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 point-to-point routing. Here we resolve a conjecture of Papadimitriou and Ratajczak that every 3-connected planar graph admits a greedy embedding into the Euclidean plane. This immediately implies that all 3-connected graphs that exclude K3,3 as a minor admit a greedy embedding into the Euclidean plane. Additionally, we provide the first non-trivial 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.

Abstract. We describe an efficient method for drawing any n-vertex 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

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 non-triangular faces as holes, by either using the nodes ’ location or using a landmark-based scheme without node location. The conformal map is computed with Ricci flow such that all the non-triangular 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 gossip-style computation. The method applies to both unit disk graph models and quasi-unit disk graph models. Simulation results are presented for these scenarios.

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.

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 near-maximal planar graph. We briefly survey the connections and analogies between these three kinds of labelings, and their uses in designing efficient geometric algorithms.

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 ill-conditioning: 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

We present a deterministic local routing scheme that is guaranteed to find a path between any pair of vertices in a halfθ6-graph whose length is at most 5 / √ 3 = 2.886... times the Euclidean distance between the pair of vertices. The half-θ6graph is identical to the Delaunay triangulation where the empty region is an equilateral triangle. Moreover, we show that no local routing scheme can achieve a better competitive spanning ratio thereby implying that our routing scheme is optimal. This is somewhat surprising because the spanning ratio of the half-θ6-graph is 2. Since every triangulation can be embedded in the plane as a half-θ6-graph using O(log n) bits per vertex coordinate via Schnyder’s embedding scheme (SODA 1990), our result provides a competitive local routing scheme for every such embedded triangulation. 1

We show an algorithm to construct a greedy drawing of every given triangulation. The algorithm relies on two main results. First, we show how to construct greedy drawings of a fairly simple class of graphs, called triangulated binary cactuses. Second, we show that every triangulation can be spanned by a triangulated binary cactus. Further, we discuss how to extend our techniques in order to prove that every triconnected planar graph admits a greedy drawing. Such a result, which proves a conjecture by Papadimitriou and Ratajczak, was independently shown by Leighton and Moitra.

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 graph drawing schemes exist for the Euclidean metric in R 2, for 3-connected planar graphs, with coordinates that can be represented succinctly, that is, with O(log n) bits. 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

A greedy embedding of a graph G = (V, E) into a metric space (X, d) is a function x: V (G) → X such that in the embedding for every pair of non-adjacent vertices x(s),x(t) there exists another vertex x(u) adjacent to x(s) which is closer to x(t) than x(s). This notion of greedy embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci. 2005), where authors conjectured that every 3-connected planar graph has a greedy embedding (possibly planar and convex) in the Euclidean plane. Recently, greedy embedding conjecture has been proved by Leighton and Moitra (FOCS 2008). However, their algorithm do not result in a drawing that is planar and convex for all 3-connected planar graph in the Euclidean plane. In this work we consider the planar convex greedy embedding conjecture and make some progress. We derive a new characterization of planar convex greedy embedding that given a 3-connected planar graph G = (V, E), an embedding x: V → R 2 of G is a planar convex greedy embedding if and only if, in the embedding x, weight of the maximum weight spanning tree (T) and weight of the minimum weight spanning tree (MST) satisfies wt(T)/wt(MST) ≤ (|V | − 1) 1−δ, for some 0 < δ ≤ 1. In order to present this result we define a notion of weak greedy embedding. For β ≥ 1 a β–weak greedy embedding of a graph is a planar embedding x: V (G) → X such that for every pair of non-adjacent vertices x(s),x(t) there exists a vertex x(u) adjacent to x(s) such that distance between x(u) and x(t) is at most β times the distance between x(s) and x(t). We show that any three connected planar graph G = (V, E) has a β– weak greedy planar convex embedding in the Euclidean plane with β ∈ [1,2 √ 2 · d(G)], where d(G) is the ratio of maximum and minimum distance between pair of vertices in the embedding of G. Finally, we also show that this bound is tight for well known Tutte embedding of 3-connected planar graphs in the Euclidean plane- which is planar and convex.