Euclidean Greedy Drawings of Trees

@article{Nllenburg2017EuclideanGD,
  title={Euclidean Greedy Drawings of Trees},
  author={Martin N{\"o}llenburg and Roman Prutkin},
  journal={Discrete \& Computational Geometry},
  year={2017},
  volume={58},
  pages={543-579}
}
Greedy embedding (or drawing) is a simple and efficient strategy to route messages in wireless sensor networks. For each source-destination pair of nodes s, t in a greedy embedding there is always a neighbor u of s that is closer to t according to some distance metric. The existence of greedy embeddings in the Euclidean plane $${\mathbb {R}}^2$$R2 is known for certain graph classes such as 3-connected planar graphs. We completely characterize the trees that admit a greedy embedding in… 
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References

SHOWING 1-10 OF 30 REFERENCES
Some Results on Greedy Embeddings in Metric Spaces
  • Ankur Moitra, F. Leighton
  • Computer Science, Mathematics
    2008 49th Annual IEEE Symposium on Foundations of Computer Science
  • 2008
TLDR
This work resolves a conjecture of Papadimitriou and Ratajczak that every 3-connected planar graph admits a greedy embedding into the Euclidean plane and proves a combinatorial condition that guarantees nonembeddability.
On Planar Greedy Drawings of 3-Connected Planar Graphs
TLDR
It is proved that every 3-connected planar graph admits a planar greedy drawing, which constitutes a natural intermediate step towards a proof of the convex greedy embedding conjecture.
Greedy Drawings of Triangulations
TLDR
It is shown, using the Knaster–Kuratowski–Mazurkiewicz Theorem, that some drawing of G belonging to this class is greedy, and a whole class of drawings of any given triangulation G is obtained.
Succinct Greedy Drawings Do Not Always Exist
TLDR
It is shown that there exist greedy‐drawable graphs that do not admit any greedy drawing in which the Cartesian coordinates have less than a polynomial number of bits.
Succinct Strictly Convex Greedy Drawing of 3-Connected Plane Graphs
TLDR
It is shown that every 3-connected plane graph has a drawing on the R2 plane that is succinct, planar, strictly convex, and is greedy with respect to a metric function based on parameters derived from Schnyder woods.
Succinct Greedy Geometric Routing Using Hyperbolic Geometry
We describe a method for performing greedy geometric routing for any n-vertex simple connected graph G in the hyperbolic plane, so that a message M between any pair of vertices may be routed by
On a Conjecture Related to Geometric Routing
Geographic Routing Using Hyperbolic Space
  • Robert D. Kleinberg
  • Computer Science
    IEEE INFOCOM 2007 - 26th IEEE International Conference on Computer Communications
  • 2007
TLDR
A scalable and reliable point-to-point routing algorithm for ad hoc wireless networks and sensor-nets, and it is proved that the greedy routing strategy makes a consistent choice of the node responsible for the address, irrespective of the source address of the request.
Self-approaching Graphs
TLDR
It is shown that there are efficient algorithms to test if a polygonal path is self-approaching inℝ2 and ℝ3, but it is NP-hard to testif a given graph drawing in �”3 has a self- Approaching uv-path.
An Algorithm to Construct Greedy Drawings of Triangulations
TLDR
An algorithm to construct a greedy drawing of every given triangulation is shown, which proves a conjecture by Papadimitriou and Ratajczak and was independently shown by Leighton and Moitra.
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