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Greedy Heuristics and an Evolutionary Algorithm for the BoundedDiameter Minimum Spanning Tree Problem
 Proceedings of the 2003 ACM Symposium on Applied Computing
, 2003
"... bound D, the boundeddiameter minimum spanning tree problem seeks a spanning tree on G of lowest weight in which no path between two vertices contains more than D edges. This problem is NPhard for 4 1, where n is the number of vertices in G. An existing greedy heuristic for the problem, called ..."
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Cited by 35 (13 self)
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bound D, the boundeddiameter minimum spanning tree problem seeks a spanning tree on G of lowest weight in which no path between two vertices contains more than D edges. This problem is NPhard for 4 1, where n is the number of vertices in G. An existing greedy heuristic for the problem, called OTTC, is based on Prim's algorithm. OTTC usually yields poor results on instances in which the triangle inequality approximately holds; it always uses the lowestweight edges that it can, but such edges do not in general connect the interior nodes of lowweight boundeddiameter trees. A new randomized greedy heuristic builds a boundeddiameter spanning tree from its center vertex or vertices. It chooses each next vertex at random but attaches the vertex with the lowestweight eligible edge. This algorithm is faster than OTTC and yields substantially better solutions on Euclidean instances. An evolutionary algorithm encodes spanning trees as lists of their edges, augmented with their center vertices. It applies operators that maintain the diameter bound and always generate valid o#spring trees. These operators are e#cient, so the algorithm scales well to larger problem instances. On 25 Euclidean instances of up to 1 000 vertices, the EA improved substantially on solutions found by the randomized greedy heuristic.
Variable neighborhood search for the bounded diameter minimum spanning tree problem
 Proceedings of the 18th Mini Euro Conference on Variable Neighborhood Search
, 2005
"... The bounded diameter minimum spanning tree problem is an NPhard combinatorial optimization problem with applications in various fields like communication network design. We propose a general variable neighborhood search approach for it, utilizing four different types of neighborhoods. They were d ..."
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Cited by 14 (9 self)
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The bounded diameter minimum spanning tree problem is an NPhard combinatorial optimization problem with applications in various fields like communication network design. We propose a general variable neighborhood search approach for it, utilizing four different types of neighborhoods. They were designed in a way enabling an efficient incremental evaluation and search for the best neighboring solution. An experimental comparison on instances with complete graphs with up to 1000 nodes indicates that this approach consistently outperforms the so far leading evolutionary algorithms with respect to solution quality and computation time.
RandomTree Diameter and the DiameterConstrained MST
 MST,” Congressus Numerantium
, 2000
"... A minimum spanning tree (MST) with a small diameter is required in numerous practical situations. It is needed, for example, in distributed mutual exclusion algorithms in order to minimize the number of messages communicated among processors per critical section. Understanding the behavior of tre ..."
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Cited by 9 (1 self)
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A minimum spanning tree (MST) with a small diameter is required in numerous practical situations. It is needed, for example, in distributed mutual exclusion algorithms in order to minimize the number of messages communicated among processors per critical section. Understanding the behavior of tree diameter is useful, for example, in determining an upper bound on the expected number of links between two arbitrary documents on the World Wide Web. The DiameterConstrained MST (DCMST) problem can be stated as follows: given an undirected, edgeweighted graph G with n nodes and a positive integer k, find a spanning tree with the smallest weight among all spanning trees of G which contain no path with more than k edges. This problem is known to be NPcomplete, for all values of k; 4 k #n  2). In this paper, we investigate the behavior of the diameter of MST in randomlyweighted complete graphs (in ErdsRnyi sense) and explore heuristics for the DCMST problem. For the case when the diameter bound k is smallindependent of n, we present a onetimetreeconstruction (OTTC) algorithm. It constructs a DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to the tree at each stage of the tree construction. This algorithm is fast and easily parallelizable. We also present a second algorithm that outperforms OTTC for larger values of k. It starts by generating an unconstrained MST and iteratively refines it by replacing edges, one by one, in the middle of long paths in the spanning tree until there is no path left with more than k edges. As expected, the performance of this heuristic is determined by the diameter of the unconstrained MST in the given graph. We discuss convergence, relative merits, and implementation of t...
Computing A DiameterConstrained Minimum Spanning Tree
, 2001
"... In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameterconstrained minimum spanning tree (DCMST) of a given undirected, edgeweighted graph, G, is the smallestweight spanning tree of all spanning trees of G which contain no path wi ..."
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Cited by 8 (0 self)
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In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameterconstrained minimum spanning tree (DCMST) of a given undirected, edgeweighted graph, G, is the smallestweight spanning tree of all spanning trees of G which contain no path with more than k edges, where k is a given positive integer. The problem of finding a DCMST is NPcomplete for all values of k; 4 k (n  2), except when all edgeweights are identical. A DCMST is essential for the efficiency of various distributed mutual exclusion algorithms, where it can minimize the number of messages communicated among processors per critical section. It is also useful in linear lightwave networks, where it can minimize interference in the network by limiting the traffic in the network lines. Another practical application requiring a DCMST arises in data compression, where some algorithms compress a file utilizing a tree datastructure, and decompress a path in the tree to access a record. A DCMST helps such algorithms to be fast without sacrificing a lot of storage space. We present a survey of the literature on the DCMST problem, study the expected diameter of a random labeled tree, and present five new polynomialtime algorithms for an approximate DCMST. One of our new algorithms constructs an approximate DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to iii the tree in each stage of the construction. Three other new algorithms start with an unconstrained minimum spanning tree, and iteratively refine it into an approximate DCMST. We also present an algorithm designed for the special case when the diameter is required to be no more than 4. Such a diameter4 tree is also used for evaluating the quality of o...
Neighbourhood Searches for the Bounded Diameter . . .
, 2006
"... We consider the Bounded Diameter Minimum Spanning Tree problem and describe four neighbourhood searches for it. They are used as local improvement strategies within a variable neighbourhood search (VNS), an evolutionary algorithm (EA) utilising a new encoding of solutions, and an ant colony optimisa ..."
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Cited by 4 (0 self)
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We consider the Bounded Diameter Minimum Spanning Tree problem and describe four neighbourhood searches for it. They are used as local improvement strategies within a variable neighbourhood search (VNS), an evolutionary algorithm (EA) utilising a new encoding of solutions, and an ant colony optimisation (ACO). We compare the performance in terms of effectiveness between these three hybrid methods on a suite of popular benchmark instances, which contains instances too large to solve by current exact methods. Our results show that the EA and the ACO outperform the VNS on almost all used benchmark instances. Furthermore, the ACO yields most of the time better solutions than the EA in longterm runs, whereas the EA dominates when the computation time is strongly restricted.
AllOptical Network Topologies based on Expander Graphs
 J. High Speed Networks
, 1995
"... Alloptical networks are networks for which all data paths remain optical from input to output. We discuss a class of wavelength division multiple access (WDMA) networks that are homogeneous in the sense that each node contains both an input/output port and a switch. We focus on permutation routing ..."
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Cited by 1 (1 self)
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Alloptical networks are networks for which all data paths remain optical from input to output. We discuss a class of wavelength division multiple access (WDMA) networks that are homogeneous in the sense that each node contains both an input/output port and a switch. We focus on permutation routing problem and first present a lower bound on the number of wavelengths required for permutation routing as a function of the size and degree of the network. We then consider a network topology based on expander graphs and derive an expression for the number of wavelengths that are sufficient for the permutation routing problem. We present these numbers as asymptotic upper bounds for very large networks. 1 Introduction Optical fiber transmission is rapidly becoming the standard transmission medium for networks, and can provide the required data rate, error rate, and delay performance for future networks. However, data rates are currently limited by the need to convert the optical signals on t...
A New Hybrid Genetic Algorithm for Solving the Bounded Diameter Minimum Spanning Tree Problem
"... Abstract — In this paper, a new hybrid genetic algorithm – known as HGA – is proposed for solving the Bounded Diameter Minimum Spanning Tree (BDMST) problem. We experiment with HGA on two sets of benchmark problem instances, both Euclidean and NonEuclidean. On the Euclidean problem instances, HGA i ..."
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Abstract — In this paper, a new hybrid genetic algorithm – known as HGA – is proposed for solving the Bounded Diameter Minimum Spanning Tree (BDMST) problem. We experiment with HGA on two sets of benchmark problem instances, both Euclidean and NonEuclidean. On the Euclidean problem instances, HGA is shown to outperform the previous best two Genetic Algorithms (GAs) reported in the BDMST literature, while on the NonEuclidean problem instance, HGA performs comparably with these two GAs. T I.
On the boundedhop MST problem on . . .
, 2006
"... The dDim hhops MST problem is defined as follows: Given a set S of points in the ddimensional Euclidean space and s ∈ S, find a minimumcost spanning tree for S rooted at s with height at most h. We investigate the problem for any constant h and d> 0. We prove the first non trivial lower bound on ..."
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The dDim hhops MST problem is defined as follows: Given a set S of points in the ddimensional Euclidean space and s ∈ S, find a minimumcost spanning tree for S rooted at s with height at most h. We investigate the problem for any constant h and d> 0. We prove the first non trivial lower bound on the solution cost for almost all Euclidean instances (i.e. the lowerbound holds with high probability). Then we introduce an easytoimplement, fast divide et impera heuristic and we prove that its solution cost matches the lower bound.
BoundedDiameter MST Instances with Hybridization of MultiObjective EA
"... The Bounded Diameter (a.k.a Diameter Constraint) Minimum Spanning Tree (BDMST/DCMST) is a wellknown combinatorial optimization problem. In this paper, we recast a few wellknown heuristics, which are evolved for BDMST problem to a BiObjective Minimum Spanning Tree (BOMST) problem and then obtain P ..."
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The Bounded Diameter (a.k.a Diameter Constraint) Minimum Spanning Tree (BDMST/DCMST) is a wellknown combinatorial optimization problem. In this paper, we recast a few wellknown heuristics, which are evolved for BDMST problem to a BiObjective Minimum Spanning Tree (BOMST) problem and then obtain Pareto fronts. After examining Pareto fronts, it is concluded that none of the heuristics provides the superior solution across the complete range of the diameter. We have used a MultiObjective Evolutionary Algorithm (MOEA) approach, Pareto Converging Genetic Algorithm (PCGA), to improve the Pareto front for BOMST, which in turn provides better solution for BDMST instances. We have considered edgeset encoding to represent MST and then applied recombination operators having strong heritability and mutation operators having negligible complexity to improve the solutions. Analysis of MOEA solutions confirms the improvement of Pareto front solutions across the complete range of the diameter over Pareto front solutions generated from individual heuristics. We have considered multiisland scheme using InterIsland rank histogram and performed multiple run of the algorithm to avoid from trapping into localoptimal solutionset.
Exploiting Hierarchical Clustering for Finding Bounded Diameter Minimum Spanning Trees on Euclidean Instances
"... The bounded diameter minimum spanning tree problem is an NPhard combinatorial optimization problem arising, for example, in network design when quality of service is of concern. There exist various exact and metaheuristic approaches addressing this problem, whereas fast construction heuristics are ..."
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The bounded diameter minimum spanning tree problem is an NPhard combinatorial optimization problem arising, for example, in network design when quality of service is of concern. There exist various exact and metaheuristic approaches addressing this problem, whereas fast construction heuristics are primarily based on Prim’s minimum spanning tree algorithm and fail to produce reasonable solutions in particular on large Euclidean instances. A method based on hierarchical clustering to guide the construction process of a diameter constrained tree is presented. Solutions obtained are further refined using a greedy randomized adaptive search procedure. Based on the idea of clustering we also designed a new neighborhood search for this problem. Especially on large Euclidean instances with a tight diameter bound the results are excellent. In this case the solution quality can also compete with that of a leading metaheuristic, whereas the computation only needs a fraction of the time. Categories and Subject Descriptors