A note on the minimum label spanning tree

We give a tight analysis of the greedy algorithm introduced by Krumke and Wirth for the minimum label spanning tree problem. The algorithm is shown to be a (ln(n−1)+1)-approximation for any graph with n nodes (n>1), which improves the known performance guarantee 2lnn+1.     

A biased random-key genetic algorithm for the capacitated minimum spanning tree problem

This paper focuses on the capacitated minimum spanning tree (CMST) problem. Given a central processor and a set of remote terminals with specified demands for traffic that must flow between the central processor and terminals, the goal is to design a minimum cost network to carry this demand. Potential links exist between any pair of terminals and between the central processor and the terminals. Each potential link can be included in the design at a given cost. The CMST problem is to design a minimum-cost network connecting the terminals with the central processor so that the flow on any arc of the network is at most Q. A biased random-key genetic algorithm (BRKGA) is a metaheuristic for combinatorial optimization which evolves a population of random vectors that encode solutions to the combinatorial optimization problem. This paper explores several solution encodings as well as different strategies for some steps of the algorithm and finally proposes a BRKGA heuristic for the CMST problem. Computational experiments are presented showing the effectiveness of the approach: Seven new best-known solutions are presented for the set of benchmark instances used in the experiments.     

A mixed integer linear formulation for the minimum label spanning tree problem

In this paper we deal with the minimum label spanning tree problem. This is a relevant problem with applications such as telecommunication networks or electric networks, where each edge is assigned with a label (such as a color) and it is intended to determine a spanning tree with the minimum number of different labels. We introduce some mixed integer formulations for this problem and prove that one of their relaxations always gives the optimal value. Finally we present and discuss the results of computational experiments.     

Rectilinear steiner tree heuristics and minimum spanning tree algorithms using geographic nearest neighbors

We study the application of the geographic nearest neighbor approach to two problems. The first problem is the construction of an approximately minimum length rectilinear Steiner tree for a set ofn points in the plane. For this problem, we introduce a variation of a subgraph of sizeO(n) used by YaO [31] for constructing minimum spanning trees. Using this subgraph, we improve the running times of the heuristics discussed by Bern [6] fromO(n ² log n) toO(n log² n). The second problem is the construction of a rectilinear minimum spanning tree for a set ofn noncrossing line segments in the plane. We present an optimalO(n logn) algorithm for this problem. The rectilinear minimum spanning tree for a set of points can thus be computed optimally without using the Voronoi diagram. This algorithm can also be extended to obtain a rectilinear minimum spanning tree for a set of nonintersecting simple polygons.The results in this paper are a part of Y. C. Yee's Ph.D. thesis done at SUNY at Albany. He was supported in part by NSF Grants IRI-8703430 and CCR-8805782. S. S. Ravi was supported in part by NSF Grants DCI-86-03318 and CCR-89-05296.     

内部节点受限的最小生成树问题算法研究

研究内部节点受限的最小生成树问题：给定一个赋权无向完全图[G=V,E],假定[w:E→R+]为边集[E]的权重函数且满足三角不等式,给定点集[V]的一个子集[RR?V],目标是寻找图[G]的一个满足[R]中的点皆为内部顶点的权重最小的生成树。由于该问题是[NP-]困难的,提出了一个伪多项式时间最优算法,设计了一个近似比为2的多项式时间近似算法,并且给出例子以说明该近似比是紧的。     

On the red/blue spanning tree problem

A geometric spanning tree of a point set S is a tree whose vertex set is S and whose edge set is a set of non-crossing straight line segments with endpoints in S. Given a set of red points and a set of blue points in the plane, the red/blue spanning tree problem is to find a geometric spanning tree for red points and a geometric spanning tree for blue points such that the number of crossing points of the two trees is a minimum. If no three points are collinear, we show that the minimum number of crossing points is completely determined by the number of maximal red (or blue) chains on the convex hull of all red points and blue points. We design an optimal algorithm for constructing a geometric spanning tree of all the red points and a geometric spanning tree of all the blue points with the minimum number of crossing points. If collinear points are allowed, we prove that the problem of deciding whether there exists a geometric spanning path of all the red points and a geometric spanning path of all the blue points without crossing is NP-complete.     

A simpler minimum spanning tree verification algorithm

The problem considered here is that of determining whether a given spanning tree is a minimal spanning tree. In 1984 Komlós presented an algorithm which required only a linear number of comparisons, but nonlinear overhead to determine which comparisons to make. We simplify his algorithm and give a linear-time procedure for its implementation in the unit cost RAM model. The procedure uses table lookup of a few simple functions, which we precompute in time linear in the size of the tree.     

Approximating minimum cocolorings

A cocoloring of a graph G is a partition of the vertex set of G such that each set of the partition is either a clique or an independent set in G. Some special cases of the minimum cocoloring problem are of particular interest.We provide polynomial-time algorithms to approximate a minimum cocoloring on graphs, partially ordered sets and sequences. In particular, we obtain an efficient algorithm to approximate within a factor of 1.71 a minimum partition of a partially ordered set into chains and antichains, and a minimum partition of a sequence into increasing and decreasing subsequences.     

最小比率生成树的竞争决策算法

最小比率生成树是找出目标函数形式为两个线性函数比值最小的生成树,例如总代价与总收益比值最小的生成树。当不限制分母的符号时,这是一个NP-hard问题。在分析最小比率生成树数学性质的基础上,提出了最小比率生成树的竞争决策算法。为了防止算法陷入局部最优,采用edge_exchange操作来增加算法的搜索范围。为了验证算法的有效性,采用无关和相关两种策略产生测试数据,并使用Delphi 7.0实现了算法的具体步骤。     

基于最小生成树NSGA-2算法的改进

多目标进化算法（MOEA）的一个关键就是保持解的分布度,提出了一种用最小生成树的边的权值来表示个体聚集距离的方法,并且对NSGA-2的交叉算子和变异率进行了改进。实验结果表明,与NSGA-2相比该方法（MST-NSGA-2）在解的分布度上有较大的提高,并且有着良好的收敛性。     

Bounded latency spanning tree reconfiguration

One of the main obstacles to the adoption of Ethernet technology in carrier-grade metropolitan and wide-area networks is the large recovery latency, in case of failure, due to spanning tree reconfiguration. In this paper we present a technique called Bounded Latency Spanning Tree Reconfiguration (BLSTR), which guarantees worst case recovery latency in the case of single faults by adopting a time-bounded bridge port reconfiguration mechanism and by eliminating the bandwidth-consuming station discovery phase that follows reconfiguration. BLSTR does not replace the Rapid and Multiple Spanning Tree reconfiguration protocols, which remain in control of network reconfiguration, whereas it operates in parallel with them.     

The realization problem for Euclidean minimum spanning trees is NP-hard

We show that the problem of determining whether a tree can be drawn so that it is the Euclidean minimum spanning tree of the locations of its vertices is NP-hard.Partially written while this author was visiting the University of Newcastle.     

A limit characterization for the number of spanning trees of graphs

In this paper we propose a limit characterization of the behaviour of classes of graphs with respect to their number of spanning trees. Let {G_n} be a sequence of graphs G₀,G₁,G₂,… that belong to a particular class. We consider graphs of the form K_n−G_n that result from the complete graph K_n after removing a set of edges that span G_n. We study the spanning tree behaviour of the sequence {K_n−G_n} when n→∞ and the number of edges of G_n scales according to n. More specifically, we define the spanning tree indicator ({G_n}), a quantity that characterizes the spanning tree behaviour of {K_n−G_n}. We derive closed formulas for the spanning tree indicators for certain well-known classes of graphs. Finally, we demonstrate that the indicator can be used to compare the spanning tree behaviour of different classes of graphs (even when their members never happen to have the same number of edges).     

Lower bounds and exact algorithms for the quadratic minimum spanning tree problem

We address the quadratic minimum spanning tree problem (QMSTP), the problem of finding a spanning tree of a connected and undirected graph such that a quadratic cost function is minimized. We first propose an integer programming formulation based on the reformulation–linearization technique (RLT). We then use the idea of partitioning spanning trees into forests of a given fixed size and obtain a QMSTP reformulation that generalizes the RLT model. The reformulation is such that the larger the size of the forests, the stronger lower bounds provided. Thus, a hierarchy of formulations is obtained. At the lowest hierarchy level, one has precisely the RLT formulation, which is already stronger than previous formulations in the literature. The highest hierarchy level provides the convex hull of integer feasible solutions for the problem. The formulations introduced here are not compact, so the direct evaluation of their linear programming relaxation bounds is not practical. To overcome that, we introduce two lower bounding procedures based on Lagrangian relaxation. These procedures are embedded into two parallel branch-and-bound algorithms. As a result of our study, several instances in the literature were solved to optimality for the first time.     

A highly asynchronous minimum spanning tree protocol

Summary In this paper, we present an efficient distributed protocol for constructing a minimum-weight spanning tree (MST). Gallager, Humblet and Spira [5] proposed a protocol for this problem with time and message complexitiesO(N logN) andO(E+NlogN) respectively. A protocol withO(N) time complexity was proposed by Awerbuch [1]. We show that the time complexity of the protocol in [5] can also be expressed asO((D+d) logN), whereD is the maximum degree of a node andd is a diameter of the MST and therefore this protocol performs better than the protocol in [1] wheneverD+d<N/logN. We give a protocol which requiresO(min(N, (D+d)logN)) time andO(E+NlogNlogN/loglogN) messages. The protocol constructs a minimum spanning tree by growing disjoint subtrees of the MST (which are referred to asfragments). Fragments having the same minimum-weight outgoing edge are combined until a single fragment which spans the entire network remains. The protocols in [5] and [1] enforce a balanced growth of fragments. We relax the requirement of balanced growth and obtain a highly asynchronous protocol. In this protocol, fast growing fragments combine more often and there-fore speed up the execution. Gurdip Singh received the B. Tech degree in Computer Science from Indian Institute of Technology, New Delhi in 1986 and the M.S. and Ph.D. degrees in Computer Science from State University of New York at Stony Brook in 1989 and 1991 respectively. Since 1991, he has been an Assistant Professor in the Department of Computing and Information Sciences at Kansas State University. His research interest include design and verification of distributed algorithms, communication networks and distributed shared memories. Arthur Bernstein received his PhD in Electrical Engineering from Columbia University. He is currently a professor in the Computer Science Department at the State University of New York at Stony Brook. His research interests center around concurrency in programming and data-base systems.This work was supported by NSF under grants CCR8701671, CCR8901966 and CCR9211621     

基于最小生成树的图像分割

基于最小生成树思想,给出了一种利用改进的最小生成树进行图像分割的方案,减少了最小生成树的构建过程,对初分割的结果利用NNG算法进行合并。该方案节约了分割时间,并且对分割后的图像进行了有效的合并,达到了较好的分割效果。     

求解度约束最小生成树的新的快速算法

针对度约束最小生成树问题,提出了一种新的快速算法。新的快速算法分为两个主要部分,第一部分从一棵最小生成树出发,构造一棵度约束树。第二部分设计了一种改进策略,从第一部分求得的度约束树出发,每次去掉树的一条边,将顶点按照连通性划分成两个集合,在不违反度约束的情况下,从这两个集合构成的边割中,选择一条权值减少最大的边添加到图中。通过大量的数值实验表明新的快速算法性能良好。     

Parallel algorithms for minimal spanning trees of directed graphs

The main results of this paper are efficient parallel algorithms, MSP and LOCATE, for computing minimal spanning trees and locating minimal paths in directed graphs, respectively. Algorithm MSP has time complexityO(log³ n) usingO(n ³/logn) processors, while LOCATE has time complexityO(logn) usingO(n ²) processors. Algorithm MSP is derived from sequential algorithms, when the unbounded parallelism model is used.     

Solving diameter‐constrained minimum spanning tree problems by constraint programming

The diameter‐constrained minimum spanning tree problem consists in finding a minimum spanning tree of a given graph, subject to the constraint that the maximum number of edges between any two vertices in the tree is bounded from above by a given constant. This problem typically models network design applications where all vertices communicate with each other at a minimum cost, subject to a given quality requirement. We propose alternative formulations using constraint programming that circumvent weak lower bounds yielded by most mixed‐integer programming formulations. Computational results show that the proposed formulation, combined with an appropriate search procedure, solves larger instances and is faster than other approaches in the literature.