On Succinct Greedy Drawings of Plane Triangulations and 3-Connected Plane Graphs View Full Text


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Article Info

DATE

2012-08-25

AUTHORS

Xin He, Huaming Zhang

ABSTRACT

Geometric routing by using virtual locations is an elegant way for solving network routing problems. In its simplest form, greedy routing, a message is simply forwarded to a neighbor that is closer to the destination. It has been an open conjecture whether every 3-connected plane graph has a greedy drawing in the Euclidean plane R2 (by Papadimitriou and Ratajczak in Theor. Comp. Sci. 344(1):3–14, 2005). Leighton and Moitra (Discrete Comput. Geom. 44(3):686–705, 2010) recently settled this conjecture positively. One main drawback of this approach is that the coordinates of the virtual locations require Ω(nlogn) bits to represent (the same space usage as traditional routing table approaches). This makes greedy routing infeasible in applications.In this paper, we show that the classical Schnyder drawing in R2 of plane triangulations is greedy with respect to a simple natural metric function H(u,v) over R2 that is equivalent to Euclidean metric DE(u,v) (in the sense that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D_{E}(u,v) \leq H(u,v) \leq2\sqrt{2}D_{E}(u,v)$\end{document}). The drawing uses two integer coordinates between 0 and 2n−5, which can be represented by logn bits. We also show that the classical Schnyder drawing in R2 of 3-connected plane graphs is weakly greedy with respect to the same metric function H(∗,∗). The drawing uses two integer coordinates between 0 and f (where f is the number of internal faces of G). More... »

PAGES

531-544

References to SciGraph publications

  • 2003-06. Geodesic Embeddings and Planar Graphs in ORDER
  • 2009. Succinct Greedy Geometric Routing in the Euclidean Plane in ALGORITHMS AND COMPUTATION
  • 2009-10-20. Some Results on Greedy Embeddings in Metric Spaces in DISCRETE & COMPUTATIONAL GEOMETRY
  • 2010. Schnyder Greedy Routing Algorithm in THEORY AND APPLICATIONS OF MODELS OF COMPUTATION
  • 2008-01-01. On the Efficiency of a Local Iterative Algorithm to Compute Delaunay Realizations in EXPERIMENTAL ALGORITHMS
  • 1999-04. Output-Sensitive Reporting of Disjoint Paths in ALGORITHMICA
  • 2009-12-29. Greedy Drawings of Triangulations in DISCRETE & COMPUTATIONAL GEOMETRY
  • 2007-09-12. Schnyder Woods and Orthogonal Surfaces in DISCRETE & COMPUTATIONAL GEOMETRY
  • 2009-04-22. Schnyder Woods for Higher Genus Triangulated Surfaces, with Applications to Encoding in DISCRETE & COMPUTATIONAL GEOMETRY
  • 2001-03. Convex Drawings of Planar Graphs and the Order Dimension of 3-Polytopes in ORDER
  • 2007-02-09. Convex Drawings of 3-Connected Plane Graphs in ALGORITHMICA
  • 2009. On Convex Greedy Embedding Conjecture for 3-Connected Planar Graphs in FUNDAMENTALS OF COMPUTATION THEORY
  • 1989-12. Planar graphs and poset dimension in ORDER
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00453-012-9682-y

    DOI

    http://dx.doi.org/10.1007/s00453-012-9682-y

    DIMENSIONS

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