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Spanning Trees and Spanners
[chapter]

2000
*
Handbook of Computational Geometry
*

We survey results in

doi:10.1016/b978-044482537-7/50010-3
fatcat:gitonpgfozgfribivszd6gf5cy
*geometric*network design theory, including algorithms for constructing*minimum**spanning**trees*and low-dilation graphs. ... Such problems include*the**minimum**spanning**tree*, maximum*spanning**tree*,*minimum**diameter**spanning**tree*, bounded degree*spanning**trees*(such as*the*traveling salesman path), and*the*k-point*minimum**spanning*... Is it possible to construct*the*exact*minimum*dilation*geometric**spanning**tree*, or an*approximation*to it, in polynomial time? Does*the**minimum*dilation*spanning**tree*have any edge crossings? ...##
###
Computing a (1+ε)-Approximate Geometric Minimum-Diameter Spanning Tree

2003
*
Algorithmica
*

Given a set P of points in

doi:10.1007/s00453-003-1056-z
fatcat:kwpcjtoukbgsld3yzjawtxf73e
*the*plane, a*geometric**minimum*-*diameter**spanning**tree*(GMDST) of P is a*spanning**tree*of P such that*the*longest path through*the**tree*is minimized. ... In this paper, we present an*approximation*algorithm that generates a*tree*whose*diameter*is no more than (1 + ) times that of a GMDST, for any > 0. ... We call this special version of*the*GMDST a restricted*geometric**minimum**diameter**spanning**tree*(RGMDST). ...##
###
Evolutionary multiobjective combinatorial optimization (EMCO)

2008
*
Proceedings of the 2008 GECCO conference companion on Genetic and evolutionary computation - GECCO '08
*

*minimum*

*spanning*

*tree*(MST) for a given complete graph minimizing simultaneously edge cost and

*diameter*of

*the*

*tree*. ...

*Diameter*-Cost

*Minimum*

*Spanning*

*Tree*Problem 60 Analysis of search space Let

*the*cost of unconstrained MST is `C' and

*diameter*is `D'. ...

##
###
On shortest networks for classes of points in the plane
[chapter]

1991
*
Lecture Notes in Computer Science
*

In contrast, a class solution for

doi:10.1007/3-540-54891-2_8
fatcat:u2qyovetuzcy7kbwgcpcgduyaa
*the**minimum**diameter**spanning**tree*problem can be computed in time O(jPj 3 ). ... For a given network optimization problem, such as nding a*minimum**spanning**tree*or nding a*minimum**diameter**spanning**tree*, we study*the*problem of choosing a subset P 0 of P that contains at least one ... First, consider*the*problem of computing*the**geometrical**minimum**diameter**spanning**tree*gMDST for a Euclidean graph without classes. ...##
###
Geometric Minimum Diameter Minimum Cost Spanning Tree Problem
[chapter]

2009
*
Lecture Notes in Computer Science
*

In this paper we consider bi-criteria

doi:10.1007/978-3-642-10631-6_30
fatcat:6fmvagdysferdhpxff3rqwerba
*geometric*optimization problems, in particular,*the**minimum**diameter**minimum*cost*spanning**tree*problem and*the**minimum*radius*minimum*cost*spanning**tree*problem for ...*The*former problem is to construct a*minimum**diameter**spanning**tree*among all possible*minimum*cost*spanning**trees*, while*the*latter is to construct a*minimum*radius*spanning**tree*among all possible*minimum*... A*spanning**tree*of a graph G(V, E) that minimizes its*diameter*is called*the**minimum**diameter**spanning**tree*.*The**geometric*version of this problem (GMDST) is defined as follows. ...##
###
Page 4113 of Mathematical Reviews Vol. , Issue 94g
[page]

1994
*
Mathematical Reviews
*

In contrast, a class solution for

*the**minimum**diameter**spanning**tree*problem can be computed in time O(|P|?). ... For a given network optimization problem, such as finding a*minimum**spanning**tree*or finding a*minimum**diameter**spanning**tree*, we study*the*problem of choosing a subset P’ of P that contains at least one ...##
###
On the Longest Spanning Tree with Neighborhoods
[chapter]

2018
*
Lecture Notes in Computer Science
*

Given a set of n compact neighborhoods in R d , select a point in each neighborhood, so that

doi:10.1007/978-3-319-78455-7_2
fatcat:4ynu7lazdrbahix3y5t6em7lqi
*the*longest*spanning**tree*on these points (as vertices) has maximum length. ... Here we give an*approximation*algorithm with ratio 0.511, which represents*the*first, albeit small, improvement beyond 1/2. ... Computing*the**minimum*or maximum Euclidean*spanning**trees*of a point set are classical problems in a*geometric*setting [13, 14] . ...##
###
On the Longest Spanning Tree with Neighborhoods
[article]

2020
*
arXiv
*
pre-print

Given a set of n compact neighborhoods in R^d, select a point in each neighborhood, so that

arXiv:1712.03297v2
fatcat:2cijvri46fcidbarzxgivn2eem
*the*longest*spanning**tree*on these points (as vertices) has maximum length. ... Here we give an*approximation*algorithm with ratio 0.511, which represents*the*first, albeit small, improvement beyond 1/2. ... Computing*the**minimum*or maximum Euclidean*spanning**trees*of a point set are classical problems in a*geometric*setting [14, 16] . ...##
###
Page 10472 of Mathematical Reviews Vol. , Issue 2004m
[page]

2004
*
Mathematical Reviews
*

Summary: “Let P be a set of 7 points in

*the*plane.*The**geometric**minimum*-*diameter**spanning**tree*(MDST) of P is a*tree*that*spans*P and minimizes*the*Euclidean length of*the*longest path. ... ; Utrecht); Park, Sang-Min; Shin, Chan-Su; Wolff, Alexander (D-KLRH-L; Karlsruhe) Facility location and*the**geometric**minimum*-*diameter**spanning**tree*. ...##
###
K-clustering in wireless ad hoc networks

2002
*
Proceedings of the second ACM international workshop on Principles of mobile computing - POMC '02
*

*The*first phase constructs a

*spanning*

*tree*of

*the*network and

*the*second phase then partitions

*the*

*spanning*

*tree*into subtrees with bounded

*diameters*. ... For

*the*special family of graphs that represent ad hoc wireless networks, modeled as unit disk graphs, we introduce a two phase distributed polynomial time and message complexity

*approximation*solution ...

*The*second phase partitions

*the*

*spanning*

*tree*into subtrees whose

*diameter*is k bounded, yielding a k-clustering of

*the*network. ...

##
###
K-clustering in wireless ad hoc networks

2002
*
Proceedings of the second ACM international workshop on Principles of mobile computing - POMC '02
*

*The*first phase constructs a

*spanning*

*tree*of

*the*network and

*the*second phase then partitions

*the*

*spanning*

*tree*into subtrees with bounded

*diameters*. ... For

*the*special family of graphs that represent ad hoc wireless networks, modeled as unit disk graphs, we introduce a two phase distributed polynomial time and message complexity

*approximation*solution ...

*The*second phase partitions

*the*

*spanning*

*tree*into subtrees whose

*diameter*is k bounded, yielding a k-clustering of

*the*network. ...

##
###
On the Shortest Separating Cycle
[article]

2019
*
arXiv
*
pre-print

*approximation*with respect to

*the*

*minimum*length can be computed in polynomial time. ... Moreover, a O(n^2/3)-

*approximation*of a separating polyhedron of

*minimum*perimeter can be found in polynomial time. ...

*The*author is grateful to an anonymous reviewer for his careful reading of

*the*manuscript and pertinent remarks. ...

##
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Improved approximation ratios for two Euclidean maximum spanning tree problems
[article]

2020
*
arXiv
*
pre-print

(i) Longest noncrossing

arXiv:2010.03870v1
fatcat:rnyyvtnbgjdbzg5palvy3mclpm
*spanning**tree*: Given a set of points in*the*plane,*the*goal is to find a maximum-length noncrossing*spanning**tree*. ... (ii) Longest*spanning**tree*with neighborhoods: Given a collection of regions (neighborhoods) in*the*plane,*the*goal is to select a point in each neighborhood so that*the*longest*spanning**tree*on selected ...*The*well-known*minimum**spanning**tree*(Min-ST) problem asks for a*spanning**tree*of a weighted graph, with*minimum*total edge weight. ...##
###
Low-stretch greedy embedding heuristics

2012
*
2012 Proceedings IEEE INFOCOM Workshops
*

In this paper we study how topological and

doi:10.1109/infcomw.2012.6193497
dblp:conf/infocom/CvetkovskiC12
fatcat:a4vsg6drpzc2bndhra3bcbb7fm
*geometric*properties of embedded graphs influence*the*hop stretch. ... It is desirable that graph embeddings also yield low hop stretch of*the*greedy over*the*shortest paths. ... ACKNOWLEDGMENTS This work was supported by*the*National Science Foundation under Grant no. CNS-1018266. ...##
###
The minimum-area spanning tree problem

2006
*
Computational geometry
*

We prove that

doi:10.1016/j.comgeo.2006.03.001
fatcat:34ihgrl2hnelxd6jfotettxiiy
*the*Euclidean*minimum**spanning**tree*of P is a constant-factor*approximation*for MAST. ... "area", where*the*area of a*spanning**tree*T is*the*area of*the*union of*the*n − 1 disks whose*diameters*are*the*edges in T . ... Conclusion We introduced*the**Minimum*-Area*Spanning**Tree*(MAST) problem, and proved that*the**minimum**spanning**tree*is a constant-factor*approximation*of*the**minimum*-area*spanning**tree*. ...
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