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Spanning Trees and Spanners [chapter]

David Eppstein
2000 Handbook of Computational Geometry  
We survey results in 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?  ... 
doi:10.1016/b978-044482537-7/50010-3 fatcat:gitonpgfozgfribivszd6gf5cy

Computing a (1+ε)-Approximate Geometric Minimum-Diameter Spanning Tree

Michael J. Spriggs, J. Mark Keil, Sergei Bespamyatnikh, Michael Segal, Jack Snoeyink
2003 Algorithmica  
Given a set P of points in 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).  ... 
doi:10.1007/s00453-003-1056-z fatcat:kwpcjtoukbgsld3yzjawtxf73e

Evolutionary multiobjective combinatorial optimization (EMCO)

Rajeev Kumar
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'.  ... 
doi:10.1145/1388969.1389079 dblp:conf/gecco/Kumar08 fatcat:acsog2ea35eijaww2ai7ozm4qi

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

Edmund Ihler, Gabriele Reich, Peter Widmayer
1991 Lecture Notes in Computer Science  
In contrast, a class solution for 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.  ... 
doi:10.1007/3-540-54891-2_8 fatcat:u2qyovetuzcy7kbwgcpcgduyaa

Geometric Minimum Diameter Minimum Cost Spanning Tree Problem [chapter]

Dae Young Seo, D. T. Lee, Tien-Ching Lin
2009 Lecture Notes in Computer Science  
In this paper we consider bi-criteria 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.  ... 
doi:10.1007/978-3-642-10631-6_30 fatcat:6fmvagdysferdhpxff3rqwerba

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]

Ke Chen, Adrian Dumitrescu
2018 Lecture Notes in Computer Science  
Given a set of n compact neighborhoods in R d , select a point in each neighborhood, so that 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] .  ... 
doi:10.1007/978-3-319-78455-7_2 fatcat:4ynu7lazdrbahix3y5t6em7lqi

On the Longest Spanning Tree with Neighborhoods [article]

Ke Chen, Adrian Dumitrescu
2020 arXiv   pre-print
Given a set of n compact neighborhoods in R^d, select a point in each neighborhood, so that 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] .  ... 
arXiv:1712.03297v2 fatcat:2cijvri46fcidbarzxgivn2eem

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

Yaacov Fernandess, Dahlia Malkhi
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.  ... 
doi:10.1145/584490.584497 dblp:conf/pomc/FernandessM02 fatcat:oma6qvh2krdbhb5bqipfeafygu

K-clustering in wireless ad hoc networks

Yaacov Fernandess, Dahlia Malkhi
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.  ... 
doi:10.1145/584495.584497 fatcat:msxz7us2ufhgdkt7t4akqii3dq

On the Shortest Separating Cycle [article]

Adrian Dumitrescu
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.  ... 
arXiv:1912.01541v1 fatcat:2s76gps46jbxfmnpciidzadyxa

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

Ahmad Biniaz
2020 arXiv   pre-print
(i) Longest noncrossing 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.  ... 
arXiv:2010.03870v1 fatcat:rnyyvtnbgjdbzg5palvy3mclpm

Low-stretch greedy embedding heuristics

Andrej Cvetkovski, Mark Crovella
2012 2012 Proceedings IEEE INFOCOM Workshops  
In this paper we study how topological and 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.  ... 
doi:10.1109/infcomw.2012.6193497 dblp:conf/infocom/CvetkovskiC12 fatcat:a4vsg6drpzc2bndhra3bcbb7fm

The minimum-area spanning tree problem

Paz Carmi, Matthew J. Katz, Joseph S.B. Mitchell
2006 Computational geometry  
We prove that 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.  ... 
doi:10.1016/j.comgeo.2006.03.001 fatcat:34ihgrl2hnelxd6jfotettxiiy
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