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On spanning trees with low crossing numbers [chapter]

Emo Welzl
1992 Lecture Notes in Computer Science  
Every set S of n points in the plane has a spanning tree such that no line disjoint from S has more than O( p n) i n tersections with the tree (where the edges are embedded as straight line segments).  ...  We review the proof of this result (originally proved by Bernard Chazelle and the author in a more general setting), point at some methods for constructing such a tree, and describe some algorithmic and  ...  One of the combinatorial tools developed for this problem are spanning trees with low crossing numbers. Let S be set of n points in the plane.  ... 
doi:10.1007/3-540-55488-2_30 fatcat:nyfoo35mtveipfy6a4pw773ibe

Spanning trees with low crossing number

Jiří Matoušek
1991 RAIRO - Theoretical Informatics and Applications  
SPANNING TREES WITH LOW CROSSING NUMBER (*) by Jiri MATOUSEK ( X ) Abstract. -Let P be a point set in the plane and T a spanning tree on P, whose edges are realized by segments.  ...  It is a basis for the algorithm finding a spanning tree with low crossing number in [14] .  ... 
doi:10.1051/ita/1991250201031 fatcat:eareco4eyngmjeqkwgccfiioru

DAG Drawing from an Information Visualization Perspective [chapter]

G. Melançon, I. Herman
2000 Eurographics  
This paper compares two popular layout strategies, one based on the extraction of a spanning tree, the other based on edge crossing minimization of directed acyclic graphs.  ...  When dealing with a graph, any visualization strategy must rely on a layout procedure at least to initiate the process.  ...  With low density graphs, the number of dummies introduced in the layout process is low, as shown on Figure 3 .  ... 
doi:10.1007/978-3-7091-6783-0_1 fatcat:g5hwgf3tjnc7rf5as42deujjr4

DAG Drawing from an Information Visualization Perspective [article]

G. Melancon, I. Herman
2000 EUROVIS 2005: Eurographics / IEEE VGTC Symposium on Visualization  
This paper compares two popular layout strategies, one based on the extraction of a spanning tree, the other based on edge crossing minimization of directed acyclic graphs.  ...  When dealing with a graph, any visualization strategy must rely on a layout procedure at least to initiate the process.  ...  With low density graphs, the number of dummies introduced in the layout process is low, as shown on Figure 3 .  ... 
doi:10.2312/vissym/vissym00/003-012 fatcat:gv2fi3xjwrhw5mir2gtqxcc7j4

Compiling Crossing-free Geometric Graphs with Connectivity Constraint for Fast Enumeration, Random Sampling, and Optimization [article]

Yu Nakahata, Takashi Horiyama, Shin-ichi Minato, Katsuhisa Yamanaka
2020 arXiv   pre-print
Our algorithms achieve O(6^n) time and O(4^n) time for compiling crossing-free spanning trees and crossing-free spanning cycles, respectively.  ...  To achieve this, Wettstein proposed specialized techniques for crossing-free spanning trees and crossing-free spanning cycles and invented compiling algorithms running in O(7.044^n) time and O(5.619^n)  ...  We define st(P ) and sc(P ) as the numbers of crossing-free spanning trees and crossing-free spanning cycles on P , respectively.  ... 
arXiv:2001.08899v1 fatcat:egscnm6krff73ffumzij5mttty

Fast enumeration algorithms for non-crossing geometric graphs

Naoki Katoh, Shin-ichi Tanigawa
2008 Proceedings of the twenty-fourth annual symposium on Computational geometry - SCG '08  
A non-crossing geometric graph is a graph embedded on a given set of points in the plane with non-crossing straight line segments.  ...  Therefore, by enumerating spanning trees in every triangulation, we will obtain all non-crossing spanning trees.  ...  Acknowledgment The first author is supported by the project New Horizons in Computing, Grant-in-Aid for Scientific Research on Priority Areas, NEXT Japan.  ... 
doi:10.1145/1377676.1377733 dblp:conf/compgeom/KatohT08 fatcat:q3gke3hji5gqpdpsdr6gndgimq

On the Minimum Density Interconnection Tree Problem

C. J. Alpert, J. Cong, A. B. Kahng, G. Robins, M. Sarrafzadeh
1994 VLSI design (Print)  
We present two efficient heuristics for constructing low-density spanning trees and prove that their outputs are on average within small constants of optimal with respect to both tree cost and density.  ...  Furthermore, the minimum density objective can be transparently combined with a number of previous interconnection objectives (e.g., minimizing tree radius or skew) without affecting solution quality with  ...  We are also able to address the case of low-density Steiner trees, while previous work on trees with low stabbing number could address only spanning constructions.  ... 
doi:10.1155/1994/20983 fatcat:d3jlorbkqzh5rmyqsz6nbi4vhe

Fast Enumeration Algorithms for Non-crossing Geometric Graphs

Naoki Katoh, Shin-Ichi Tanigawa
2009 Discrete & Computational Geometry  
A non-crossing geometric graph is a graph embedded on a set of points in the plane with noncrossing straight line segments.  ...  Therefore, by enumerating spanning trees in every triangulation, we will obtain all non-crossing spanning trees since every non-crossing spanning tree is a subgraph of some triangulation.  ...  One of the anonymous referees pointed out how to improve the result of Theorem 3.7 from 1.41 n 2 tri(P ) ≤ cg(P ) to 1.52 n−1 tri(P ) ≤ cg(P ) by a simple idea. We greatly appreciate this comment.  ... 
doi:10.1007/s00454-009-9164-4 fatcat:smkajngqqzamnpkwv6mq26dm4e

Multitree-like Graph Layering Crossing Optimization

Radek Mařík
2019 Proceedings of the 14th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications  
We propose a new method of how the order of subtrees selected by the driving spanning tree can be derived from the actual edge crossings.  ...  We improve a method of multitree-like graph visualization using a spanning tree-driven layout technique with constraints specified by layers and the ordering of groups of nodes within layers.  ...  To ensure a low number of edge crossings an edge e r j incident with the subtree T j having the least number of edge crossings K j is selected.  ... 
doi:10.5220/0007345302330240 dblp:conf/grapp/Marik19 fatcat:b26h55anljdhjm5y6gups6v7vy

Comparative study of fully pre-cross-connected protection architectures for transparent optical networks

Aden Grue, Wayne D. Grover, Matthieu Clouqueur, Dominic A. Schupke, John Doucette, Brian Forst, Diane Onguetou, Dimitri Baloukov
2007 2007 6th International Workshop on Design and Reliable Communication Networks  
The architectures considered were p-cycles, failure independent pathprotecting (FIPP) p-cycles, demand-wise shared protection (DSP), pre-cross-connected trails (PXTs), span-protecting p-trees, and path-protecting  ...  p-trees.  ...  spans) and maximum tree node degree of 4 (i.e. maximum number of tree spans incident on any tree node).  ... 
doi:10.1109/drcn.2007.4762269 fatcat:gvvd7ahmcjg57cvygpp2h6mqbi

A Spanning Tree-based Genetic Algorithm for Distribution Network Reconfiguration [article]

Mukesh Gautam, Narayan Bhusal, Mohammed Benidris, Sushil J. Louis
2020 arXiv   pre-print
This paper presents a spanning tree-based genetic algorithm (GA) for the reconfiguration of electrical distribution systems with the objective of minimizing active power losses.  ...  The proposed method searches spanning trees of potential configurations and finds the optimal spanning tree using a genetic algorithm in two steps.  ...  Spanning Tree Search Algorithm For a graph G with V vertices and E edges, a spanning tree can be defined as a subset of the graph, which has a minimum number of edges (say, E s ) connecting all vertices  ... 
arXiv:2008.07908v1 fatcat:yacvxjnxozbedjgcjqpmqjk42m

Simple Proofs of Classical Theorems in Discrete Geometry via the Guth–Katz Polynomial Partitioning Technique

Haim Kaplan, Jiří Matoušek, Micha Sharir
2012 Discrete & Computational Geometry  
We apply this method to obtain new and simple proofs of two well known results: the Szemerédi-Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers  ...  We also mention some generalizations and extensions, such as the Pach-Sharir bound on the number of incidences with algebraic curves of bounded degree.  ...  Spanning trees with low crossing number. Let P be a finite set of points in R 2 .  ... 
doi:10.1007/s00454-012-9443-3 fatcat:puwz54wxjnhfnp4eevprtnwpb4

Implementation of loop co-tree basis functions for EFIE formulation at low frequency

Seung Mo Seo
2019 IET Microwaves, Antennas & Propagation  
The EFIE with Rao-Wilton-Glisson (RWG) basis functions suffer from low-frequency instability. Even for an appropriate frequency, the EFIE with RWG basis functions is ill-posed.  ...  The approach is shown to be very effective for resolving both the low-frequency instability and the ill-condition of the MoM matrix with a generalised minimal residual method.  ...  The spanning tree is not unique, but the number of the edges in the tree is same.  ... 
doi:10.1049/iet-map.2018.5485 fatcat:ubdm5z4pona25n63rhsc7k3r3q

Spanning Trees Crossing Few Barriers

Tetsuo Asano, Mark de Berg, Otfried Cheong, Leonidas J. Guibas, Jack Snoeyink, Hisao Tamaki
2003 Discrete & Computational Geometry  
We consider the problem of finding low-cost spanning trees for sets of n points in the plane, where the cost of a spanning tree is defined as the total number of intersections of tree edges with a given  ...  , then there is always a spanning tree of cost O(m); (iii) if the barriers are disjoint convex objects, then there is always a spanning tree of cost O(n + m).  ...  Combining our analysis with known results on BSPs will then give us low-cost spanning trees for so-called uncluttered scenes (defined below).  ... 
doi:10.1007/s00454-003-2853-5 fatcat:iultrxv235fhdpaekinjhxmv4y

An Efficient Parallel Biconnectivity Algorithm

Robert E. Tarjan, Uzi Vishkin
1985 SIAM journal on computing (Print)  
A general algorithmic technique that simplifies and improves computation of various functions on trees is introduced.  ...  An alternative implementation runs in O(n2/p) time and O(n2) space using any number p <= n/log n of processors, on a concurrent-read, exclusive-write parallel RAM.  ...  Suppose we number the vertices in preorder with respect to any spanning tree; identify vertices by number; compute nd(v), the number of descendants of the vertex, for each vertex v; and compute the low  ... 
doi:10.1137/0214061 fatcat:6meu4wtajjfxhjagzuzhpcjrcy
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