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Euclidean Greedy Drawings of Trees

2017
*
Discrete & Computational Geometry
*

(Graph

doi:10.1007/s00454-017-9913-8
fatcat:ergvcu4j2rhgjmyppzaw5o4uya
*Drawing*2009) and is a further step in characterizing the graphs that admit*Euclidean**greedy*embeddings. ... The existence*of**greedy*embeddings in the*Euclidean*plane R 2 is known for certain graph classes such as 3-connected planar graphs. ... M.N. received financial support by the 'Concept for the Future'*of*KIT within the framework*of*the German Excellence Initiative". ...##
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Euclidean Greedy Drawings of Trees
[article]

2013
*
arXiv
*
pre-print

(Graph

arXiv:1306.5224v1
fatcat:fsy6br5yxnau5mtcjpljggnpp4
*Drawing*2009) and is a further step in characterizing the graphs that admit*Euclidean**greedy*embeddings. ... The existence*of**greedy*embeddings in the*Euclidean*plane R^2 is known for certain graph classes such as 3-connected planar graphs. ... M.N. received financial support by the 'Concept for the Future'*of*KIT within the framework*of*the German Excellence Initiative". ...##
###
Succinct Greedy Graph Drawing in the Hyperbolic Plane
[article]

2008
*
arXiv
*
pre-print

Our algorithm produces

arXiv:0806.0341v1
fatcat:wntdkv5slffxtftaz5rwrizbiq
*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 ... 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 ... Papadimitriou and Ratajczak [20] continue this line*of*work on*greedy**drawings*, studying*greedy*schemes that are guaranteed to work, and they conjecture that*Euclidean**greedy**drawings*exist for any graph ...##
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Succinct Greedy Graph Drawing in the Hyperbolic Plane
[chapter]

2009
*
Lecture Notes in Computer Science
*

We describe a method for producing a

doi:10.1007/978-3-642-00219-9_3
fatcat:sgqjl4cvjndunapbssjisqghem
*greedy*embedding*of*any nvertex simple graph G in the hyperbolic plane, so that a message M between any pair*of*vertices may be routed by having each vertex that receives ... Our algorithm produces succinct*drawings*, where vertex positions are represented using O(log n) bits and distance comparisons may be performed efficiently using these representations. ... It is not clear whether either*of*these*greedy**drawings*in*Euclidean*spaces run in polynomial time, however. ...##
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Drawing a tree as a minimum spanning tree approximation

2012
*
Journal of computer and system sciences (Print)
*

edge in the path connecting them. (1 + ε)-EMST

doi:10.1016/j.jcss.2011.06.001
fatcat:ugvpsgnni5bsvnknm3pppztx4m
*drawings*are good approximations*of**Euclidean*minimum spanning*trees*. ... As a byproduct*of*one*of*our techniques, we improve the best known area upper bound for*Euclidean*minimum spanning*tree*realizations*of*complete binary*trees*. ... This investigation naturally leads to the following question: Which are those*trees*that have an EMST*drawing*, i.e. a straight-line*drawing*that is also a*Euclidean*minimum spanning*tree**of*the set*of*...##
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Greedy Convex Embeddings for Sensor Networks

2009
*
2009 International Conference on Parallel and Distributed Computing, Applications and Technologies
*

However, their algorithm does not result in a

doi:10.1109/pdcat.2009.86
dblp:conf/pdcat/BerchenkoT09
fatcat:judgsrasejciphg7vkc3wyrz3y
*drawing*that is planar and convex in the*Euclidean*plane for all 3-connected planar graphs. ... (possibly planar and convex) in the*Euclidean*plane. ... One can study the space*of*all equiprobable 3-connected planar graphs, or, perhaps more interestingly, some realistic model*of*communication networks. ...##
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Greedy Convex Embeddings for Sensor Networks

2009
*
Frontiers in Neuroinformatics
*

However, their algorithm does not result in a

doi:10.3389/conf.neuro.01.2009.16.104
fatcat:pddrbe2sbrapvo376cmvn5sele
*drawing*that is planar and convex in the*Euclidean*plane for all 3-connected planar graphs. ... (possibly planar and convex) in the*Euclidean*plane. ... One can study the space*of*all equiprobable 3-connected planar graphs, or, perhaps more interestingly, some realistic model*of*communication networks. ...##
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Succinct Greedy Drawings Do Not Always Exist
[chapter]

2010
*
Lecture Notes in Computer Science
*

Hence, a natural and practical problem is the one

doi:10.1007/978-3-642-11805-0_17
fatcat:v247zawbsrdj3gbecnoyrqhmcq
*of*constructing*greedy**drawings*in the plane using few bits for representing vertex Cartesian coordinates and using the*Euclidean*distance as a metric. ... A*greedy**drawing*is a graph*drawing*containing a distance-decreasing path for every pair*of*nodes. A path (v0, v1, . . . , vm)*Greedy**drawings*easily support geographic*greedy*routing. ... Given a*greedy**drawing*Γ*of*a*tree*T , any subtree*of*T is represented in Γ by a*greedy**drawing*. ...##
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Complete combinatorial characterization of greedy-drawable trees
[article]

2022
*
arXiv
*
pre-print

A (

arXiv:2203.04664v1
fatcat:oeqs5srqjrbttf3mfwikuyhxni
*Euclidean*)*greedy**drawing**of*a graph is a*drawing*in which, for any two vertices s,t (s ≠ t), there is a neighbor vertex*of*s that is closer to t than to s in the*Euclidean*distance. ...*Greedy**drawings*are important in the context*of*message routing in networks, and graph classes that admit*greedy**drawings*have been actively studied. Nöllenburg and Prutkin (Discrete Comput. ... That is, a (*Euclidean*)*greedy**drawing**of*a graph G is a*drawing**of*G in which for any two vertices s, t (s = t)*of*G, there is a neighbor vertex u*of*s with d(s, t) > d(u, t), where d is the*Euclidean*...##
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A Note on the Area Requirement of Euclidean Greedy Embeddings of Christmas Cactus Graphs
[article]

2016
*
arXiv
*
pre-print

This

arXiv:1605.09244v1
fatcat:krjdqflhqfc6jnyikuwlqhe4t4
*drawing*style is motivated by*greedy*geometric routing in wireless sensor networks. ... An*Euclidean**greedy*embedding*of*a graph is a straight-line embedding in the plane, such that for every pair*of*vertices s and t, the vertex s has a neighbor v with smaller distance to t than s. ... In the presented encoding scheme for the vertex coordinates, their positions in the*Euclidean*plane are not stored explicitly, and the*drawings*might still have exponential size. Angelini et al. ...##
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On Self-Approaching and Increasing-Chord Drawings of 3-Connected Planar Graphs
[chapter]

2014
*
Lecture Notes in Computer Science
*

Nöllenburg and Prutkin [14] characterized

doi:10.1007/978-3-662-45803-7_40
fatcat:niq2sdxqdfcurmanpittkmopwi
*trees*admitting a*Euclidean**greedy*embedding. ... We show that in the*Euclidean*plane, triangulations admit increasing-chord*drawings*, and for planar 3-*trees*we can ensure planarity. ... Papadimitriou and Ratajczak conjectured that every 3-connected planar graph admits a*greedy*embedding into the*Euclidean*plane [16] . ...##
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Some Results on Greedy Embeddings in Metric Spaces

2009
*
Discrete & Computational Geometry
*

Here we resolve a conjecture

doi:10.1007/s00454-009-9227-6
fatcat:asu4bmoiizcsvawzoyguug74qa
*of*Papadimitriou and Ratajczak that every 3-connected planar graph admits a*greedy*embedding into the*Euclidean*plane. ... 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. ... A graph that admits a*greedy*embedding into the*Euclidean*plane, and yet no spanning*tree**of*this graph admits such an embedding. ...##
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Some Results on Greedy Embeddings in Metric Spaces

2008
*
2008 49th Annual IEEE Symposium on Foundations of Computer Science
*

Here we resolve a conjecture

doi:10.1109/focs.2008.18
dblp:conf/focs/MoitraL08
fatcat:wsptm7s6tncn7i6jaokijkvify
*of*Papadimitriou and Ratajczak that every 3-connected planar graph admits a*greedy*embedding into the*Euclidean*plane. ... 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. ... A graph that admits a*greedy*embedding into the*Euclidean*plane, and yet no spanning*tree**of*this graph admits such an embedding. ...##
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A generalized greedy routing algorithm for 2-connected graphs

2010
*
Theoretical Computer Science
*

In this paper, we introduce a version

doi:10.1016/j.tcs.2010.08.032
fatcat:rhmdw6abznd5zjiclt2u37ezra
*of**greedy*routing which we call generalized*greedy*routing algorithm. ... We also generalize this result to arbitrary*trees*. Both algorithms are natural and simple to be implemented. ... When l is sufficiently large, it is easy to argue that K 2,l does not have*greedy**drawing*in any fixed dimension*Euclidean*space R m (otherwise, K l/2 would have a*greedy**drawing*in R m , a contradiction ...##
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Self-Approaching Graphs
[article]

2013
*
arXiv
*
pre-print

; (2) we can characterize the

arXiv:1306.5460v1
fatcat:asenvb4zendero6fuz2shkqfx4
*trees*that have self-approaching*drawings*; (3) for any given set*of*terminal points in the plane, we can find a linear sized network that has a self-approaching path between ... This is a more stringent condition than a*greedy**drawing*(where only the distance between vertices on the path and the destination vertex must decrease), and guarantees that the*drawing*is a 5.33-spanner ... This work was done as part*of*an Algorithms Problem Session at the University*of*Waterloo, and we thank the other participants for helpful discussions. ...
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