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11
A random walk construction of uniform spanning trees and uniform labelled trees
 SIAM Journal on Discrete Mathematics
, 1990
"... Abstract A random walk on a finite graph can be used to construct a uniformrandom spanning tree. We show how random walk techniques can be applied to the study of several properties of the uniform randomspanning tree: the proportion of leaves, the distribution of degrees, and the diameter. ..."
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Cited by 80 (3 self)
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Abstract A random walk on a finite graph can be used to construct a uniformrandom spanning tree. We show how random walk techniques can be applied to the study of several properties of the uniform randomspanning tree: the proportion of leaves, the distribution of degrees, and the diameter.
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...
The Asymptotic Distribution of the Diameter of a Random Mapping
, 2002
"... The asymptotic distribution of the diameter of the digraph of a uniformly distributed random mapping of an nelement set to itself is represented as the distribution of a functional of a reflecting Brownian bridge. This yields a formula for the Mellin transform of the asymptotic distribution, ge ..."
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Cited by 5 (3 self)
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The asymptotic distribution of the diameter of the digraph of a uniformly distributed random mapping of an nelement set to itself is represented as the distribution of a functional of a reflecting Brownian bridge. This yields a formula for the Mellin transform of the asymptotic distribution, generalizing the evaluation of its mean by Flajolet and Odlyzko (1990).
Branched Polymers
"... A branched polymer is a connected configuration of nonoverlapping unit balls in space. Building on and from the work of Brydges and Imbrie, we give an elementary calculation of the volume of the space of branched polymers of order n in the plane and in 3space. Our development reveals some more gen ..."
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Cited by 4 (0 self)
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A branched polymer is a connected configuration of nonoverlapping unit balls in space. Building on and from the work of Brydges and Imbrie, we give an elementary calculation of the volume of the space of branched polymers of order n in the plane and in 3space. Our development reveals some more general identities, and allows exact random sampling. In particular we show that a random 3dimensional branched polymer of order n has diameter of order √ n. 1
THE DISTRIBUTION OF HEIGHT AND DIAMETER IN RANDOM NONPLANE BINARY TREES
"... ABSTRACT. This study is dedicated to precise distributional analyses of the height of nonplane unlabelled binary trees (“Otter trees”), when trees of a given size are taken with equal likelihood. The height of a rooted tree of size n is proved to admit a limiting theta distribution, both in a centr ..."
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Cited by 3 (0 self)
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ABSTRACT. This study is dedicated to precise distributional analyses of the height of nonplane unlabelled binary trees (“Otter trees”), when trees of a given size are taken with equal likelihood. The height of a rooted tree of size n is proved to admit a limiting theta distribution, both in a central and local sense, as well as obey moderate as well as large deviations estimates. The approximations obtained for height also yield the limiting distribution of the diameter of unrooted trees. The proofs rely on a precise analysis, in the complex plane and near singularities, of generating functions associated with trees of bounded height.
Biased mutation operators for subgraphselection problems
 IN IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION
, 2004
"... Many graph problems seek subgraphs of minimum weight that satisfy a set of constraints. Examples include the minimum spanning tree problem (MSTP), the degreeconstrained minimum spanning tree problem (dMSTP), and the traveling salesman problem (TSP). Lowweight edges predominate in optimum solution ..."
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Cited by 2 (0 self)
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Many graph problems seek subgraphs of minimum weight that satisfy a set of constraints. Examples include the minimum spanning tree problem (MSTP), the degreeconstrained minimum spanning tree problem (dMSTP), and the traveling salesman problem (TSP). Lowweight edges predominate in optimum solutions to such problems, and the performance of evolutionary algorithms (EAs) is often improved by biasing variation operators to favor these edges. We investigate the impact of biased edgeexchange mutation. In a largescale empirical investigation, we study the distributions of edges in optimum solutions of the MSTP, the dMSTP, and the TSP in terms of the edges ’ weightbased ranks. We approximate these distributions by exponential functions and derive approximately optimal probabilities for selecting edges to be incorporated into candidate solutions during mutation. A theoretical analysis of the expected running time
A sharp threshold for minimum boundeddepth and boundeddiameter spanning trees and Steiner trees in random networks
, 2008
"... ..."
Extremal Statistics on NonCrossing Configurations †
"... Abstract. We obtain several properties of extremal statistics in noncrossing configurations with n vertices. We prove that the maximum degree and the largest component are of logarithmic order, and the diameter is of order √ n. The proofs are based on singularity analysis, an application of the fir ..."
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Abstract. We obtain several properties of extremal statistics in noncrossing configurations with n vertices. We prove that the maximum degree and the largest component are of logarithmic order, and the diameter is of order √ n. The proofs are based on singularity analysis, an application of the first and second moment method and on the analysis of iterated functions. Résumé. On obtient des propriétés de paramètres extrémales dans les configurations sans croisement avec n sommets. On démontre que le degré maximal et la composante plus large sont d’ordre logarithmique, et le diamètre est d’ordre √ n. Les preuves utilisent l’analyse de singularités, une application de la méthode du premier et second moment, et l’analyse de fonctions itérées.