5 Conclusion
We have deﬁned a new concept, a variant of the **minimum** **spanning** **tree** problem, where each edge in the graph belongs to some contractor, and each contractor has his/her own cost function for paving a subset of the edges. We proved that the general optimization problem for this concept is NP-hard, even when restricted to cardinality cost functions, i.e. cost functions which depend only on the number of edges, and the number of contractors is relatively large Ω(log n). We gave a characterisation of when a **tree**, with a prescribed number of edges from each color class exists, and a new eﬃcient algorithm for ﬁnding such a **tree** when k = O(log n). Finally, we proved that the polytope of feasible solutions for cardinality cost functions values is integral.

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© 2011 Elsevier Inc. All rights reserved.
1. Introduction
The Euclidean **minimum** **spanning** **tree** of a set points in 2D and in 3D is among the most fundamental and hence most studied geometric structures (see, e.g. [9,17]). In their seminal paper, Monma and Suri [16] initiated the investigation of the combinatorial properties of the Euclidean **minimum** **spanning** trees in the plane. This investigation naturally leads to the following question: Which are those trees that have an EMST drawing, i.e. a straight-line drawing that is also a Euclidean **minimum** **spanning** **tree** of the set of its vertices? This question is of interest not only in geometric graph theory, but also in graph drawing.

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1. INTRODUCTION
The graph theory is effectively used to tackle may optimization problems. The classical graph theory is not well sufficient to deal with the problems where the data related to the graphs such as weight attached to the vertices, edges, the nature of connectivity (attributes) between the vertices are imprecise due to incomplete information .The fuzzy set theory has been used to deal with these type of problems where fuzzy weights are attached to the weighted graphs and different ranking methods are used to rank these fuzzy numbers to deal with the impreciseness. Fuzzy set theory is also ineffective if proper information regarding the nature of connectivity between the vertices is lacking. T.He et al [1],[2] have developed rough graph where they have applied rough set theory to attributed graph. Liang et al [3] has proposed edge rough graph to overcome some deficiencies of rough graph as proposed by T. He. S.P.Mohanty et al [4] have used the concept of T. He to find the **minimum** **spanning** **tree** (MST) of a rough graph where fuzzy numbers are attached as weights to the edges and the idea was explained through one numerical example to find the possibly conflicting MST.

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We recall that among the various methods for the solution of the unconstrained problem an efficient way to find the **minimum** **spanning** **tree** is based on the simple procedure of choosing one after the other an edge of **minimum** weight that has not be chosen yet and does not create cycles if added to the previously chosen edges. This technique is known as the “greedy algorithm”. There are problems for which the greedy algorithm works and problems for which it does not.

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combining pixel values in a considered window according to a given kernel. Others, like the median, are not linear and rely on ordering data and choosing a representative value.
There are also more complex filters, like the bilateral, which combines the concept of physical and colour distance with the gaussian filtering, to obtain and edge-preserving trans- formation. From this idea, Stawiaski and Meyer [42], developed their adaptive image filtering, based on **minimum** **spanning** trees. The algorithm considers a small window and the neigh- bourhood of pixels in it: a graph G = (V, E, W ) is built with these pixels considering as a weight between i and j the absolute difference between their grey levels. The graph obtained in this way is then used to create an MST that follows the shape of the image because similar pixels, which usually lies on the same side of an edge, have similar grey values and the MST algorithm will connect them. All the trees obtained with the window in different positions of the image, are then fused into a single one. The final image will transform the pixel values along the **minimum** **spanning** **tree** to delete most of the noise. As a result, the final signal to noise ratio of the image is almost doubled with respect to a traditional filtering technique.

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Email: seiji@nda.ac.jp (S. Kataoka), yamada.144b@gmail.com (T.Yamada)
Abstract
We formulate the **minimum** **spanning** **tree** problem with resource allocation (MSTRA) in two ways, as discrete and continuous optimization problems (d-MSTRA/c-MSTRA), prove these to be N P-hard, and present algorithms to solve these problems to optimality. We reformulate d-MSTRA as the knapsack constrained **minimum** **spanning** **tree** problem, and solve this prob- lem using a previously published branch-and-bound algorithm. By applying a ‘peg test’, the size of d-MSTRA is (significantly) reduced. To solve c- MSTRA, we introduce the concept of f -f ractional solution, and prove that an optimal solution can be found within this class of solutions. Based on this fact, as well as conditions for ‘pruning’ subproblems, we develop an enu- merative algorithm to solve c-MSTRA to optimality. We implement these algorithms in ANSI C programming language and, through extensive numer- ical tests, evaluate the performance of the developed codes on various types of instances.

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ACO algorithms have in particular shown to be successful in solving problems from combinatorial optimization. In contrast to many applications, first theoretical estimations of the runtime of such algorithms for the optimization of pseudo-boolean functions have been obtained only recently. In the case of combinatorial optimization problems, the construction graphs used are more related to the problem at hand. For the first time, the effect of such graphs have been investigated by rigorous runtime analyses. We have considered a simple ACO algorithm 1-ANT for the well- known **minimum** **spanning** **tree** problem. In the case of the Broder-based construction procedure a polynomial, but relatively large, upper bound has been proven. In addition, it has been shown that heuristic information can mislead the algorithm such that an optimal solution is not found within a polynomial number of steps with high probability. In the case of the Kruskal-based construction procedure, the upper bound obtained shows that this construction graph leads to a better optimization process than the 1-ANT and simple evolutionary algorithms in the context of the optimization of pseudo-boolean functions. In addition, a large influence of heuristic information makes the algorithm mimic Kruskal’s algorithm for the **minimum** **spanning** **tree** problem. All analyses provide insight into the guided random walks that the 1-ANT performs in order to create solutions of our problem.

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September 12, 2001
Abstract
The Generalized **Minimum** **Spanning** **Tree** problem denoted by GMST is a variant of the classical **Minimum** **Spanning** **Tree** problem in which nodes are partitioned into clusters and the problem calls for a **minimum** cost **tree** **spanning** at least one node from each cluster. A different version of the problem, called E-GMST arises when exactly one node from each cluster has to be visited. Both GMST problem and E-GMST problem are NP-hard problems. In this paper, we model GMST problem and E-GMST problem as integer linear programs and study the facial structure of the corresponding polytopes.

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dealing with medium to large size graphs, efficient algorithms, which provide good, but not necessarily optimal solutions, are required. The first two heuris- tics for the QMSTP were given in [1]. These algo- rithms are constructive by nature and, therefore, are not able to provide good enough solutions for larger graphs. Zhou and Gen [21] presented a genetic algo- rithm (GA) in which the Prüfer number [16] to encode a **spanning** **tree** was adopted. They reported com- putational results for 17 test instances. Soak, Corne and Ahn [18] developed another genetic algorithm which employed a decoder-based redundant encod- ing strategy. They have shown that their GA imple- mentation outperforms genetic algorithm using the Prüfer number representation. More recently, Cor- done and Passeri [2] have applied a tabu search tech- nique to solve the QMSTP. At each iteration of their algorithm, the search is performed in the 1-exchange neighborhood, which consists of all **spanning** trees that can be obtained from the current **spanning** **tree** by replacing one of its edges with a non-**tree** edge. In [10], Öncan and Punnen have presented a local search algorithm with tabu thresholding. In this al- gorithm, the same neighborhood structure as in [2] is used. An artificial bee colony algorithm for solv- ing the QMSTP was proposed by Sundar and Singh [19]. In the last phase, the algorithm makes a call to a local search procedure. This algorithm compares favourably with earlier evolutionary approaches. Gao and Lu [5] introduced the fuzzy quadratic **minimum** **spanning** **tree** problem. It is formulated as expected value model, chance-constrained programming and dependent-chance programming according to differ- ent criteria. In [5], a genetic algorithm using Prüfer number representation for solving this problem was developed.

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The present work proposes decentralize distributed algorithm that constructs a **minimum** **spanning** **tree** for cognitive radio network. All the nodes have equal priority to calculate the MST in CRN. The actions of SUs are distributed and it is local and separable in cognitive radio network. The MST grows from both ends of the nodes connected by an edge. However, when the algorithm terminates, the resultant **tree** at the end of computation is unique. In the algorithm, each secondary user node is uniquely identiﬁed by the identiﬁer (id). Each SU node maintains a local channel set. The id and the LCS are to be transmitted among the nodes using message. From LCS, at least one from CCC set is used to create edges among the SU nodes that carry some cost. At each step, sub graph information is broadcasted to all neighbors except from where the message has been received. The receiver node send acknowledgement to the sender.

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That is the bad news. The good news is that you can solve it very easily by the algorithm described below without even using a computer.
A Remarkably Simple Algorithm
Starting with no links in the network, each step of the algorithm selects one new link to insert from the list of potential links. As described below, the algorithm continues in this way until every node is touched by a link, at which point the selected nodes form a **minimum** **spanning** **tree**.

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doi: 10.17706/jcp.10.1.45-56
Abstract: **Minimum** **spanning** **tree** is a **minimum**-cost **spanning** **tree** connecting the whole network, but it couldn't be directly obtained on uncertain graph. In this paper, we define the reliability as the existence probability of all **minimum** **spanning** trees and present an algorithm for evaluating reliability of the **minimum** **spanning** **tree** on uncertain graph. The time complexity of the algorithm is O(Nmn), where n, m and N stand for the number of vertices, edges and **minimum** **spanning** trees, respectively. Because this algorithm spends more time finding **minimum** **spanning** **tree**, we propose an improved algorithm whose time complexity is O(Nm). The improved algorithm uses disjoint set data structure so that the average time complexity on finding a new **minimum** **spanning** **tree** is O(m/n). The two algorithms are analyzed in detail and the experiment results agree with theoretical analysis.

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1,2,3
Department of Mathematics, College of Engineering and Technology, Techno Campus, Ghatikia, Bhubaneswar-751003, India
Abstract––The **minimum** **spanning** **tree** of a connected graph has many applications in different field of knowledge. In many real world problems the input data are often imprecise due to incomplete or non-obtainable information. This paper is intended to find **minimum** **spanning** **tree** on a rough graph where the edges have fuzzy weights. The Boruvka algorithm has been used to find the **minimum** **spanning** **tree** and signed distance ranking method has been used for ranking fuzzy numbers.

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The first results with respect to the runtime of a simple ACO algorithm have been obtained for the optimization of pseudo-Boolean functions [ 3 , 4 , 11 , 15 ]. Many combinatorial optimization problems can be considered as the optimization of a specific pseudo-Boolean function. Especially in the case of polynomially solvable problems, we cannot hope that more or less general search heuristics outperform the best-known algorithms for a specific problem. Nevertheless, it is interesting to analyze them on such problems as this shows how the heuristics work and therefore improve the understanding of these, in practice successful, algorithms. A basic evolutionary algorithm called ( 1 + 1 ) EA has been considered for a wide class of combinatorial optimization problems in the context of optimizing a pseudo-Boolean function. All results with respect to the ( 1 + 1 ) EA transfer to a simple ACO algorithm called 1-ANT in this context [ 15 ]. This includes runtime bounds on some of the best-known polynomially solvable combinatorial optimization problems such as maximum matching, and the **minimum** **spanning** **tree** (MST) problem. In the case of NP-hard problems, the result of Witt [ 18 ] on the partition problem transfers to the 1-ANT.

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The present research analyzes the dynamic evolution of linkages between global government bond markets, as well as the impact of the euro on EMU markets, using a methodology drawn from the econophysics literature. Originally suggested by Mantegna [12]
**minimum** **spanning** **tree** (MST) analysis involves transforming the correlation matrix of asset returns into distances to produce a connected graph. The procedure provides a parsimonious representation of the correlations between markets and is particularly suitable for extracting the most important information concerning linkages when a large number of markets is under consideration. It also provides an additional tool to measure financial integration, in terms of the

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Grid2= Distance of the nodes from the BS is more from 15m to 35m
Grid3= Distance of the nodes from the BS is more than 35m Groups are framed inside the level where part nodes are associated with the CHs with the assistance of **Minimum** **Spanning** **Tree** (MST). A **Spanning** Prim's Algorithm is utilized for the MST development between the nodes and CHs and in addition between CHs of various grids. A crossing **tree** of a chart is an associated sub-diagram in which there are no cycles. The least crossing **tree** for a given chart is a base cost spreading over **tree** for that diagram.

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Here, we recognize (again) the high correlation among moments of correlation coefficients and distances among stocks traded in Indonesia.
We move now to the analysis of the ultrametricity space of stocks. We do the previously elaborated Kruskal’s algorithm to the distance matrix in order to have the simple locally **minimum** **spanning** **tree** of the stocks. Fig- ure 3 shows the yielded ultrametric spaces of four trading years. The figure is produced by using Pajek, the software for analysis and visualization of large networks [1]. Geometrically we can see that in 2001 TLKM and ASII are highly connected to other stocks showing that the fluctuations of both stocks is frequently referred by some particular stocks. This pattern is rela- tively persistent in the following year (2002). However, in the year 2003, the pattern is changed since the both stocks are no longer highly connected to other stocks. In stead, there are apparently some connections among some sectors: consumer goods industries and the financial sector. Eventually, in 2004, the previous geometric pattern seems to dwindle. There seems no single dominant fluctuation reference and the stock connects to each other arbitrarily; nevertheless, several stocks seems still to be connected with the same sectors.

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Performanace of Improved **Minimum** **Spanning** **Tree** Based on Clustering Technique
P.Sampurnima α , J Srinivas σ & Harikrishna ρ
Abstract - Clustering technique is one of the most important and basic tool for data mining. Cluster algorithms have the ability to detect clusters with irregular boundaries, **minimum** **spanning** **tree**-based clustering algorithms have been widely used in practice. In such clustering algorithms, the search for nearest objects in the construction of **minimum** **spanning** trees is the main source of computation and the standard solutions take O(N 2 ) time. In this paper, we present a fast **minimum** **spanning** **tree**-inspired clustering algorithm, which, by using an efficient implementation of the cut and the cycle property of the **minimum** **spanning** trees, can have much better performance than O(N 2 ).

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© 2011 Elsevier B.V. All rights reserved.
1. Introduction
A **spanning** **tree** of a connected undirected graph G with real edge weights is a **minimum** **spanning** **tree** if its weights, i.e., the total weight of its edges, is minimal among all total weights of **spanning** trees of G. Constructing a **minimum** **spanning** **tree** is a well-known fundamental problem in network analysis with numerous applications. Besides the importance of the problem in its own right, the problem arises in solutions of other problems (see, e.g., [ 24 ] for a nice survey on results from the earliest known algorithm of Borůvka [ 11 ] to the invention of Fibonacci heaps and [ 2 ] for a survey and empirical study on various **minimum** **spanning** **tree** algorithms). Since modern applications require huge graphs, explicit representations by adjacency matrices or adjacency lists may cause conflicts with memory limitations. If time and space resources do not suffice to consider individual vertices, one way out could be to deal with sets of vertices and edges represented by their characteristic functions. Ordered binary decision diagrams, denoted OBDDs, introduced by Bryant in 1986 [ 13 ], are well suited for the representation and manipulation of Boolean functions, therefore, a research branch has emerged which is concerned with the design and analysis of so-called symbolic algorithms for classical graph problems on OBDD-represented graph instances (see, e.g., [ 3 , 4 , 21 – 23 , 25 , 34 – 38 ], and [ 42 , 43 ]). Symbolic algorithms have to solve problems on a given graph instance by efficient functional operations offered by the OBDD data structure. Although the worst-case complexity of symbolic algorithms is usually worse than that of corresponding explicit algorithms, symbolic algorithms have already been successfully applied as heuristics in areas like model checking and circuit verification (see, e.g., [ 15 ]). The main purpose is not to beat explicit algorithms on graphs which can be represented explicitly but to solve problems for very large structured graphs in reasonable time and space. The maximum flow problem in 0–1 networks has been one of the first classical

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email: {Marc.Reimann, Marco.Laumanns}@ifor.math.ethz.ch
Abstract
The problem of finding a Capacitated **Minimum** **Spanning** **Tree** asks for connecting a set of client nodes to a root node through a **minimum** cost **tree** network, subject to capacity constraints on all links. This paper reports on our design, implementation and performance evaluation of a hybrid Ant Colony Optimization algorithm for finding Capacitated **Minimum** **Spanning** Trees. Our Ant Colony Optimization algorithm is based on two important problem characteristics, namely the close relationship of the Capacitated **Minimum** **Spanning** **Tree** Problem with both the Capacitated Vehicle Routing Problem and the **Minimum** **Spanning** **Tree** Problem and hybridizes the Savings based Ant System with the algorithm of Prim, which is used to solve the subproblems of finding minimal **spanning** trees exactly. To assess the performance of our implementation we perform a computational study on a set of well known benchmark instances. For these instances our results show both the effectiveness and the efficiency of our algorithm when compared to several other state-of-the-art techniques.

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