scholarly journals PROXIMITY DRAWINGS OF HIGH-DEGREE TREES

2013 ◽  
Vol 23 (03) ◽  
pp. 213-230
Author(s):  
FERRAN HURTADO ◽  
GIUSEPPE LIOTTA ◽  
DAVID R. WOOD
Keyword(s):  
Spanning Tree ◽  
Maximum Degree ◽  
Minimum Spanning Tree ◽  
Bounded Degree ◽  
Natural Properties ◽  
Vertex Set ◽  
High Degree ◽  
The Given

A drawing of a given (abstract) tree that is a minimum spanning tree of the vertex set is considered aesthetically pleasing. However, such a drawing can only exist if the tree has maximum degree at most 6. What can be said for trees of higher degree? We approach this question by supposing that a partition or covering of the tree by subtrees of bounded degree is given. Then we show that if the partition or covering satisfies some natural properties, then there is a drawing of the entire tree such that each of the given subtrees is drawn as a minimum spanning tree of its vertex set.

scholarly journals DELAUNAY AND DIAMOND TRIANGULATIONS CONTAIN SPANNERS OF BOUNDED DEGREE

2009 ◽  
Vol 19 (02) ◽  
pp. 119-140 ◽  
Author(s):  
PROSENJIT BOSE ◽  
MICHIEL SMID ◽  
DAMING XU
Keyword(s):  
Unit Disk ◽  
Delaunay Triangulation ◽  
Maximum Degree ◽  
Time Algorithm ◽  
Connected Subgraph ◽  
Unit Disk Graph ◽  
Bounded Degree ◽  
Vertex Set ◽  
Range Of Values ◽  
Degree Bound

Given a triangulation G, whose vertex set V is a set of n points in the plane, and given a real number γ with 0 < γ < π, we design an O(n)-time algorithm that constructs a connected subgraph G' of G with vertex set V whose maximum degree is at most 14 + ⌈2π/γ⌉. If G is the Delaunay triangulation of V, and γ = 2π/3, we show that G' is a t-spanner of V (for some constant t) with maximum degree at most 17, thereby improving the previously best known degree bound of 23. If G is a triangulation satisfying the diamond property, then for a specific range of values of γ dependent on the angle of the diamonds, we show that G' is a t-spanner of V (for some constant t) whose maximum degree is bounded by a constant dependent on γ. If G is the graph consisting of all Delaunay edges of length at most 1, and γ = π/3, we show that a modified version of the algorithm produces a plane subgraph G' of the unit-disk graph which is a t-spanner (for some constant t) of the unit-disk graph of V, whose maximum degree is at most 20, thereby improving the previously best known degree bound of 25.

scholarly journals Spanning Trees of Bounded Degree

10.37236/1577 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Andrzej Czygrinow ◽  
Genghua Fan ◽  
Glenn Hurlbert ◽  
H. A. Kierstead ◽  
William T. Trotter
Keyword(s):  
Spanning Tree ◽  
Maximum Degree ◽  
Additional Assumption ◽  
Spanning Trees ◽  
Independent Set ◽  
Hamilton Cycle ◽  
Bounded Degree ◽  
Hamilton Path ◽  
Classic Theorem

Dirac's classic theorem asserts that if ${\bf G}$ is a graph on $n$ vertices, and $\delta({\bf G})\ge n/2$, then ${\bf G}$ has a hamilton cycle. As is well known, the proof also shows that if $\deg(x)+\deg(y)\ge(n-1)$, for every pair $x$, $y$ of independent vertices in ${\bf G}$, then ${\bf G}$ has a hamilton path. More generally, S. Win has shown that if $k\ge 2$, ${\bf G}$ is connected and $\sum_{x\in I}\deg(x)\ge n-1$ whenever $I$ is a $k$-element independent set, then ${\bf G}$ has a spanning tree ${\bf T}$ with $\Delta({\bf T})\le k$. Here we are interested in the structure of spanning trees under the additional assumption that ${\bf G}$ does not have a spanning tree with maximum degree less than $k$. We show that apart from a single exceptional class of graphs, if $\sum_{x\in I}\deg(x)\ge n-1$ for every $k$-element independent set, then ${\bf G}$ has a spanning caterpillar ${\bf T}$ with maximum degree $k$. Furthermore, given a maximum path $P$ in ${\bf G}$, we may require that $P$ is the spine of ${\bf T}$ and that the set of all vertices whose degree in ${\bf T}$ is $3$ or larger is independent in ${\bf T}$.

scholarly journals CONSTRUCTING DEGREE-3 SPANNERS WITH OTHER SPARSENESS PROPERTIES

1996 ◽  
Vol 07 (02) ◽  
pp. 121-135 ◽  
Author(s):  
GAUTAM DAS ◽  
PAUL J. HEFFERNAN
Keyword(s):  
Euclidean Space ◽  
Shortest Path ◽  
Spanning Tree ◽  
Maximum Degree ◽  
Euclidean Distance ◽  
Minimum Spanning Tree ◽  
Dimensional Euclidean Space ◽  
Edge Weight

Let V be any set of n points in k-dimensional Euclidean space. A subgraph of the complete Euclidean graph is a t-spanner if for all u and υ in V, the length of the shortest path from u to υ in the spanner is at most t times the Euclidean distance between u and υ. We show that for any δ>1, there exists a t-spanner (where t is a constant that depends only on δ and k) with the following properties: its maximum degree is 3, it has at most n·δ edges, its total edge weight is at most O(1) times the weight of the minimum spanning tree of V, and it can be constructed in O(n log n) time. The constants implicit in the O-notation depend on δ and k.

FUZZY RANDOM DEGREE-CONSTRAINED MINIMUM SPANNING TREE PROBLEM

2007 ◽  
Vol 15 (supp02) ◽  
pp. 107-115
Author(s):  
WEI LIU ◽  
CHENGJING YANG
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Programming Models ◽  
Intelligent Algorithm ◽  
Hybrid Intelligent Algorithm ◽  
Random Weights ◽  
Fuzzy Random Programming ◽  
Minimum Spanning Tree Problem ◽  
The Given ◽  
Fuzzy Random

The degree-constrained minimum spanning tree problem (dc-MST) is to find a minimum spanning tree of the given graph, subject to constraints on node degrees. This paper investigates the dc-MST problem with fuzzy random weights. Three concepts are presented: expected fuzzy random dc-MST, (α,β)-dc-MST and the most chance dc-MST according to different optimization requirement. Correspondingly, by using the concepts as decision criteria, three fuzzy random programming models for dc-MST are given. Finally, a hybrid intelligent algorithm is designed to solve these models, and some numerical examples are provided to illustrate its effectiveness.

scholarly journals Algorithms for Euclidean Degree Bounded Spanning Tree Problems

2019 ◽  
Vol 29 (02) ◽  
pp. 121-160 ◽  
Author(s):  
Patrick J. Andersen ◽  
Charl J. Ras
Keyword(s):  
Approximation Algorithms ◽  
Spanning Tree ◽  
Maximum Degree ◽  
Euclidean Plane ◽  
Minimum Spanning Tree ◽  
Computational Experiments ◽  
Optimal Solutions ◽  
New Type ◽  
Disjoint Sets ◽  
Set Of Points

Given a set of points in the Euclidean plane, the Euclidean [Formula: see text]-minimum spanning tree ([Formula: see text]-MST) problem is the problem of finding a spanning tree with maximum degree no more than [Formula: see text] for the set of points such the sum of the total length of its edges is minimum. Similarly, the Euclidean [Formula: see text]-minimum bottleneck spanning tree ([Formula: see text]-MBST) problem, is the problem of finding a degree-bounded spanning tree for a set of points in the plane such that the length of the longest edge is minimum. When [Formula: see text], these two problems may yield disjoint sets of optimal solutions for the same set of points. In this paper, we perform computational experiments to compare the accuracies of a variety of heuristic and approximation algorithms for both these problems. We develop heuristics for these problems and compare them with existing algorithms. We also describe a new type of edge swap algorithm for these problems that outperforms all the algorithms we tested.

Online Minimum Spanning Tree with Advice

2018 ◽  
Vol 29 (04) ◽  
pp. 505-527
Author(s):  
Maria Paola Bianchi ◽  
Hans-Joachim Böckenhauer ◽  
Tatjana Brülisauer ◽  
Dennis Komm ◽  
Beatrice Palano
Keyword(s):  
Lower Bound ◽  
Weight Function ◽  
Competitive Ratio ◽  
Spanning Tree ◽  
Online Algorithm ◽  
Minimum Spanning Tree ◽  
Optimal Solution ◽  
Bounded Degree ◽  
Advice Complexity ◽  
Graph Classes

In the online minimum spanning tree problem, a graph is revealed vertex by vertex; together with every vertex, all edges to vertices that are already known are given, and an online algorithm must irrevocably choose a subset of them as a part of its solution. The advice complexity of an online problem is a means to quantify the information that needs to be extracted from the input to achieve good results. For a graph of size [Formula: see text], we show an asymptotically tight bound of [Formula: see text] on the number of advice bits to produce an optimal solution for any given graph. For particular graph classes, e.g., with bounded degree or a restricted edge weight function, we prove that the upper bound can be drastically reduced; e.g., [Formula: see text] advice bits allow to compute an optimal result if the weight function equals the Euclidean distance; if the graph is complete and has two different edge weights, even a logarithmic number suffices. Some of these results make use of the optimality of Kruskal’s algorithm for the offline setting. We also study the trade-off between the number of advice bits and the achievable competitive ratio. To this end, we perform a reduction from another online problem to obtain a linear lower bound on the advice complexity for any near-optimal solution. Using our results finally allows us to give a lower bound on the expected competitive ratio of any randomized online algorithm for the problem, even on graphs with three different edge weights.

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