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Optimal algorithms to embed trees in a point set
[chapter]

1996
*
Lecture Notes in Computer Science
*

*In*

*the*

*degree*-

*constrained*

*embedding*problem we are given a set of n

*points*P where each

*point*is assigned a positive

*degree*and

*the*

*degrees*sum to 2n − 2 and are asked to embed a

*tree*

*in*P using straight ...

*In*

*the*rootedtree

*embedding*problem we are given a rooted

*tree*T with n nodes and a set of n

*points*P with one designated

*point*p and are asked to find a straight-line

*embedding*of T

*into*P with

*the*root ... Acknowledgments We thank

*the*referees for their suggestions on improving

*the*presentation and organisation of this paper. ...

##
###
Optimal Algorithms to Embed Trees in a Point Set

1997
*
Journal of Graph Algorithms and Applications
*

*In*

*the*

*degree*-

*constrained*

*embedding*problem we are given a set of n

*points*P where each

*point*is assigned a positive

*degree*and

*the*

*degrees*sum to 2n − 2 and are asked to embed a

*tree*

*in*P using straight ...

*In*

*the*rootedtree

*embedding*problem we are given a rooted

*tree*T with n nodes and a set of n

*points*P with one designated

*point*p and are asked to find a straight-line

*embedding*of T

*into*P with

*the*root ... Acknowledgments We thank

*the*referees for their suggestions on improving

*the*presentation and organisation of this paper. ...

##
###
Optimal Algorithms to Embed Trees in a Point Set
[chapter]

2002
*
Graph Algorithms and Applications I
*

*In*

*the*

*degree*-

*constrained*

*embedding*problem we are given a set of n

*points*P where each

*point*is assigned a positive

*degree*and

*the*

*degrees*sum to 2n − 2 and are asked to embed a

*tree*

*in*P using straight ...

*In*

*the*rootedtree

*embedding*problem we are given a rooted

*tree*T with n nodes and a set of n

*points*P with one designated

*point*p and are asked to find a straight-line

*embedding*of T

*into*P with

*the*root ... Acknowledgments We thank

*the*referees for their suggestions on improving

*the*presentation and organisation of this paper. ...

##
###
Planar embedding of trees on point sets without the general position assumption

2015
*
Turkish Journal of Mathematics
*

We also give some results when

doi:10.3906/mat-1410-12
fatcat:kc4egztq4bb5hmotl2g6czatvy
*the*problem is limited to*degree*-*constrained**trees*and*point*sets having constant number of collinear*points*. ...*The*problem of*point*-set*embedding*of a planar graph G on a*point*set P*in**the**plane*is defined as finding a straight-line planar drawing of G such that*the*nodes of G are mapped one to one on*the**points*... Therefore,*the*complexity of*the*problem is still open if we limit*the*problem to*the**degree*-*constrained**trees*or limit*the*maximum number of collinear*points**in**the**point*set to a constant. ...##
###
Page 1921 of Mathematical Reviews Vol. , Issue 97C
[page]

1997
*
Mathematical Reviews
*

*In*

*the*

*degree*-

*constrained*

*embedding*problem we are given a set of n

*points*P where each

*point*is assigned a positive

*degree*and

*the*

*degrees*sum to 2m — 2 and are asked to find an

*embedding*of a

*tree*

*in*...

*In*

*the*rooted-

*tree*

*embedding*problem we are given a rooted-

*tree*JT with n nodes and a set of n

*points*P with one designated

*point*p and are asked to find a straight-line

*embedding*of T

*into*P with

*the*...

##
###
Computing Steiner points for gradient-constrained minimum networks

2010
*
Discrete Optimization
*

A

doi:10.1016/j.disopt.2009.10.001
fatcat:lydvisnoyrg6dngfkya7wasz4q
*degree*-3 Steiner*point*is labelled minimal if*the*total length of*the*three adjacent edges is minimized for a given labelling. ... Let T g be a gradient-*constrained*minimum network, that is, a minimum length network spanning a given*point*set*in*3-dimensional space with edges that are*constrained*to have gradients no more than an ...*The*Steiner*points**in*LMGCNs can be classified*into*different types, called labellings, according to whether*the*gradients of their incident edges are less than, equal to or greater than m when*embedded*...##
###
Semi-bipartite Graph Visualization for Gene Ontology Networks
[chapter]

2010
*
Lecture Notes in Computer Science
*

*In*this paper we present algorithms which test

*in*polynomial time if a given hypergraph H has a planar support that is (i) a path, (ii) a cycle, (iii) a

*tree*, or (iv) a

*tree*where

*the*maximal

*degree*of ... We also show that

*the*planar slope number of every planar partial 3-

*tree*and also every

*plane*partial 3-

*tree*is at most 2 O(∆) .

*In*particular, we answer

*the*question of Dujmović et al. ...

##
###
Overlay addressing and routing system based on hyperbolic geometry

2011
*
2011 IEEE Symposium on Computers and Communications (ISCC)
*

Local knowledge routing schemes based on virtual coordinates taken from

doi:10.1109/iscc.2011.5983793
dblp:conf/iscc/CassagnesTBM11
fatcat:rxmfwtup7bd7bm66ipxanjtreu
*the*hyperbolic*plane*have attracted considerable interest*in*recent years. ...*In*this paper, we propose a new approach for seizing*the*power of*the*hyperbolic geometry. ... This particular tiling splits*the*hyperbolic*plane**in*distinct spaces and constructs our*embedded**tree*. ...##
###
Point-set embeddings of plane 3-trees

2012
*
Computational geometry
*

We prove an Ω(n log n) lower bound on

doi:10.1016/j.comgeo.2011.09.002
fatcat:u5tgclkngneqfbzil5aiy3iata
*the*time complexity for finding a*point*-set*embedding*of a*plane*3-*tree*. ...*In*this paper, we give an O (n 2 ) time algorithm to decide whether a*plane*3-*tree*admits a*point*-set*embedding*on a given set of*points*or not, and find an*embedding*if it exists. ... Acknowledgements This work is done*in*Graph Drawing & Information Visualization Laboratory of*the*Department of CSE, BUET established under*the*project "Facility Upgradation for Sustainable Research on ...##
###
Polynomial area bounds for MST embeddings of trees

2011
*
Computational geometry
*

*In*their seminal paper on geometric minimum spanning

*trees*, Monma and Suri (1992) [31] showed how to embed any

*tree*of maximum

*degree*5 as a minimum spanning

*tree*

*in*

*the*Euclidean

*plane*. ...

*In*this paper, we show how to construct MST

*embeddings*of arbitrary

*trees*of maximum

*degree*3 and 4 within polynomial area. ...

*The*initial results on complete binary

*trees*were achieved together with Roberto Tamassia. A preliminary and short version of this paper appeared

*in*[21] . ...

##
###
Point-Set Embeddings of Plane 3-Trees
[chapter]

2011
*
Lecture Notes in Computer Science
*

We prove an Ω(n log n) lower bound on

doi:10.1007/978-3-642-18469-7_29
fatcat:4au3pwcbszawzm35espqszfiiy
*the*time complexity for finding a*point*-set*embedding*of a*plane*3-*tree*. ...*In*this paper, we give an O (n 2 ) time algorithm to decide whether a*plane*3-*tree*admits a*point*-set*embedding*on a given set of*points*or not, and find an*embedding*if it exists. ... Acknowledgements This work is done*in*Graph Drawing & Information Visualization Laboratory of*the*Department of CSE, BUET established under*the*project "Facility Upgradation for Sustainable Research on ...##
###
A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem
[article]

2015
*
arXiv
*
pre-print

We introduce

arXiv:1506.05673v1
fatcat:pemapktu45a6hj3u4k44dhnfpa
*the*cd-*tree*data structure and give a new characterization of c-planarity. It leads to efficient algorithms for c-planarity testing*in**the*following cases. ... Moreover,*the*cd-*tree*reveals interesting connections between c-planarity and planarity with constraints on*the*order of edges around vertices. ... Conclusion*In*this paper we have introduced*the*cd-*tree*and we have shown that it can be used to reformulate*the*classic c-planarity problem as a*constrained**embedding*problem. ...##
###
Point-Set Embedding of Trees with Edge Constraints
[chapter]

2008
*
Lecture Notes in Computer Science
*

Given a graph G with n vertices and a set S of n

doi:10.1007/978-3-540-77537-9_14
fatcat:s7xjcoxusvgxxhkvczcvllmsoy
*points**in**the**plane*, a*point*-set*embedding*of G on S is a planar drawing such that each vertex of G is mapped to a distinct*point*of S. ... We concentrate on*trees*and show how to compute*the*output*in*O(n 2 log n) time and with at most 1 + 2 k/2 bends per edge, where k is*the*number of vertices of*the*given subdrawing. ... Introduction Let G be a planar graph with n vertices and let S be a set of n*points**in**the**plane*. ...##
###
Fast geometric approximation techniques and geometric embedding problems

1992
*
Theoretical Computer Science
*

We give

doi:10.1016/0304-3975(92)90252-b
fatcat:wtb6nhryhrb7vptjokdrm3n4hm
*the*example of approximating*the*convex hull of a set of*points**in**the**plane*. ..' pn > of II*points**in*R", and an undirected graph G = (V, E) on n vertices. ... We give fast approximation algorithms for*embedding*these graphs on*the*line and*in**the**plane**in*several metrics. ...*Trees*Our EMST approximation algorithm yields an approximation algorithm for*embedding*bounded-*degree**trees*, such as a complete binary*tree*, onto*points*P*in**the**plane*. ...##
###
Optimal Orthogonal Drawings of Planar 3-Graphs in Linear Time
[article]

2019
*
arXiv
*
pre-print

., a planar graph with vertex

arXiv:1910.11782v1
fatcat:bsfhm3dcsnbo7p4wu7vkhvj3j4
*degree*at most three), what is*the*best computational upper bound to compute a bend-minimum planar orthogonal drawing of G*in**the*variable*embedding*setting? ...*In*this setting*the*algorithm can choose among*the*exponentially many planar*embeddings*of G*the*one that leads to an orthogonal drawing with*the*minimum number of bends. ... SPQR-*Trees*. Let G be a biconnected graph. An SPQR-*tree*T of G represents*the*decomposition of G*into*its triconnected components and can be computed*in*linear time [8, 17, 19] . ...
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