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Euclidean Greedy Drawings of Trees

Martin Nöllenburg, Roman Prutkin
2017 Discrete & Computational Geometry  
(Graph 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".  ... 
doi:10.1007/s00454-017-9913-8 fatcat:ergvcu4j2rhgjmyppzaw5o4uya

Euclidean Greedy Drawings of Trees [article]

Martin Nöllenburg, Roman Prutkin
2013 arXiv   pre-print
(Graph 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".  ... 
arXiv:1306.5224v1 fatcat:fsy6br5yxnau5mtcjpljggnpp4

Succinct Greedy Graph Drawing in the Hyperbolic Plane [article]

David Eppstein, Michael T. Goodrich
2008 arXiv   pre-print
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  ...  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  ... 
arXiv:0806.0341v1 fatcat:wntdkv5slffxtftaz5rwrizbiq

Succinct Greedy Graph Drawing in the Hyperbolic Plane [chapter]

David Eppstein, Michael T. Goodrich
2009 Lecture Notes in Computer Science  
We describe a method for producing a 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.  ... 
doi:10.1007/978-3-642-00219-9_3 fatcat:sgqjl4cvjndunapbssjisqghem

Drawing a tree as a minimum spanning tree approximation

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Henk Meijer
2012 Journal of computer and system sciences (Print)  
edge in the path connecting them. (1 + ε)-EMST 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  ... 
doi:10.1016/j.jcss.2011.06.001 fatcat:ugvpsgnni5bsvnknm3pppztx4m

Greedy Convex Embeddings for Sensor Networks

Yakir Berchenko, Mina Teicher
2009 2009 International Conference on Parallel and Distributed Computing, Applications and Technologies  
However, their algorithm does not result in a 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.  ... 
doi:10.1109/pdcat.2009.86 dblp:conf/pdcat/BerchenkoT09 fatcat:judgsrasejciphg7vkc3wyrz3y

Greedy Convex Embeddings for Sensor Networks

Teicher Mina
2009 Frontiers in Neuroinformatics  
However, their algorithm does not result in a 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.  ... 
doi:10.3389/conf.neuro.01.2009.16.104 fatcat:pddrbe2sbrapvo376cmvn5sele

Succinct Greedy Drawings Do Not Always Exist [chapter]

Patrizio Angelini, Giuseppe Di Battista, Fabrizio Frati
2010 Lecture Notes in Computer Science  
Hence, a natural and practical problem is the one 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.  ... 
doi:10.1007/978-3-642-11805-0_17 fatcat:v247zawbsrdj3gbecnoyrqhmcq

Complete combinatorial characterization of greedy-drawable trees [article]

Hiroyuki Miyata, Reiya Nosaka
2022 arXiv   pre-print
A (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  ... 
arXiv:2203.04664v1 fatcat:oeqs5srqjrbttf3mfwikuyhxni

A Note on the Area Requirement of Euclidean Greedy Embeddings of Christmas Cactus Graphs [article]

Roman Prutkin
2016 arXiv   pre-print
This 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.  ... 
arXiv:1605.09244v1 fatcat:krjdqflhqfc6jnyikuwlqhe4t4

On Self-Approaching and Increasing-Chord Drawings of 3-Connected Planar Graphs [chapter]

Martin Nöllenburg, Roman Prutkin, Ignaz Rutter
2014 Lecture Notes in Computer Science  
Nöllenburg and Prutkin [14] characterized 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] .  ... 
doi:10.1007/978-3-662-45803-7_40 fatcat:niq2sdxqdfcurmanpittkmopwi

Some Results on Greedy Embeddings in Metric Spaces

Tom Leighton, Ankur Moitra
2009 Discrete & Computational Geometry  
Here we resolve a conjecture 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.  ... 
doi:10.1007/s00454-009-9227-6 fatcat:asu4bmoiizcsvawzoyguug74qa

Some Results on Greedy Embeddings in Metric Spaces

Ankur Moitra, Tom Leighton
2008 2008 49th Annual IEEE Symposium on Foundations of Computer Science  
Here we resolve a conjecture 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.  ... 
doi:10.1109/focs.2008.18 dblp:conf/focs/MoitraL08 fatcat:wsptm7s6tncn7i6jaokijkvify

A generalized greedy routing algorithm for 2-connected graphs

Huaming Zhang, Xin He
2010 Theoretical Computer Science  
In this paper, we introduce a version 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  ... 
doi:10.1016/j.tcs.2010.08.032 fatcat:rhmdw6abznd5zjiclt2u37ezra

Self-Approaching Graphs [article]

Soroush Alamdari, Timothy M. Chan, Elyot Grant, Anna Lubiw, Vinayak Pathak
2013 arXiv   pre-print
; (2) we can characterize the 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.  ... 
arXiv:1306.5460v1 fatcat:asenvb4zendero6fuz2shkqfx4
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