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Showing papers on "Prim's algorithm published in 2008"


Journal ArticleDOI
TL;DR: A new method of solving graph problems related to Vertex Cover by enumerating and expanding appropriate sets of nodes and faster algorithms for solving subclasses of the Steiner tree problem on graphs are investigated.
Abstract: We present a new method of solving graph problems related to Vertex Cover by enumerating and expanding appropriate sets of nodes. As an application, we obtain dramatically improved runtime bounds for two variants of the Vertex Cover problem. In the case of Connected Vertex Cover, we take the upper bound from O *(6 k ) to O *(2.7606 k ) without large hidden factors. For Tree Cover, we show a complexity of O *(3.2361 k ), improving over the previous bound of O *((2k) k ). In the process, faster algorithms for solving subclasses of the Steiner tree problem on graphs are investigated.

74 citations


Proceedings ArticleDOI
Mikkel Thorup1
17 May 2008
TL;DR: A simple and fast deterministic algorithm for the minimum k-way cut problem in a capacitated graph, that is, finding a set of edges with minimum total capacity whose removal splits the graph into at least k components, which essentially match the O(n(2-o(1))k) running time of the Monto Carlo (no correctness guarantee) randomized algorithm of Karger and Stein.
Abstract: We present a simple and fast deterministic algorithm for the minimum k-way cut problem in a capacitated graph, that is, finding a set of edges with minimum total capacity whose removal splits the graph into at least k components. The algorithm packs O(mk3 log n) trees. Each new tree is a minimal spanning tree with respect to the edge utilizations, and the utilization of an edge is the number of times it has been used in previous spanning trees divided by its capacity. We prove that each minimum k-way cut is crossed at most 2k-2 times by one of the trees. We can enumerate all such cuts in ~O(n2k) time, which is hence the running time of our algorithm producing all minimum k-way cuts. The previous fastest deterministic algorithm of Kamidoi et al. [SICOMP'06] took O(n(4+o(1))k) time, so this is a near-quadratic improvement. Moreover, we essentially match the O(n(2-o(1))k) running time of the Monto Carlo (no correctness guarantee) randomized algorithm of Karger and Stein [JACM'96].

68 citations


Proceedings ArticleDOI
20 Feb 2008
TL;DR: A general relation between the edge uncertainty and the vertex uncertainty versions of a problem is given and used to derive a 4-update competitive algorithm for the minimum spanning tree problem in the vertices uncertainty model, which is the best possible among deterministic algorithms.
Abstract: We consider the minimum spanning tree problem in a setting where informa- tion about the edge weights of the given graph is uncertain. Initially, for each edge e of the graph only a set Ae, called an uncertainty area, that contains the actual edge weight we is known. The algorithm can 'update' e to obtain the edge weight we 2 Ae. The task is to output the edge set of a minimum spanning tree after a minimum number of updates. An algorithm is k-update competitive if it makes at most k times as many updates as the optimum. We present a 2-update competitive algorithm if all areas Ae are open or trivial, which is the best possible among deterministic algorithms. The condition on the areas Ae is to exclude degenerate inputs for which no constant update competitive algorithm can exist. Next, we consider a setting where the vertices of the graph correspond to points in Euclidean space and the weight of an edge is equal to the distance of its endpoints. The location of each point is initially given as an uncertainty area, and an update reveals the exact location of the point. We give a general relation between the edge uncertainty and the vertex uncertainty versions of a problem and use it to derive a 4-update competitive algorithm for the minimum spanning tree problem in the vertex uncertainty model. Again, we show that this is best possible among deterministic algorithms.

48 citations


Journal ArticleDOI
TL;DR: This work proposes a new approximation Minimum Routing Cost Tree algorithm that has time complexity lower than the fastest known approximation algorithm and provides a spanning tree with the same routing cost in practice.

40 citations


Journal ArticleDOI
TL;DR: Some general methods for reducing exponentially the consumption of random bits in comparison-based algorithms are developed, and using this algorithm as a guide, a unified view of the existing “nongreedy” minimum spanning tree algorithms is presented.
Abstract: For many fundamental problems there exist randomized algorithms that are asymptotically optimal and are superior to the best-known deterministic algorithm. Among these are the minimum spanning tree (MST) problem, the MST sensitivity analysis problem, the parallel connected components and parallel minimum spanning tree problems, and the local sorting and set maxima problems. (For the first two problems there are provably optimal deterministic algorithms with unknown, and possibly superlinear, running times.) One downside of the randomized methods for solving these problems is that they use a number of random bits linear in the size of input. In this article we develop some general methods for reducing exponentially the consumption of random bits in comparison-based algorithms. In some cases we are able to reduce the number of random bits from linear to nearly constant, without affecting the expected running time.Most of our results are obtained by adjusting or reorganizing existing randomized algorithms to work well with a pairwise or O(1)-wise independent sampler. The prominent exception, and the main focus of this article, is a linear-time randomized minimum spanning tree algorithm that is not derived from the well-known Karger-Klein-Tarjan algorithm. In many ways it resembles more closely the deterministic minimum spanning tree algorithms based on soft heaps. Further, using our algorithm as a guide, we present a unified view of the existing “nongreedy” minimum spanning tree algorithms. Concepts from the Karger-Klein-Tarjan algorithm, such as F-lightness, MST verification, and sampled graphs, are related to the concepts of edge corruption, subgraph contractibility, and soft heaps, which are the basis of the deterministic MST algorithms of Chazelle and Pettie-Ramachandran.

26 citations


Journal ArticleDOI
TL;DR: This paper calculates exact formulas for the tree congestion and spanning tree congestion for various families of graphs, including grids and complete bipartite graphs.

25 citations


Journal ArticleDOI
TL;DR: This paper introduces the prize-collecting generalized minimum spanning tree problem, and describes several heuristic strategies, including local search and a genetic algorithm, as well as presenting a simple and computationally efficient branch-and-cut algorithm.
Abstract: We introduce the prize-collecting generalized minimum spanning tree problem. In this problem a network of node clusters needs to be connected via a tree architecture using exactly one node per cluster. Nodes in each cluster compete by offering a payment for selection. This problem is NP-hard, and we describe several heuristic strategies, including local search and a genetic algorithm. Further, we present a simple and computationally efficient branch-and-cut algorithm. Our computational study indicates that our branch-and-cut algorithm finds optimal solutions for networks with up to 200 nodes within two hours of CPU time, while the heuristic search procedures rapidly find near-optimal solutions for all of the test instances.

19 citations


Proceedings ArticleDOI
13 Apr 2008
TL;DR: This paper proposes a novel algorithm, which generates sparse obstacle-avoiding spanning graphs efficiently, and designs a fast algorithm for the minimum terminal spanning tree construction, which is the bottleneck step of several existing approaches in terms of running time.
Abstract: Obstacle-avoiding Steiner tree construction is a fundamental problem in VLSI physical design. In this paper, we provide a new approach for rectilinear Steiner tree construction in the presence of obstacles. We propose a novel algorithm, which generates sparse obstacle-avoiding spanning graphs efficiently. We design a fast algorithm for the minimum terminal spanning tree construction, which is the bottleneck step of several existing approaches in terms of running time. We adopt an edge-based heuristic, which enables us to perform both local and global refinement, leading to Steiner trees with small lengths. The time complexity of our algorithm is O(nlogn). Hence, our technique is the most efficient one to the best of our knowledge. Experimental results on various benchmarks show that our algorithm achieves 25.8 times speedup on average, while the average length of the resulting obstacle-avoiding rectilinear Steiner trees is only 1.58% larger than the best existing solution

17 citations


Proceedings ArticleDOI
18 Nov 2008
TL;DR: In the EPACOR, when a node needs to deliver data to the sink, ant colony systems are used to establish the route with optimal or sub-optimal power consumption, and meanwhile, learning mechanism is embedded to predict the energy consumption of neighboring nodes when the node chooses a neighboring node added to the route.
Abstract: Routing algorithms play important roles in wireless sensor networks (WSNs). Usually, nodes in a WSN run on battery with limited power. Hence, routing with efficient power consumption is becoming a critical issue for WSNs. In this paper, a routing algorithm, referred to as Energy Prediction and Ant Colony Optimization Routing (EPACOR), is proposed. In the EPACOR, when a node needs to deliver data to the sink, ant colony systems are used to establish the route with optimal or sub-optimal power consumption, and meanwhile, learning mechanism is embedded to predict the energy consumption of neighboring nodes when the node chooses a neighboring node added to the route. The EPACOR is compared both with the MST (Minimal Spanning Tree)-based routing algorithm following the Prim algorithm and with the Least Energy Tree (LET)-based routing algorithm following the Dijkstra algorithm. Numeric experiment shows that the EPACOR has the best network lifetime among the three while keeping energy consumption in low level.

16 citations


Proceedings ArticleDOI
06 Apr 2008
TL;DR: A partheno-genetic algorithm based on tree structure encoding method for distribution network optimal planning is proposed, and two new genetic operators such as shift operator and redistribution operator are proposed to perfect the tree structure encode partheniogenesis algorithm.
Abstract: A partheno-genetic algorithm based on tree structure encoding method for distribution network optimal planning is proposed, and two new genetic operators such as shift operator and redistribution operator for tree structure encoding partheno-genetic algorithm are proposed in this paper, after the reason that the convergence of bintree structure encoding genetic algorithm can be generalized to tree structure encoding genetic algorithm has been given. Prim algorithm is employed to create initial radial networks, which are better than entirely random initial schemes. The advantages of partheno-genetic algorithm are fully utilized in distribution network optimal planning. All schemes in the solving process are always naturally being radial mode, no need for inspection of connectivity and being radial in connection of the networks. The determination of conducting wire section, the treatment of street bifurcate nodes and the method of expansion planning for distribution networks are discussed. The examples of distribution network planning show that the method is feasible and efficient. Finally, in the conclusion part, mutation operator has been proposed to perfect the tree structure encoding partheno-genetic algorithm.

13 citations


Book ChapterDOI
15 Dec 2008
TL;DR: A new Particle Swarm Optimization algorithm for solving the d -MST problem using some new methods for selecting vector of particles and results are reported to show the efficiency of the algorithm.
Abstract: Given a connected, weighted, undirected graph G=(V, E) and a bound d . The Degree-Constrained Minimum Spanning Tree problem (DCMST or d -MST) seeks the spanning tree with smallest weight in which no vertex have degree more than d . This problem is NP-hard with d ***2. This paper proposes a new Particle Swarm Optimization algorithm for solving the d -MST problem. The proposed algorithm uses some new methods for selecting vector of particles. Results of computational experiments are reported to show the efficiency of the algorithm.

Journal ArticleDOI
TL;DR: Two new metaheuristic approaches for the leaf-constrained minimum spanning tree problem are proposed, one is an ant-colony optimization (ACO) algorithm, whereas the other is a tabu search based algorithm.
Abstract: Given an undirected, connected, weighted graph, the leaf-constrained minimum spanning tree (LCMST) problem seeks a spanning tree of the graph with smallest weight among all spanning trees of the graph, which contains at least l leaves. In this paper we have proposed two new metaheuristic approaches for the LCMST problem. One is an ant-colony optimization (ACO) algorithm, whereas the other is a tabu search based algorithm. Similar to a previously proposed genetic algorithm, these metaheuristic approaches also use the subset coding that represents a leaf-constrained spanning tree by the set of its interior vertices. Our new approaches perform well in comparison with two best heuristics reported in the literature for the problem — the subset-coded genetic algorithm and a greedy heuristic.

Book ChapterDOI
20 Aug 2008
TL;DR: This paper describes the reasons about why it is beneficial to combine with graph theory and board game and descants three graph theories: Dijkstra's, Prim's, and Kruskal's minimum spanning tree.
Abstract: This paper describes the reasons about why it is beneficial to combine with graph theory and board game. Forbye, it also descants three graph theories: Dijkstra's, Prim's, and Kruskal's minimum spanning tree. Then it would describe the information about the board game we choose and how to combine the game with before-mentioned three graph theories. At last, we would account for the advantage of combining with these three graph theories and the game specifically.

Journal ArticleDOI
TL;DR: A polynomial time algorithm is proposed for the minimum degree spanning tree problem in directed acyclic graphs that starts with an arbitrary spanning tree, and iteratively reduces the number of vertices of maximum degree to result in an optimal tree.

Proceedings ArticleDOI
17 Dec 2008
TL;DR: A perturbation based local search for the minimum routing cost spanning tree problem is proposed and compared against three methods reported in the literature - two genetic algorithms and a stochastic hill climber.
Abstract: Given a connected, weighted, and undirected graph, the minimum routing cost spanning tree problem seeks on this graph a spanning tree of minimum routing cost, where routing cost is defined as the sum of the costs of all the paths connecting two distinct vertices in a spanning tree. In this paper we have proposed a perturbation based local search for this problem. We have compared our approach against three methods reported in the literature - two genetic algorithms and a stochastic hill climber.Computational results show the effectiveness of our approach.

11 Dec 2008
TL;DR: In this paper, a new tabu-search-based approximate solution method that does not iteratively solve minimum spanning tree problems is proposed, which provides a good performance in terms of accuracy over those of existing methods for relatively high cardinality k.
Abstract: This paper considers k-minimum spanning tree problems. An existing solution algorithm based on tabu search, which was proposed by Katagiri et al., includes an iterative solving procedure of minimum spanning tree (MST) problems for subgraphs to obtain a local optimal solution of k-minimum spanning tree problems. This article provides a new tabu-searchbased approximate solution method that does not iteratively solve minimum spanning tree problems. Results of numerical experiments show that the proposed method provides a good performance in terms of accuracy over those of existing methods for relatively high cardinality k.

01 Jan 2008
TL;DR: This paper proposes a new approach using Genetic Algorithm that reduces the computational complexity by selecting the best spanning tree in a stochastic manner and is very time consuming, especially when the Metro Ethernet network is large.
Abstract: †Summary IEEE Spanning Tree Protocol (STP) is a layer-2 protocol which ensures a loop free topology in Metro Ethernet networks. It is based on Minimum Spanning Tree solution that involves determining the links which can join all the nodes of a network together such that the sum of the costs of the chosen links is minimized. In STP, all customers need to use the same spanning tree and there isn’t any traffic engineering mechanism for load balancing. This results in uneven load distribution and bottlenecks, especially close to the root. A solution for this problem is using the multi-criteria Minimum Spanning Tree by considering criterions such as load balance over links and switches. In our previous work, the algorithm was based on computation of the total cost for each possible spanning tree and then selection the best one with minimum total cost. This algorithm is very time consuming, especially when our Metro Ethernet network is large. In this paper, we propose a new approach using Genetic Algorithm. It reduces the computational complexity by selecting the best spanning tree in a stochastic manner.

Proceedings ArticleDOI
T. Makabe1, T. Takenaka1
17 Nov 2008
TL;DR: The results of simulation experiments are given which demonstrate the superiority of the proposed composite weighted Steiner tree algorithm compared with the distributed minimum hop tree (DMHT) algorithm, from the viewpoint of the light-tree request blocking probability.
Abstract: We propose novel tree construction algorithms for multicast communication in photonic networks. Since multicast communications consume many more link resources than unicast communications, effective algorithms for route selection and wavelength allocation are required. We propose novel tree construction algorithms called the weighted Steiner tree (WST) algorithm and the composite weighted Steiner tree (CWST) algorithm. Since the proposed Weighted Steiner Tree algorithm takes account of the number of wavelengths available for each link, this algorithm can select those links which have larger number of available wavelengths. By using this link selection technique, this algorithm can accept many more multicast requests than other multicast tree construction algorithms. However, under certain delay constraints, the blocking characteristics of the proposed weighted Steiner tree algorithm deteriorate since some light paths between source and destination use long hops and cannot satisfy the delay constraint. In order to adapt the approach to the delay-sensitive environments, we have devised the composite weighted Steiner tree algorithm comprising the weighted Steiner tree algorithm and the Dijkstra algorithm for use in a delay constrained environment such as an IPTV application. In this paper, we also give the results of simulation experiments which demonstrate the superiority of the proposed composite weighted Steiner tree algorithm compared with the distributed minimum hop tree (DMHT) algorithm, from the viewpoint of the light-tree request blocking probability.

Proceedings ArticleDOI
22 Apr 2008
TL;DR: A new chain topology is proposed and solved minimum spanning chain (MSC) by linear programming (LP) and results of simulation analysis confirmed that the proposed chain solution is very useful routing topology in USNs.
Abstract: Many routing topologies have been proposed to extend lifetime of ubiquitous sensor network (USN) consisting of sensor nodes with limited energy. If measurement data is collected periodically for every nodes and perfect fusion during data aggregation, minimum spanning tree (MST) and chains have been two major topologies. MST minimizes total transmission energy and chain consumes energy very evenly throughout the nodes. Various publications such as LEACH, PEGASIS and PEDAP announced many good USN topologies, there still is little study about the topology itself especially in terms of energy consumption excluding USN system-related factors. This paper includes the following study. We proposed a new chain topology and solved minimum spanning chain (MSC) by linear programming (LP). Then we calculated and analysed topology length and standard deviation/coefficient of variation of link lengths through simulation. As results of simulation analysis we confirmed that the proposed chain solution is very useful routing topology in USNs.

Journal ArticleDOI
TL;DR: Novel mathematical properties for DCMST are indicated which lead to a new reduction algorithm that can significantly reduce the size of the problem.
Abstract: The degree-constrained minimum spanning tree (DCMST) is an NP-hard problem in graph theory. It consists of finding a spanning tree whose vertices should not exceed some given maximum degrees and whose total edge length is minimal. In this paper, novel mathematical properties for DCMST are indicated which lead to a new reduction algorithm that can significantly reduce the size of the problem. Also an algorithm for DCMST to solve the smaller problem is presented which has been preprocessed by reduction algorithm.

Posted Content
TL;DR: In this paper, a new topological vertex nder is presented which combines ideas of the well-established ZVTOP algorithm with a novel minimum spanning tree approach, and a preliminary performance study with simulated e + e! q q q events at a centre of mass energy of p s = 91:2 GeV shows that the new approach is competitive with existing vertex nDer algorithms.
Abstract: A new topological vertex nder is presented which combines ideas of the well-established ZVTOP algorithm with a novel minimum spanning tree approach A preliminary performance study with simulated e + e ! q q events at a centre of mass energy of p s = 91:2 GeV shows that the new approach is competitive with existing vertex nder algorithms

Patent
06 Aug 2008
TL;DR: In this article, a multicast routing method based on a distance complete ant colony algorithm is proposed, in which the ants accomplish the constitution of a solution, the updating of the entire information elements is implemented, the information elements on the side of the most preferable tree in history are reinforced.
Abstract: The present invention discloses a multicast routing method based on a distance complete ant colony algorithm. Firstly, a distance complete graph is established for a multicast routing network; then multicast trees are constructed randomly based on the ant colony algorithm and a Prim algorithm, wherein, the setting of heuristic information makes the algorithm to be more disposed to select target nodes. A redundant detection and a correction are carried out on the generated multicast trees and the updating of local information elements is implemented. Lastly, after all of the ants accomplish the constitution of a solution, the updating of the entire information elements is implemented, the information elements on the side of the most preferable tree in history are reinforced. By simulating the test result and the comparison with the algorithms of the same type, it shows that the distance complete ant colony algorithm of the present invention is capable of solving the problem of multicast routing more quickly.

Proceedings Article
23 Jul 2008
TL;DR: An algorithm for finding minimum flow in networks with arcs capacities using the dynamic tree implementation and an example for this algorithm is presented.
Abstract: We present an algorithm for finding minimum flow in networks with arcs capacities using the dynamic tree implementation As a final part, we present an example for this algorithm

Proceedings ArticleDOI
18 Jun 2008
TL;DR: A hybrid intelligent algorithm combined genetic algorithm and stochastic simulation is given, and the Prufer encoding schemes represented spanning trees are adopted in order to solve the problem of minimum spanning trees in uncertain networks.
Abstract: This paper considers the problem of minimum spanning trees in uncertain networks in which the edge weights are random variables. We propose the concept of expected minimum spanning tree and formulate the model according to expected value. In order to solve the model, a hybrid intelligent algorithm combined genetic algorithm and stochastic simulation is given, and the Prufer encoding schemes represented spanning trees are adopted. The algorithm has been proved to be useful for solving practical problems by a numerical example.

Journal ArticleDOI
TL;DR: Tight bounds on the minimum cost of spanning trees are obtained in the special case where the barriers are interior disjoint line segments that form a convex subdivision of the plane and there is a point in every cell of the subdivision.
Abstract: Given m points (sites) and n obstacles (barriers) in the plane, we address the problem of finding a straight line minimum cost spanning tree on the sites, where the cost is proportional to the number of intersections (crossings) between tree edges and barriers. If the barriers are infinite lines, it is known that there is a spanning tree such that every barrier is crossed by $O(\sqrt{m}\,)$tree edges, and this bound is asymptotically optimal. Asano et al. showed that if the barriers are pairwise disjoint line segments, then there is a spanning tree such that every barrier crosses at most 4 tree edges and so the total cost is at most 4n. Lower bound constructions are known with 3 crossings per barrier and 2n total cost. We obtain tight bounds on the minimum cost of spanning trees in the special case where the barriers are interior disjoint line segments that form a convex subdivision of the plane and there is a point in every cell of the subdivision. In particular, we show that there is a spanning tree such that every barrier crosses at most 2 tree edges, and there is a spanning tree of total cost 5n/3. Both bounds are the best possible.

Proceedings Article
07 Nov 2008
TL;DR: An optimal algorithm to solve the problem for tree topology graphs in polynomial time is proposed, where n is the number of nodes in a k-terminal graph, with the Goldberg-Tarjan's network flow algorithm.
Abstract: Geometrical k-cut problem has numerous applications, particularly in clustering-related setups such as task assignment and VLSI cell placement. This problem is NP-hard in general. We propose an optimal algorithm to solve the problem for tree topology graphs in polynomial time. The time complexity of the algorithm is O(kn3), where n is the number of nodes in a k-terminal graph, with the Goldberg-Tarjan's network flow algorithm.

Book ChapterDOI
01 Jan 2008
TL;DR: An algorithm of surface reconstruction is proposed to reconstruct surface of defective bone in this paper, and according to the constructed topo- logical relation, 3D geometric model of defectiveBone is recon- structed.
Abstract: 3D model reconstruction of defective bone is of great practical value in the application for medical diagnosis of bone, plastic, prosthesis surgery planning and so on. Based on Delaunay triangulation, an algorithm of surface reconstruction is proposed to reconstruct surface of defective bone in this paper. In the algorithm, data of CT 2D-images are input to build the space point dataset. The bounding box theory is applied to search k-nearest neighbors of pi, and its normal vector is computed. K-nearest neighbor points of pi are projected on the tangent plane, and its Delaunay 2D-neighbors are chosen from those projected points. Delaunay 3D-neighbors of pi are gotten by mapping Delaunay 2D-neighbors. Delaunay 3D-neighbors are searched and matched to construct triangular mesh of surface. Then, the topology of Delaunay 3Dneighbors is reconstructed to identify the feature of hole. Subsequently, a weighted undirected connected graph is generated. EMST of this graph is generated by using Prim algorithm. After searching EMST based on the depth-first principle, the normal vector is adjusted to orient the same direction of surface. Lastly, according to the constructed topological relation, 3D geometric model of defective bone is reconstructed. The validity and reliability of algorithm are verified by computing a typical example, and its time and space complexity is lower.

01 Jan 2008
TL;DR: A hierarchical minimum spanning tree algorithm is proposed which is first applied to clusters of similar images and then it is applied to the average images of the clusters, which outperforms the previous image set compression algorithms for image sets which are not very similar.
Abstract: A number of minimum spanning tree algorithms have been proposed for lossy compression of image sets. In these algorithms, a complete graph is constructed from the entire image set and possibly an average image, and a minimum spanning tree is used to determine which difference images to encode. In this paper, we propose a hierarchical minimum spanning tree algorithm in which the minimum spanning tree algorithm is first applied to clusters of similar images and then it is applied to the average images of the clusters. It is shown that the new algorithm outperforms the previous image set compression algorithms for image sets which are not very similar, especially at lower bitrates. Furthermore, the computational requirement for a hierarchical minimum spanning tree is significantly lower than the previous minimum spanning tree algorithms when the cost of clustering can be neglected.

01 Jan 2008
TL;DR: A distributed algorithm A for G is a set of local algorithms that include instructions for sending and receiving messages along the edges of the network, and its message complexity is the total number of messages sent over any execution of the algorithm, in the worst case.
Abstract: Consider a communication network, modeled by an undirected weighted graph G D .V; E/, where jV j D n; jEj D m. Each vertex of V represents a processor of unlimited computational power; the processors have unique identity numbers (ids), and they communicate via the edges of E by sending messages to each other. Also, each edge e 2 E has associated a weight w.e/, known to the processors at the endpoints of e. Thus, a processor knows which edges are incident to it and their weights, but it does not know any other information about G. The network is asynchronous: each processor runs at an arbitrary speed, which is independent of the speed of other processors. A processor may wake up spontaneously or when it receives a message from another processor. There are no failures in the network. Each message sent arrives at its destination within a finite but arbitrary delay. A distributed algorithm A for G is a set of local algorithms, one for each processor of G, that include instructions for sending and receiving messages along the edges of the network. Assuming that A terminates (i.e., all the local algorithms eventually terminate), its message complexity is the total number of messages sent over any execution of the algorithm, in the worst case. Its time complexity is the worst-case execution time, assuming processor steps take negligible time, and message delays are normalized to be at most 1 unit. A minimum spanning tree (MST) of G is a subset E 0 of E such that the graph T D .V; E 0/ is a tree (connected and acyclic) and its total weight, w.E 0/ D P e2E0 w.e/, is as small as possible.

Journal Article
TL;DR: It is suggested to solve the full bottleneck Steiner tree problem is to find a tree S from the undirected graph G, with all the vertices in the given points being leaves and the weights of the maximum edges being minimum.
Abstract: It is suggested to solve the full bottleneck Steiner tree problem is to find a tree S from the undirected graph G,with all the vertices in the given points being leaves and the weights of the maximum edges being minimumAfter giving out the definition of full bottleneck Steiner tree,we use methed of transforming,decomposing and combining to give out a polynominal algorithm for solving the problemComplexity of time in that algorithm is given,and examples are presented to show the validity of the algorithm