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17
Representing Trees in Genetic Algorithms
 Proceedings of the First IEEE Conference on Evolutionary Computation
, 1994
"... We consider the problem of representing trees (undirected, cyclefree graphs) in Genetic Algorithms. This problem arises, among other places, in the solution of network design problems. After comparing several commonly used representations based on their usefulness in genetic algorithms, we describe ..."
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Cited by 46 (1 self)
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We consider the problem of representing trees (undirected, cyclefree graphs) in Genetic Algorithms. This problem arises, among other places, in the solution of network design problems. After comparing several commonly used representations based on their usefulness in genetic algorithms, we describe a new representation and show it to be superior in almost all respects to the others. In particular, we show that our representation covers the entire space of solutions, produces only viable offspring, and possesses locality, all necessary features for the effective use of a genetic algorithm. We also show that the representation will reliably produce very good, if not optimal, solutions even when the problem definition is changed. I. Introduction In this paper, we consider the problem of representing trees in genetic algorithms. A tree is an undirected graph which contains no closed cycles. There are many optimization problems which can be phrased in terms of finding the optimal tree wit...
Coalescent Random Forests
 J. COMBINATORIAL THEORY A
, 1998
"... Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ..."
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Cited by 38 (18 self)
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Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitude...
Enumerations Of Trees And Forests Related To Branching Processes And Random Walks
 Microsurveys in Discrete Probability, number 41 in DIMACS Ser. Discrete Math. Theoret. Comp. Sci
, 1997
"... In a GaltonWatson branching process with offspring distribution (p 0 ; p 1 ; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the first passage time to \Gammak for a random walk started at 0 which takes steps of size j with probability p ..."
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Cited by 38 (15 self)
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In a GaltonWatson branching process with offspring distribution (p 0 ; p 1 ; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the first passage time to \Gammak for a random walk started at 0 which takes steps of size j with probability p j+1 for j \Gamma1. The formula for this distribution is a probabilistic expression of the Lagrange inversion formula for the coefficients in the power series expansion of f(z) k in terms of those of g(z) for f(z) defined implicitly by f(z) = zg(f(z)). The Lagrange inversion formula is the analytic counterpart of various enumerations of trees and forests which generalize Cayley's formula kn n\Gammak\Gamma1 for the number of rooted forests labeled by a set of size n whose set of roots is a particular subset of size k. These known results are derived by elementary combinatorial methods without appeal to the Lagrange formula, which is then obtained as a byproduct. This approach unifies an...
An Approach To A Problem In Network Design Using Genetic Algorithms
, 1995
"... In the work of communications network design there are several recurring themes: maximizing flows, finding circuits, and finding shortest paths or minimal cost spanning trees, among others. Some of these problems appear to be harder than others. For some, effective algorithms exist for solving them, ..."
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Cited by 31 (0 self)
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In the work of communications network design there are several recurring themes: maximizing flows, finding circuits, and finding shortest paths or minimal cost spanning trees, among others. Some of these problems appear to be harder than others. For some, effective algorithms exist for solving them, for others, tight bounds are known, and for still others, researchers have few clues towards a good approach. One of these latter, nastier problems arises in the design of communications networks: the Optimal Communication Spanning Tree Problem (OCSTP). First posed by Hu in 1974, this problem has been shown to be in the family of NPcomplete problems. So far, a good, generalpurpose approximation algorithm for it has proven elusive. This thesis describes the design of a genetic algorithm for finding reliably good solutions to the OCSTP. The genetic algorithm approach was thought to be an appropriate choice since they are computationally simple, provide a powerful parallel search capability...
Bijections for Cayley trees, spanning trees, and their qanalogues
 Journal of Combinatorial Theory
, 1986
"... We construct a family of extremely simple bijections that yield Cayley’s famous formula for counting trees. The weight preserving properties of these bijections furnish a number of multivariate generating functions for weighted Cayley trees. Essentially the same idea is used to derive bijective pro ..."
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Cited by 16 (3 self)
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We construct a family of extremely simple bijections that yield Cayley’s famous formula for counting trees. The weight preserving properties of these bijections furnish a number of multivariate generating functions for weighted Cayley trees. Essentially the same idea is used to derive bijective proofs and qanalogues for the number of spanning trees of other graphs, including the complete bipartite and complete tripartite graphs. These bijections also allow the calculation of explicit formulas for the expected number of various statistics on Cayley trees. 0 1986 Academic Press, Inc. Let 9?,, denote the set of Cayley trees on n vertices, i.e., the set of simple graphs T = (V, E) with no cycles where the vertex set V = {l,..., n} and E is the set of edges. We let G & denote the set of rooted Cayley trees on II vertices where vertex i is the root. Cayley’s famous formula [ 1) for the number of Cayley trees is IKI,Il =Kr+l,il =(n+ lY ’ for n 2 1 and i = l,..., n + 1. (0.1) There are a number of analytic proofs of (0.1) in the literature [a]. Priifer [3] was the first to give a bijective proof of (0.1), and more recently Joyal [4] constructed an elegant encoding for birooted Cayley trees from which (0.1) follows.
A bijective proof for the number of labelled qtrees
 Ars Combinatoria
, 1988
"... We giv e abijective proof that the number of vertex labeled qtrees on n vertices is given by ⎛ n ⎞ ⎝ q ⎠ [ qn − q2 n − q − 2 + 1] The bijection transforms each pair ( S, f) where S is a qelement subset of an nset, and f is a function mapping an ( n − q − 2)set to a ( qn − q 2 + 1)set into a la ..."
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Cited by 4 (0 self)
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We giv e abijective proof that the number of vertex labeled qtrees on n vertices is given by ⎛ n ⎞ ⎝ q ⎠ [ qn − q2 n − q − 2 + 1] The bijection transforms each pair ( S, f) where S is a qelement subset of an nset, and f is a function mapping an ( n − q − 2)set to a ( qn − q 2 + 1)set into a labeled qtree on n nodes by a cutandpaste process. As a special case, q = 1 yields a new bijective proof of Cayley’s formula for labeled trees. The general bijection also provides for the enumeration of labeled qtrees in which a specified subset of the vertices forms a clique. 1.
Enumeration of Some Labelled Trees
"... . In this paper we are interesting in the enumeration of rooted labelled trees according to the relationship between the root and its sons. Let Tn;k be the family of Cayley trees on [n] such that the root has exactly k smaller sons. In a first time we give a bijective proof of the fact that jTn+1;k ..."
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Cited by 3 (0 self)
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. In this paper we are interesting in the enumeration of rooted labelled trees according to the relationship between the root and its sons. Let Tn;k be the family of Cayley trees on [n] such that the root has exactly k smaller sons. In a first time we give a bijective proof of the fact that jTn+1;k j = \Gamma n k \Delta n n\Gammak . Moreover, we use the family Tn+1;0 of Cayley trees for which the root is smaller than all its sons to give combinatorial explanations of various identities involving n n . We rely this family to the enumeration of minimal factorization of the ncycle (1; 2; : : : ; n) as a product of transpositions. Finally, we use the fact that jTn+1;0 j = n n to prove bijectively that there are 2n n ordered alternating trees on [n + 1]. R' esum' e. Dans cet article nous nous int'eressons `a l"enum'eration d'arbres 'etiquet'es enracin'es, en consid'erant un nouveau param`etre relatif `a l'ordre existant entre la racine et ses fils. Soit donc Tn;k la famille d...
Forest volume decompositions and AbelCayleyHurwitz multinomial expansions
, 2001
"... This paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitz's multinomial identities, involving polynomials defined by sums over all subsets of a finite set. The identities are interpreted as decompositions of forest volumes define ..."
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Cited by 2 (0 self)
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This paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitz's multinomial identities, involving polynomials defined by sums over all subsets of a finite set. The identities are interpreted as decompositions of forest volumes defined by the enumerator polynomials of sets of rooted labeled forests. These decompositions involve the following basic forest volume formula, which is a refinement of Cayley's multinomial expansion: for R ` S the polynomial enumerating outdegrees of vertices of rooted forests labeled by S whose set of roots is R, with edges directed away from the roots, is ( P r2R x r )( P s2S x s ) jS j\GammajRj\Gamma1
Some Remarks on Sparsely Connected IsomorphismFree Labeled Graphs
, 2000
"... . Given a set = fH1 ..."
On Encodings of Spanning Trees
, 2007
"... Deo and Micikevicius recently gave a new bijection for spanning trees of complete bipartite graphs. In this paper we devise a generalization of Deo and Micikevicius’s method, which is also a modification of Olah’s method for encoding the spanning trees of any complete multipartite graph K(n1,...,nr) ..."
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Cited by 1 (0 self)
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Deo and Micikevicius recently gave a new bijection for spanning trees of complete bipartite graphs. In this paper we devise a generalization of Deo and Micikevicius’s method, which is also a modification of Olah’s method for encoding the spanning trees of any complete multipartite graph K(n1,...,nr). We also give a bijection between the spanning trees of a planar graph and those of any of its planar duals. Finally we discuss the possibility of bijections for spanning trees of DeBriujn Graphs, Cubes, and regular graphs such as the Petersen graph that have integer eigenvalues.