Broadcasting secure messages via optimal independent spanning trees in folded hypercubes

Fault-tolerant broadcasting and secure message distribution are important issues for network applications. It is a common idea to design multiple spanning trees with a specific property in the underlying graph of a network to serve as a broadcasting scheme or a distribution protocol for receiving high levels of fault-tolerance and security. An n-dimensional folded hypercube, denoted by FQ_n, is a strengthening variation of hypercube by adding additional links between nodes that have the furthest Hamming distance. In, [12], Ho(1990) proposed an algorithm for constructing n+1 edge-disjoint spanning trees each with a height twice the diameter of FQ_n. Yang et al. (2009), [29] recently proved that Ho’s spanning trees are indeed independent, i.e., any two spanning trees have the same root, say r, and for any other node v≠r, the two different paths from v to r, one path in each tree, are internally node-disjoint. In this paper, we provide another construction scheme to produce n+1 independent spanning trees of FQ_n, where the height of each tree is equal to the diameter of FQ_n plus one. As a result, the heights of independent spanning trees constructed in this paper are shown to be optimal.     

Minimal spanning trees

The main result is that a recursive weighted graph having a minimal spanning tree has a minimal spanning tree that is . This leads to a proof of the failure of a conjecture of Clote and Hirst (1998) concerning Reverse Mathematics and minimal spanning trees.

     

Generalized spanning trees

In this paper, we propose a definition for the Generalized Minimal Spanning Tree (GMST) of a graph. The GMST requires spanning at least one node out of every set of disjoint nodes (node partition) in a graph. The analysis of the GMST problem is motivated by real life agricultural settings related to construction of irrigation networks in desert environments. We prove that the GMST problem is NP-hard, and examine a number of heuristic solutions for this problem. Computational experiments comparing these heuristics are presented.     

Efficient spanning trees

The definition of a shortest spanning tree of a graph is generalized to that of an efficient spanning tree for graphs with vector weights, where the notion of optimality is of the Pareto type. An algorighm for obtaining all efficient spanning trees is presented.     

Chain-constrained spanning trees

Mathematical Programming - We consider the problem of finding a spanning tree satisfying a family of additional constraints. Several settings have been considered previously, the most famous being...     

Low-degree minimum spanning trees

Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that, under theL _p norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs where the maximum degree is minimized. We give the best-known bounds for the maximum MST degree for arbitraryL _p metrics in all dimensions, with a focus on the rectilinear metric in two and three dimensions. We show that for any finite set of points in the rectilinear plane an MST exists with maximum degree of at most 4, and for three-dimensional rectilinear space the maximum possible degree of a minimum-degree MST is either 13 or 14. Gabriel Robins was partially supported by NSF Young Investigator Award MIP-9457412. Jeffrey Salowe was partially supported by NSF Grants MIP-9107717 and CCR-9224789.     

Homeomorphically irreducible spanning trees

We show that if G is a graph such that every edge is in at least two triangles, then G contains a spanning tree with no vertex of degree 2 (a homeomorphically irreducible spanning tree). This result was originally asked in a question format by Albertson, Berman, Hutchinson, and Thomassen in 1979, and then conjectured to be true by Archdeacon in 2009.     

Degree-bounded minimum spanning trees

Given n points in the Euclidean plane, the degree-δ minimum spanning tree (MST) problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most δ. The problem is NP-hard for 2≤δ≤3, while the NP-hardness of the problem is open for δ=4. The problem is polynomial-time solvable when δ=5. By presenting an improved approximation analysis for Chan’s degree-4 MST algorithm [T. Chan, Euclidean bounded-degree spanning tree ratios, Discrete & Computational Geometry 32 (2004) 177-194], we show that, for any arbitrary collection of points in the Euclidean plane, there always exists a degree-4 spanning tree of weight at most times the weight of an MST.     

On Steiner trees and minimum spanning trees in hypergraphs

The bottleneck of the state-of-the-art algorithms for geometric Steiner problems is usually the concatenation phase, where the prevailing approach treats the generated full Steiner trees as edges of a hypergraph and uses an LP-relaxation of the minimum spanning tree in hypergraph (MSTH) problem. We study this original and some new equivalent relaxations of this problem and clarify their relations to all classical relaxations of the Steiner problem. In an experimental study, an algorithm of ours which is designed for general graphs turns out to be an efficient alternative to the MSTH approach.     

Transitions in geometric minimum spanning trees

We study some combinatorial and algorithmic problems associated with an arbitrary motion of input points in space. The motivation for such an investigation comes from two different sources:computer modeling andsensitivity analysis. In modeling, the dynamics enters the picture since geometric objects often model physical entities whose positions can change over time. In sensitivity analysis, the motion of the input points might represent uncertainties in the precise location of objects. The main results of the paper deal with state transitions in the minimum spanning tree when one or more of the input points move arbitrarily in space. In particular, questions of the following form are addressed: (i) How many different minimum spanning trees can arise if one point moves while the others remain fixed? (ii) When does the minimum spanning tree change its topology if all points are allowed to move arbitrarily?     

Rainbow spanning trees in abelian groups

Journal of Algebraic Combinatorics - A Ryser design $${mathcal {D}}$$ on v points is a collection of v proper subsets (called blocks) of a point-set with v points such that every two blocks...     

Linear verification for spanning trees

Given a rooted tree with values associated with then vertices and a setA of directed paths (queries), we describe an algorithm which finds the maximum value of every one of the given paths, and which uses only $$5n + n\log \frac{{\left| A \right| + n}}{n}$$ comparisons. This leads to a spanning tree verification algorithm usingO(n+e) comparisons in a graph withn vertices ande edges. No implementation is offered.     

On encodings of spanning trees

Deo and Micikevicius recently gave a new bijection for spanning trees of complete bipartite graphs. In this paper we devise a generalization of Deo and Micikevicius's method, which is also a modification of Olah's method for encoding the spanning trees of any complete multipartite graph K(n₁,…,n_r). We also give a bijection between the spanning trees of a planar graph and those of any of its planar duals. Finally we discuss the possibility of bijections for spanning trees of DeBriujn graphs, cubes, and regular graphs such as the Petersen graph that have integer eigenvalues.     

On generalized minimum spanning trees

The generalized minimum spanning tree problem consists of designing a minimum cost tree spanning several clusters. The purpose of this note is to pinpoint several inaccuracies contained in a previous publication and to propose a valid formulation for this problem.     

Finding thek smallest spanning trees

We give improved solutions for the problem of generating thek smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes timeO(m log(m, n)=k ²); for planar graphs this bound can be improved toO(n+k ²). We also show that thek best spanning trees for a set of points in the plane can be computed in timeO(min(k ² n+n logn,k ²+kn log(n/k))). Thek best orthogonal spanning trees in the plane can be found in timeO(n logn+kn log log(n/k)+k ²).