Succinct Greedy Graph Drawing in the Hyperbolic Plane

Eppstein, David; Goodrich, Michael T.

doi:10.1007/978-3-642-00219-9_3

Cited by 43 publications

(41 citation statements)

References 28 publications

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“…Namely, Eppstein and Goodrich [7] proved that every graph has a greedy drawing in the hyperbolic plane in which the vertex coordinates can be represented by O(log n) bits; Goodrich and Strash [10] proved that every triconnected planar graph has a greedy drawing in the Euclidean plane in which the vertex coordinates can be represented by O(log n) bits; Angelini et al [1] proved that there exist trees requiring exponential area in any greedy drawing (that is, there exist trees that, in any greedy drawing in the Euclidean plane, require a polynomial number of bits to represent the Cartesian coordinates of the vertices).…”

Section: Problem 1 What Are the Area Requirements Of Greedy Drawings mentioning

confidence: 99%

An Algorithm to Construct Greedy Drawings of Triangulations

Angelini¹,

Frati²,

Grilli³

2010

JGAA

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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.

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Section: Problem 1 What Are the Area Requirements Of Greedy Drawings mentioning

confidence: 99%

An Algorithm to Construct Greedy Drawings of Triangulations

Angelini¹,

Frati²,

Grilli³

2010

JGAA

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“…Many theoretical results on greedy embeddings (including those presented here) require the bit complexity of representing the greedy embedding to exceed the bit complexity of describing the adjacency matrix of the graph. In [4], Eppstein and Goodrich are able to support greedy routing (in hyperbolic space), but substantially improve the bit-complexity of representing the virtual coordinates used by the algorithm. Additionally, Goorich and Strash [7] presented variants of some embeddings given here, and were able to obtain similar improvements in the case of greedy embeddings into the Euclidean plane.…”

Section: Conjecturementioning

confidence: 99%

Some Results on Greedy Embeddings in Metric Spaces

Moitra

Leighton

2008

2008 49th Annual IEEE Symposium on Foundations of Computer Science

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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 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 K 3,3 as a minor admit a greedy embedding into the Euclidean plane. We also prove a combinatorial condition that guarantees non-embeddability. We use this result to construct graphs that can be greedily embedded into the Euclidean plane, but for which no spanning tree admits such an embedding.

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“…This completely eliminates the necessity of face routing and of course all the practical issues that come with it. This family of work includes both heuristic algorithms [23], centralized and theoretical constructions for 3-connected planar graphs [2], [9], [11], [15], [19], [21], embedding in high dimensional spaces [12], embeddings in hyperbolic spaces [11], [18], [20], embedding into circular domains (all the holes are circular) [24]. Most of existing work are mainly of theoretical interest.…”

Section: A Prior Work On Routing In Mobile Networkmentioning

confidence: 99%

Compact Conformal Map for Greedy Routing in Wireless Mobile Sensor Networks

Zeng

Zhou

et al. 2016

IEEE Trans. on Mobile Comput.

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Abstract-Motivated by mobile sensor networks as in participatory sensing applications, we are interested in developing a practical, lightweight solution for routing in a mobile network. While greedy routing is robust to mobility, location errors and link dynamics, it may get stuck in a local minimum, which then requires non-trivial recovery methods. We follow the approach taken by Sarkar et. al. [24] to find an embedding of the network such that greedy routing using the virtual coordinates guarantees delivery, thus eliminating the necessity of any recovery methods. Our new contribution is to replace the in-network computation of the embedding by a preprocessing of the domain before network deployment and encode the map of network domain to virtual coordinate space by using a small number of parameters which can be pre-loaded to all sensor nodes. As a result, the map is only dependent on the network domain and is independent of the network connectivity. Each node can directly compute or update its virtual coordinates by applying the locally stored map on its geographical coordinates. This represents the first practical solution for using virtual coordinates for greedy routing in a sensor network and could be easily extended to the case of a mobile network. Being extremely light-weight, greedy routing on the virtual coordinates is shown to be very robust to mobility, link dynamics and non-unit disk graph connectivity models.

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Succinct Greedy Graph Drawing in the Hyperbolic Plane

Cited by 43 publications

References 28 publications

An Algorithm to Construct Greedy Drawings of Triangulations

An Algorithm to Construct Greedy Drawings of Triangulations

Some Results on Greedy Embeddings in Metric Spaces

Compact Conformal Map for Greedy Routing in Wireless Mobile Sensor Networks

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