."
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Cited by 3 (1 self)
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In greedy geometric routing, messages are passed in a network embedded in a metric space according to the greedy strategy of always forwarding messages to nodes that are closer to the destination. We show that greedy geometric routing schemes exist for the Euclidean metric in R 2, for 3connected planar graphs, with coordinates that can be represented succinctly, that is, with O(log n) bits, where n is the number of vertices in the graph. Moreover, our embedding strategy introduces a coordinate system for R 2 that supports distance comparisons using our succinct coordinates. Thus, our scheme can be used to significantly reduce bandwidth, space, and header size over other recently discovered greedy geometric routing implementations for R 2. 1
Spherical Representation and Polyhedron Routing for Load Balancing in Wireless Sensor Networks
"... Abstract—In this paper we address the problem of scalable and load balanced routing for wireless sensor networks. Motivated by the analog of the continuous setting that geodesic routing on a sphere gives perfect load balancing, we embed sensor nodes on a convex polyhedron in 3D and use greedy routin ..."
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Cited by 2 (2 self)
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Abstract—In this paper we address the problem of scalable and load balanced routing for wireless sensor networks. Motivated by the analog of the continuous setting that geodesic routing on a sphere gives perfect load balancing, we embed sensor nodes on a convex polyhedron in 3D and use greedy routing to deliver messages between any pair of nodes with guaranteed success. This embedding is known to exist by the KoebeAndreevThurston Theorem for any 3connected planar graphs. In our paper we use discrete Ricci flow to develop a distributed algorithm to compute this embedding. Further, such an embedding is not unique and differs from one another by a Möbius transformation. We employ an optimization routine to look for the Möbius transformation such that the nodes are spread on the polyhedron as uniformly as possible. We evaluated the load balancing property of this greedy routing scheme and showed favorable comparison with previous schemes. I.
Regular Labelings and Geometric Structures
, 2010
"... Three types of geometric structure—grid triangulations, rectangular subdivisions, and orthogonal polyhedra— can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an associated maximal or nearmaximal planar graph. We briefly survey the ..."
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Cited by 2 (1 self)
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Three types of geometric structure—grid triangulations, rectangular subdivisions, and orthogonal polyhedra— can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an associated maximal or nearmaximal planar graph. We briefly survey the connections and analogies between these three kinds of labelings, and their uses in designing efficient geometric algorithms.
A Solution to the PapadimitriouRatajczak Conjecture
, 2009
"... Geographic Routing is a family of routing algorithms that uses geographic point locations as addresses for the purposes of routing. Such routing algorithms have proven to be both simple to implement and heuristically effective when applied to wireless sensor networks. Greedy Routing is a natural abs ..."
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Cited by 1 (0 self)
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Geographic Routing is a family of routing algorithms that uses geographic point locations as addresses for the purposes of routing. Such routing algorithms have proven to be both simple to implement and heuristically effective when applied to wireless sensor networks. Greedy Routing is a natural abstraction of this model in which nodes are assigned virtual coordinates in a metric space, and these coordinates are used to perform pointtopoint routing. Here we resolve a conjecture of Papadimitriou and Ratajczak that every 3connected planar graph admits a greedy embedding into the Euclidean plane. This immediately implies that all 3connected graphs that exclude K3,3 as a minor admit a greedy embedding into the Euclidean plane. Additionally, we provide the first nontrivial examples of graphs that admit no such embedding. These structural results provide efficiently verifiable certificates that a graph admits a greedy embedding or that a graph admits no greedy embedding into the Euclidean plane.
Computational Geometry
, 2009
"... at: 1:08pmder to explain this discrepancy between theory and practice, many authors have shown that Simplex Algorithms are efficient in expectation on randomized Linear Programs. We strengthen these results by proving a partial concentration bound for the SHADOW VERTEX Simplex Algorithm. Next, we po ..."
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at: 1:08pmder to explain this discrepancy between theory and practice, many authors have shown that Simplex Algorithms are efficient in expectation on randomized Linear Programs. We strengthen these results by proving a partial concentration bound for the SHADOW VERTEX Simplex Algorithm. Next, we point out a limitation in an algorithm that is commonly used by practitioners and suggest a way of overcoming this. Recommendation Systems are algorithms that are used to recommend goods (books, movies etc.) to users based on the similarities between their past preferences and those of other users. Low Rank Approximation is a common method used for this. We point out a common limitation of this method in the presence of illconditioning: the presence of multiple local minima. We also suggest a simple averaging based technique to overcome this limitation and show that this improves the performance of the system. Finally, we consider some basic results in convexity like Radon’s, Helly’s and Carathéodory’s theorems and generalize them to the topological plane, i.e., a plane which has the concept of a linear path that is analogous to a straight line but no notion of a metric. v CONTENTS iv