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14
Simulated brain tumor growth dynamics using a threedimensional cellular automaton
 J Theor Biol
, 2000
"... The competition between local and global driving forces is significant in a wide variety of naturally occurring branched networks. We have investigated the impact of a global minimization criterion versus a local one on the structure of spanning trees. To do so, we consider two spanning tree structu ..."
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Cited by 25 (3 self)
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The competition between local and global driving forces is significant in a wide variety of naturally occurring branched networks. We have investigated the impact of a global minimization criterion versus a local one on the structure of spanning trees. To do so, we consider two spanning tree structures the generalized minimal spanning tree (GMST) defined by Dror et al. [1] and an analogous structure based on the invasion percolation network, which we term the generalized invasive spanning tree or GIST. In general, these two structures represent extremes of global and local optimality, respectively. Structural characteristics are compared between the GMST and GIST for a fixed lattice. In addition, we demonstrate a method for creating a series of structures which enable one to span the range between these two extremes. Two structural characterizations, the occupied edge density (i.e., the fraction of edges in the graph that are included in the tree) and the tortuosity of the arcs in the trees, are shown to correlate well with the degree to which an intermediate structure resembles the GMST or GIST. Both characterizations are straightforward to determine from an image and are potentially useful tools in the analysis of the formation of network structures. 1
Variable Neighborhood Search For The DegreeConstrained Minimum Spanning Tree Problem
 Discrete Applied Mathematics
, 2001
"... . Given an undirected graph with weights associated with its edges, the degreeconstrained minimum spanning tree problem consists in finding a minimum spanning tree of the given graph, subject to constraints on node degrees. We propose a variable neighborhood search heuristic for the degreeconstrain ..."
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Cited by 15 (4 self)
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. Given an undirected graph with weights associated with its edges, the degreeconstrained minimum spanning tree problem consists in finding a minimum spanning tree of the given graph, subject to constraints on node degrees. We propose a variable neighborhood search heuristic for the degreeconstrained minimum spanning tree problem, based on a dynamic neighborhood model and using a variable neighborhood descent iterative improvement algorithm for local search. Computational experiments illustrating the effectiveness of the approach on benchmark problems are reported. Key words. Combinatorial optimization, degreeconstrained minimum spanning tree, local search, metaheuristics, variable neighborhoods 1. Introduction. Let G = (V; E) be a connected undirected graph, where V is the set of nodes and E denotes the set of edges. Given a nonnegative weight function w : E ! IR + associated with its edges and a nonnegative integer valued degree function b : V ! IN associated with its nodes, th...
A New Evolutionary Approach to the Degree Constrained Minimum Spanning Tree Problem
 IEEE Transactions on Evolutionary Computation
, 2000
"... Finding the degreeconstrained minimum spanning tree (dMST) of a graph is a well studied NPhard problem which is important in network design. We introduce a new method which improves on the best technique previously published for solving the dMST, either using heuristic or evolutionary app ..."
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Cited by 11 (0 self)
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Finding the degreeconstrained minimum spanning tree (dMST) of a graph is a well studied NPhard problem which is important in network design. We introduce a new method which improves on the best technique previously published for solving the dMST, either using heuristic or evolutionary approaches. The basis of this encoding is a spanningtree construction algorithm which we call the Randomised Primal Method (RPM), based on the wellknown Prim's algorithm [6], and an extension [4] which we call `dPrim's'. We describe a novel encoding for spanning trees, which involves using the RPM to interpret lists of potential edges to include in the growing tree. We also describe a random graph generator which produces particularly challenging dMST problems. On these and other problems, we find that an evolutionary algorithm (EA) using the RPM encoding outperforms the previous best published technique from the operations research literature, and also outperforms simulated...
A new encoding for labeled trees employing a stack and a queue
 Bulletin of the Institute of Combinatorics and its Applications
, 2002
"... A novel algorithm for encoding finite labeled trees is proposed in this paper. The algorithm establishes a onetoone mapping between trees of order n and (n – 2)tuples of the node labels. This encoding is similar to those proposed by Prüfer [8] and Neville [6], except that our encoding scheme has ..."
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Cited by 11 (3 self)
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A novel algorithm for encoding finite labeled trees is proposed in this paper. The algorithm establishes a onetoone mapping between trees of order n and (n – 2)tuples of the node labels. This encoding is similar to those proposed by Prüfer [8] and Neville [6], except that our encoding scheme has additional properties (not present in earlier codes), which allows us to efficiently determine the center(s), radius, and diameter directly from the code, without having to explicitly construct the tree. We also present conditions for modifying the proposed code in order to obtain trees at distance one from the original. A queue is used during encoding while a stack is used during decoding. Both encoding and decoding require O(n) time.
Prüferlike codes for labeled trees
 Congressus Numerantium
, 2001
"... In 1918 Prüfer showed a onetoone correspondence between nnode labeled trees and (n – 2)tuples of node labels. The proof employed a tree code, computed by iteratively deleting the leaf with the smallest label and recording its neighbor. Since then other tree codes have been proposed, based on dif ..."
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Cited by 10 (1 self)
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In 1918 Prüfer showed a onetoone correspondence between nnode labeled trees and (n – 2)tuples of node labels. The proof employed a tree code, computed by iteratively deleting the leaf with the smallest label and recording its neighbor. Since then other tree codes have been proposed, based on different node deletion sequences. These codes have different properties, interesting and useful in graph theory and computer science. In this paper we survey and classify these Prüferlike codes as well as new codes based on the parameter values of the proposed generic treeencoding algorithm and study their properties. 1.
A New Evolutionary Approach to the DegreeConstrained Minimum Spanning Tree Problem
 IEEE Transactions on Evolutionary Computation
, 1999
"... Finding the degreeconstrained minimum spanning tree (dMST) of a graph is a wellstudied NPhard problem of importance in communications network design and other networkrelated problems. In this paper we describe some previously proposed algorithms for solving the problem, and then introduce a nove ..."
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Cited by 10 (2 self)
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Finding the degreeconstrained minimum spanning tree (dMST) of a graph is a wellstudied NPhard problem of importance in communications network design and other networkrelated problems. In this paper we describe some previously proposed algorithms for solving the problem, and then introduce a novel tree construction algorithm called the Randomised Primal Method (RPM) which builds degreeconstrained trees of low cost from solution vectors taken as input. RPM is applied in three stochastic iterative search methods: simulated annealing, multistart hillclimbing, and a genetic algorithm. While other researchers have mainly concentrated on finding spanning trees in Euclidean graphs, we consider the more general case of random graph problems. We describe two random graph generators which produce particularly challenging dMST problems. On these and other problems we find that the genetic algorithm employing RPM outperforms simulated annealing and multistart hillclimbing. Our experimental ...
A Comparison of Encodings and Algorithms for Multiobjective Minimum Spanning Tree Problems
 In Proceedings of the 2001 Congress on Evolutionary Computation (CEC'01
, 1997
"... this paper we apply (appropriately modified) the best of recent methods for the (degreeconstrained) single objective MST problem to the multiobjective MST problem, and compare with a method based on Zhou and Gen's approach. Our evolutionary computation approaches, using the different encodings, inv ..."
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Cited by 8 (1 self)
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this paper we apply (appropriately modified) the best of recent methods for the (degreeconstrained) single objective MST problem to the multiobjective MST problem, and compare with a method based on Zhou and Gen's approach. Our evolutionary computation approaches, using the different encodings, involve a new populationbased variant of Knowles and Corne's PAES algorithm. We find the direct encoding to considerably outperform the Prufer encoding. And we find that a simple iterated approach, based on Prim's algorithm modified for the multiobjective MST, also significantly outperforms the Prufer encoding.
A Unified Approach to Coding Labeled Trees
 IN PROCEEDINGS OF THE 6TH LATIN AMERICAN SYMPOSIUM ON THEORETICAL INFORMATICS (LATIN ’04), LNCS 2976
, 2004
"... We consider the problem of coding labeled trees by means of strings of node labels and we present a unified approach based on a reduction of both coding and decoding to integer (radix) sorting. Applying this approach to four wellknown codes introduced by Prufer [18], Neville [17], and Deo and M ..."
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Cited by 8 (4 self)
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We consider the problem of coding labeled trees by means of strings of node labels and we present a unified approach based on a reduction of both coding and decoding to integer (radix) sorting. Applying this approach to four wellknown codes introduced by Prufer [18], Neville [17], and Deo and Micikevicius [5], we close some open problems. With respect to
Parallel algorithms for computing Prüferlike codes of labeled trees
, 2001
"... In 1918 Prüfer showed a onetoone correspondence between labeled trees on n nodes and sequences of (n2) node labels, obtained by iteratively deleting the leaf with the smallest label in the current tree and appending the label of the adjacent node to the code. In 1953 Neville proposed two additi ..."
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Cited by 4 (1 self)
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In 1918 Prüfer showed a onetoone correspondence between labeled trees on n nodes and sequences of (n2) node labels, obtained by iteratively deleting the leaf with the smallest label in the current tree and appending the label of the adjacent node to the code. In 1953 Neville proposed two additional methods for encoding a labeled tree of order n into a sequence of (n2) labels. Recently, yet another treecode was proposed by the authors. As a generalization, we define a treecode to be Prüferlike, if it is a sequence of (n2) node labels computed by iteratively deleting the leaves of the tree in some deterministic order. The four aforementioned Prüferlike codes can be computed sequentially in O(n) time. Two EREW PRAM O(log n)time algorithms for computing the Prüfer code have been proposed in the literature. Neither of these algorithms is workoptimal, since they require O(n) processors. In this paper we propose three different EREW PRAM O(log n)time algorithms for computing the treecodes proposed by Neville, and by the authors. The algorithm for Neville's third encoding requires O(n / log n) processors and therefore is workoptimal. To the best of our knowledge, no other workoptimal algorithm for computing Prüferlike treecode has been published.
Lineartime algorithms for encoding trees as sequences of node labels
, 2007
"... In this paper we present O(n)time algorithms for encoding/decoding nnode labeled trees as sequences of n − 2 node labels. All known encodings of this type are covered, including Prüferlike codes and the three codes proposed by Picciotto the happy, blob, and dandelion codes. The algorithms for Pi ..."
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Cited by 1 (0 self)
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In this paper we present O(n)time algorithms for encoding/decoding nnode labeled trees as sequences of n − 2 node labels. All known encodings of this type are covered, including Prüferlike codes and the three codes proposed by Picciotto the happy, blob, and dandelion codes. The algorithms for Picciotto’s codes are of special significance as previous publications describe suboptimal approaches requiring O(n log n) or even O(n 2) time.