An adaptive heuristic to the bounded-diameter minimum spanning tree problem

Given a graph G and a bound d?≥?2, the bounded-diameter minimum spanning tree problem seeks a spanning tree on G of minimum weight subject to the constraint that its diameter does not exceed d. This problem is NP-hard; several heuristics have been proposed to find near-optimal solutions to it in reasonable times. A decentralized learning automata-based algorithm creates spanning trees that honor the diameter constraint. The algorithm rewards a tree if it has the smallest weight found so far and penalizes it otherwise. As the algorithm proceeds, the choice probability of the tree converges to one; and the algorithm halts when this probability exceeds a predefined value. Experiments confirm the superiority of the algorithm over other heuristics in terms of both speed and solution quality.     

Degree-constrained minimum spanning tree

In this paper the problem of a degree-constrained minimum spanning tree (DCMST) is defined. The problem is formulated as a linear 0–1 integer programming problem. A primal and a dual heuristic (construction) procedure and a branch-and-bound algorithm are proposed to construct a DCMST. These procedures are illustrated with a simple example. Some computational experience with these algorithms is also reported.     

Evaluating Steiner-tree heuristics and diameter variations for application layer multicast

Latency reduction in distributed interactive applications has been studied intensively. Such applications may have stringent latency requirements and dynamic user groups. We focus on application-layer multicast with a centralized approach to the group management. The groups are organized in overlay networks that are created using graph algorithms that create a tree structure for the group. A tree has no cycles and uses a small routing table, as opposed to a connected overlay mesh.We investigated a group of spanning tree problems that are referred to as Steiner-tree problems, and we have a particular focus on reducing the diameter of a tree, which is the maximum pairwise latency in a tree. In addition, we focus on reducing the time it takes to execute membership changes. In that context, we use core-selection heuristics to find well-placed client nodes, and edge-pruning algorithms to reduce the number of edges in an otherwise fully meshed overlay. Our edge-pruning algorithms strongly connect well-placed client nodes to the remaining group members, to create new and pruned group graphs. Consequently, when a tree algorithm is applied to a pruned group graph, it is manipulated into creating trees with a smaller diameter.We devised new Steiner-tree heuristics that reduced the diameter, and also proposed new edge-pruning algorithms to make the tree construction faster. These heuristics and algorithms were implemented and analyzed experimentally along with several spanning-tree and core-selection heuristics found in the literature. We found that a full-mesh of shortest paths makes it difficult for Steiner-tree heuristics to find better trees than spanning tree algorithms. The network seen from the application layer is in fact a full mesh of shortest paths. In addition, we found that faster Steiner-tree heuristics that do not explicitly optimize the diameter are able to compete with slower heuristics that do optimize it.     

Degree constrained minimum spanning tree problem: a learning automata approach

Degree-constrained minimum spanning tree problem is an NP-hard bicriteria combinatorial optimization problem seeking for the minimum weight spanning tree subject to an additional degree constraint on graph vertices. Due to the NP-hardness of the problem, heuristics are more promising approaches to find a near optimal solution in a reasonable time. This paper proposes a decentralized learning automata-based heuristic called LACT for approximating the DCMST problem. LACT is an iterative algorithm, and at each iteration a degree-constrained spanning tree is randomly constructed. Each vertex selects one of its incident edges and rewards it if its weight is not greater than the minimum weight seen so far and penalizes it otherwise. Therefore, the vertices learn how to locally connect them to the degree-constrained spanning tree through the minimum weight edge subject to the degree constraint. Based on the martingale theorem, the convergence of the proposed algorithm to the optimal solution is proved. Several simulation experiments are performed to examine the performance of the proposed algorithm on well-known Euclidean and non-Euclidean hard-to-solve problem instances. The obtained results are compared with those of best-known algorithms in terms of the solution quality and running time. From the results, it is observed that the proposed algorithm significantly outperforms the existing method.     

A multiaccess tree algorithm with free access, interference cancellation and single signal memory requirements

Tree algorithms are a well studied class of collision resolution algorithms for solving multiple access control problems. Successive interference cancellation, which allows one to recover additional information from otherwise lost collision signals, has recently been combined with tree algorithms with blocked access [Y. Yu, G.B. Giannakis, SICTA: A 0.693 contention tree algorithm using successive interference cancellation, in: INFOCOM 2005. 24th Annual Joint Conference of the IEEE Computer and Communications Societies, Miami, USA, 2005, pp. 1908–1916], providing a substantially higher maximum stable throughput (MST): 0.693 for Poisson arrivals, given an infinite number of memory locations for storing signals. We propose a novel tree algorithm for a similar problem, but with two relaxed model assumptions: free access is supported and a single signal memory location suffices. A study of the maximal stable throughput of this algorithm is provided using matrix analytical methods; as a result, an MST of 0.5698 for Poisson arrivals is achieved. Our methodology also allows us to investigate the MST when the multiple access channel is subject to Markovian arrival processes.     

Energy-aware topology control for wireless sensor networks using memetic algorithms

Cost-effective topology control is critical in wireless sensor networks. While much research has been carried out in this aspect using various methods, no attention has been made on utilizing modern heuristics for this purpose. This paper proposes a memetic algorithm-based solution for energy-aware topology control for wireless sensor networks. This algorithm (called ToCMA), using a combination of problem-specific light-weighted local search and genetic algorithms, is able to solve the minimum energy network connectivity (MENC) this NP-hard problem in an approximated manner that performs better than the classical minimum spanning tree (MST) solution. The outcomes of ToCMA can also be utilized for various network optimization and fault-tolerant purposes.     

Concatenation-Based Greedy Heuristics for the Euclidean Steiner Tree Problem

We present a class of O(n log n) heuristics for the Steiner tree problem in the Euclidean plane. These heuristics identify a small number of subsets with few, geometrically close, terminals using minimum spanning trees and other well-known structures from computational geometry: Delaunay triangulations, Gabriel graphs, relative neighborhood graphs, and higher-order Voronoi diagrams. Full Steiner trees of all these subsets are sorted according to some appropriately chosen measure of quality. A tree spanning all terminals is constructed using greedy concatenation. New heuristics are compared with each other and with heuristics from the literature by performing extensive computational experiments on both randomly generated and library problem instances. Received October 27, 1997; revised May 7, 1998.     

Distributed MST for constant diameter graphs

This paper considers the problem of distributively constructing a minimum-weight spanning tree (MST) for graphs of constant diameter in the bounded-messages model, where each message can contain at most B bits for some parameter B. It is shown that the number of communication rounds necessary to compute an MST for graphs of diameter 4 or 3 can be as high as and , respectively. The asymptotic lower bounds hold for randomized algorithms as well. On the other hand, we observe that O(log n) communication rounds always suffice to compute an MST deterministically for graphs with diameter 2, when B = O(log n). These results complement a previously known lower bound of for graphs of diameter Ω(log n). An extended abstract of this work appears in Proceedings of 20th ACM Symposium on Principles of Distributed Computing, August 2001.     

Minimum-weight spanning tree algorithms A survey and empirical study

The minimum-weight spanning tree problem is one of the most typical and well-known problems of combinatorial optimisation. Efficient solution techniques had been known for many years. However, in the last two decades asymptotically faster algorithms have been invented. Each new algorithm brought the time bound one step closer to linearity and finally Karger, Klein and Tarjan proposed the only known expected linear-time method. Modern algorithms make use of more advanced data structures and appear to be more complicated to implement. Most authors and practitioners refer to these but still use the classical ones, which are considerably simpler but asymptotically slower. The paper first presents a survey of the classical methods and the more recent algorithmic developments. Modern algorithms are then compared with the classical ones and their relative performance is evaluated through extensive empirical tests, using reasonably large-size problem instances. Randomly generated problem instances used in the tests range from small networks having 512 nodes and 1024 edges to quite large ones with 16 384 nodes and 524 288 edges. The purpose of the comparative study is to investigate the conjecture that modern algorithms are also easy to apply and have constants of proportionality small enough to make them competitive in practice with the older ones.Scope and purposeThe minimum-weight spanning tree (MST) problem is a well-known combinatorial optimisation problem concerned with finding a spanning tree of an undirected, connected graph, such that the sum of the weights of the selected edges is minimum. The importance of this problem derives from its direct applications in the design of computer, communication, transportation, power and piping networks; from its appearance as part of solution methods to other problems to which it applies less directly such as network reliability, clustering and classification problems and from its occurrence as a subproblem in the solution of other problems like the travelling salesman problem, the multi-terminal flow problem, the matching problem and the capacitated MST problem. Although efficient solution techniques capable of solving large instances had existed, there has been sustained effort over the last two decades to invent asymptotically faster algorithms. With each new algorithm found the time bound approached linearity. Finally, an expected linear-time method has been proposed. The purpose of this work is to survey the classical and modern solution techniques and empirically compare the performance of the existing methods.     

Degree-constrained minimum spanning tree problem with uncertain edge weights

Uncertainty theory has shown great advantages in solving many nondeterministic problems, one of which is the degree-constrained minimum spanning tree (DCMST) problem in uncertain networks. Based on different criteria for ranking uncertain variables, three types of DCMST models are proposed here: uncertain expected value DCMST model, uncertain α-DCMST model and uncertain most chance DCMST model. In this paper, we give their uncertainty distributions and fully characterize uncertain expected value DCMST and uncertain α-DCMST in uncertain networks. We also discover an equivalence relation between the uncertain α-DCMST of an uncertain network and the DCMST of the corresponding deterministic network. Finally, a related genetic algorithm is proposed here to solve the three models, and some numerical examples are provided to illustrate its effectiveness.     

赋权有向图的最小生成树算法

针对赋权有向图最小生成树问题存在可行解的情况,根据树节点入度最大值为1的性质,提出赋权有向图最小生成树性质。采用反证法,调整生成树根节点到弧头的路径来证明赋权有向图MST性质的正确性。基于赋权有向图MST性质,给出改进的Prim和Kruskal算法及其时间复杂度分析。实验给出构造某赋权有向图实例最小生成树的具体步骤,表明这2种算法能正确有效地构造赋权有向图最小生成树。     

Randomized local search,evolutionary algorithms,and the minimum spanning tree problem

Randomized search heuristics, among them randomized local search and evolutionary algorithms, are applied to problems whose structure is not well understood, as well as to problems in combinatorial optimization. The analysis of these randomized search heuristics has been started for some well-known problems, and this approach is followed here for the minimum spanning tree problem. After motivating this line of research, it is shown that randomized search heuristics find minimum spanning trees in expected polynomial time without employing the global technique of greedy algorithms.     

A swarm intelligence approach to the quadratic minimum spanning tree problem

The quadratic minimum spanning tree problem (Q-MST) is an extension of the minimum spanning tree problem (MST). In Q-MST, in addition to edge costs, costs are also associated with ordered pairs of distinct edges and one has to find a spanning tree that minimizes the sumtotal of the costs of individual edges present in the spanning tree and the costs of the ordered pairs containing only edges present in the spanning tree. Though MST can be solved in polynomial time, Q-MST is NP-Hard. In this paper we present an artificial bee colony (ABC) algorithm to solve Q-MST. The ABC algorithm is a new swarm intelligence approach inspired by intelligent foraging behavior of honey bees. Computational results show the effectiveness of our approach.     

交通选线优化算法的设计与实现

将交通选线问题求解转化为最小生成树（Minimun Spanning Tree,MST）的求解,对比了经典MST求解算法,以图论为基础,采取一种求最小生成树的改进遗传算法．该算法以二进制编码表示最小树问题,用深度优先搜索算法进行图的连通性判断,并采用相应的适应度函数、单亲换位算子和单亲逆转算子及多种控制进化策略,能在一次遗传进化过程中获得一批最小生成树,可供决策部门综合评价与决策．     

Finding the k most vital edges with respect to minimum spanning tree

For a connected, undirected and weighted graph G = (V,E), the problem of finding the k most vital edges of G with respect to minimum spanning tree is to find k edges in G whose removal will cause greatest weight increase in the minimum spanning tree of the remaining graph. This problem is known to be NP-hard for arbitraryk. In this paper, we first describe a simple exact algorithm for this problem, based on t he approach of edge replacement in the minimum spanning tree of G. Next we present polynomial-time randomized algorithms that produce optimal and approximate solutions to this problem. For and , our algorithm producing optimal solution has a time complexity of O(mn) with probability of success at least , which is 0.90 for and asymptotically 1 when k goes to infinity. The algorithm producing approximate solution runs in time with probability of success at least , which is 0.998 for , and produces solution within factor 2 to the optimal one. Finally we show that both of our randomized algorithms can be easily parallelized. On a CREW PRAM, the first algorithm runs in O(n) time using processors, and the second algorithm runs in time using mn/logn processors and hence is RNC. Received 30 October 1995 / 5 November 1998     

Reconstructing a Minimum Spanning Tree after Deletion of Any Node

Abstract. Updating a minimum spanning tree (MST) is a basic problem for communication networks. In this paper we consider single node deletions in MSTs. Let G=(V,E) be an undirected graph with n nodes and m edges, and let T be the MST of G . For each node v in V , the node replacement for v is the minimum weight set of edges R(v) that connect the components of T-v . We present a sequential algorithm and a parallel algorithm that find R(v) for all V simultaneously. The sequential algorithm takes O(m log n) time, but only O(m α (m,n)) time when the edges of E are presorted by weight. The parallel algorithm takes O(log ² n) time using m processors on a CREW PRAM.     

A Two-Player Coalition Cooperative Scheme for the Bodyguard Allocation Problem

We address the bodyguard allocation problem (BAP), an optimization problem that illustrates the conflict of interest between two classes of processes with contradictory preferences within a distributed system. While a class of processes prefers to minimize its distance to a particular process called the root, the other class prefers to maximize it; at the same time, all the processes seek to build a communication spanning tree with the maximum social welfare. The two state-of-the-art algorithms for this problem always guarantee the generation of a spanning tree that satisfies a condition of Nash equilibrium in the system; however, such a tree does not necessarily produce the maximum social welfare. In this paper, we propose a two-player coalition cooperative scheme for BAP, which allows some processes to perturb or break a Nash equilibrium to find another one with a better social welfare. By using this cooperative scheme, we propose a new algorithm called FFC-BAP_S for BAP. We present both theoretical and empirical analyses which show that this algorithm produces better quality approximate solutions than former algorithms for BAP.     

An O(log n) algorithm for parallel update of minimum spanning trees

Parallel algorithms are presented for updating a minimum spanning tree when the cost of an edge changes or when a new node is inserted in the underlying graph. Our model of computation is a parallel random access machine which allows simultaneous reads but prohibits simultaneous writes into the same memory location. The algorithms described in this paper for updating a minimum spanning tree require O(log n) time and O(n²) processors. These algorithms are efficient when compared to previously known algorithms for initial construction of a minimum spanning tree that require O(log²n) time and use O(n²) processors. The two main contributions of this paper are: (i) usage of an inverted tree for updating minimum spanning trees, and (ii) discovery of an interesting property of minimum spanning trees that we exploit to develop a novel algorithm for vertex insertion update.     

遗传算法在求解最小生成树中的运用

以图论和遗传算法为基础,提出了求解最小生成树问题的遗传算法。该算法解决了常用二进制编码不能正确表达最小生成树的问题,利用Prufer数对生成树进行编码;在遗传操作中对变异算子进行了改进,避免了由于变异产生大量不可行解。从而提高了遗传算法的效率;通过数值试验,表明该算法简单,高效,收敛率高。     

Maintaining Spanning Trees of Small Diameter

Given a graph G with m edges and n nodes, a spanning tree T of G , and an edge e that is being deleted from or inserted into G , we give efficient O(n) algorithms to compute a possible swap for e that minimizes the diameter of the new spanning tree. This problem arises in high-speed networks, particularly in optical networks. Received January 1995; revised February 1997.