In this paper we explore some implications of view-ing graphs as geometric objects. This approach of-fers a new perspective on a number of graph-theoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect the met-ric 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 dis-tances between their geometric images. In this paper we develop efficient algorithms for em-bedding graphs low-dimensionally with a small distor-tion. Further algorithmic applications include: 0 A simple, unified approach to a number of prob-lems on multicommodity flows, including the Leighton-Rae Theorem [29] and some of its ex-tensions. 0 For graphs embeddable in low-dimensional spaces with a small distortion, we can find low-diameter 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 low-dimensional representations of statisti-cal data allow for meaningful and efficient cluster-ing, which is one of the most basic tasks in pattern-recognition. For the (mostly heuristic) methods used

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 coordinate-based 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.

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 inverse-power 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 multi-robot 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.

vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1

We describe how some simple properties of discrete one-forms 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 "spring-embedding" 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 non-convex in the embedding. We characterize when it is still possible to achieve an embedding, despite these boundaries being non-convex. 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.

Abstract not found

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].

In this paper we investigate the general position model for the drawing of arbitrary degree graphs in the D-dimensional (D >= 2) orthogonal grid. In this model no two vertices lie in the same grid hyperplane.

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 3-connected 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 Knaster-Kuratowski-Mazurkiewicz Theorem, that some drawing of G belonging to this class is greedy. 1 1

Let G = (V, E) be a directed graph and n denote |V|. We show that G is k-vertex connected iff for every subset X of V with IX I = k, there is an embedding of G in the (k- I)-dimensional space Rk-l, ~ : V ~Rk-l, such that no hyperplane contains k points of {~(v) \ v G V}, and for each v E V – X, f(v) is in the convex hull of {~(w) I (v, W) G E}. This result generalizes to directed graphs the notion of convex embedding of undirected graphs introduced by Linial, LOV6SZ and Wigderson in ‘Rubber bands, convex embedding and graph connectivity, ” Combinatorics 8 (1988), 91-102. Using this characterization, a directed graph can be tested for k-vertex connectivity by a Monte Carlo algo-rithm in time O((M(n) + nkf(k)). (log n)) with error probability < l/n, and by a Las Vegas algorithm in ex-pected time O((lf(n)+nM(k)).k), where M(n) denotes the number of arithmetic steps for multiplying two n x n matrices (Al(n) = 0(n2.3755)). Our Monte Carlo algo-rithm improves on the best previous deterministic and randomized time complexities for k> no. *9; e.g., for k = @, the factor of improvement is> n0.G2. Both al-gorithms have processor efficient parallel versions that run in O((log n)2) time on the EREW PRAM model of computation, using a number of processors equal to (logn) times the respective sequential time complexi-ties. Our Monte Carlo parallel algorithm improves on the number of processors used by the best previous (Monte Carlo) parallel algorithm by a factor of at least (n2/(log n)3) while having the same running time. Generalizing the notion of s-t numberings, we give a combinatorial construction of a directed s-t nulmberiug for any 2-vertex connected directed graph.