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On spanning trees with low crossing numbers
[chapter]

1992
*
Lecture Notes in Computer Science
*

Every set S of n points in the plane has a

doi:10.1007/3-540-55488-2_30
fatcat:nyfoo35mtveipfy6a4pw773ibe
*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. ...##
###
Spanning trees with low crossing number

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] . ...

##
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DAG Drawing from an Information Visualization Perspective
[chapter]

2000
*
Eurographics
*

This paper compares two popular layout strategies,

doi:10.1007/978-3-7091-6783-0_1
fatcat:g5hwgf3tjnc7rf5as42deujjr4
*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 . ...##
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DAG Drawing from an Information Visualization Perspective
[article]

2000
*
EUROVIS 2005: Eurographics / IEEE VGTC Symposium on Visualization
*

This paper compares two popular layout strategies,

doi:10.2312/vissym/vissym00/003-012
fatcat:gv2fi3xjwrhw5mir2gtqxcc7j4
*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 . ...##
###
Compiling Crossing-free Geometric Graphs with Connectivity Constraint for Fast Enumeration, Random Sampling, and Optimization
[article]

2020
*
arXiv
*
pre-print

Our algorithms achieve O(6^n) time and O(4^n) time for compiling

arXiv:2001.08899v1
fatcat:egscnm6krff73ffumzij5mttty
*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. ...##
###
Fast enumeration algorithms for non-crossing geometric graphs

2008
*
Proceedings of the twenty-fourth annual symposium on Computational geometry - SCG '08
*

A non-

doi:10.1145/1377676.1377733
dblp:conf/compgeom/KatohT08
fatcat:q3gke3hji5gqpdpsdr6gndgimq
*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. ...##
###
On the Minimum Density Interconnection Tree Problem

1994
*
VLSI design (Print)
*

We present two efficient heuristics for constructing

doi:10.1155/1994/20983
fatcat:d3jlorbkqzh5rmyqsz6nbi4vhe
*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. ...##
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Fast Enumeration Algorithms for Non-crossing Geometric Graphs

2009
*
Discrete & Computational Geometry
*

A non-

doi:10.1007/s00454-009-9164-4
fatcat:smkajngqqzamnpkwv6mq26dm4e
*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. ...##
###
Multitree-like Graph Layering Crossing Optimization

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

doi:10.5220/0007345302330240
dblp:conf/grapp/Marik19
fatcat:b26h55anljdhjm5y6gups6v7vy
*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. ...##
###
Comparative study of fully pre-cross-connected protection architectures for transparent optical networks

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-

doi:10.1109/drcn.2007.4762269
fatcat:gvvd7ahmcjg57cvygpp2h6mqbi
*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). ...##
###
A Spanning Tree-based Genetic Algorithm for Distribution Network Reconfiguration
[article]

2020
*
arXiv
*
pre-print

This paper presents a

arXiv:2008.07908v1
fatcat:yacvxjnxozbedjgcjqpmqjk42m
*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 ...##
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Simple Proofs of Classical Theorems in Discrete Geometry via the Guth–Katz Polynomial Partitioning Technique

2012
*
Discrete & Computational Geometry
*

We apply this method to obtain new and simple proofs of two well known results: the Szemerédi-Trotter theorem

doi:10.1007/s00454-012-9443-3
fatcat:puwz54wxjnhfnp4eevprtnwpb4
*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 . ...##
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Implementation of loop co-tree basis functions for EFIE formulation at low frequency

2019
*
IET Microwaves, Antennas & Propagation
*

The EFIE

doi:10.1049/iet-map.2018.5485
fatcat:ubdm5z4pona25n63rhsc7k3r3q
*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. ...##
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Spanning Trees Crossing Few Barriers

2003
*
Discrete & Computational Geometry
*

We consider the problem of finding

doi:10.1007/s00454-003-2853-5
fatcat:iultrxv235fhdpaekinjhxmv4y
*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). ...##
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An Efficient Parallel Biconnectivity Algorithm

1985
*
SIAM journal on computing (Print)
*

A general algorithmic technique that simplifies and improves computation of various functions

doi:10.1137/0214061
fatcat:6meu4wtajjfxhjagzuzhpcjrcy
*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*...
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