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23
Transfer Theorems and Asymptotic Distributional Results for mary Search Trees
, 2004
"... We derive asymptotics of moments and identify limiting distributions, under the random permutation model on mary search trees, for functionals that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the soca ..."
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Cited by 13 (7 self)
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We derive asymptotics of moments and identify limiting distributions, under the random permutation model on mary search trees, for functionals that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the socalled shape functional fall under this framework. The approach is based on establishing transfer theorems that link the order of growth of the input into a particular (deterministic) recurrence to the order of growth of the output. The transfer theorems are used in conjunction with the method of moments to establish limit laws. It is shown that (i) for small toll sequences (tn) [roughly, tn = O(n1/2)] we have asymptotic normality if m ≤ 26 and typically periodic behavior if m ≥ 27; (ii) for moderate toll sequences [roughly, tn = ω(n1/2) but tn = o(n)] we have convergence to nonnormal distributions if m ≤ m0 (where m0 ≥ 26) and typically periodic behavior if m ≥ m0 + 1; and (iii) for large toll sequences [roughly, tn = ω(n)] we have convergence to nonnormal distributions for all values of m.
On qfunctional equations and excursion moments
, 2005
"... We analyse qfunctional equations arising from treelike combinatorial structures, which are counted by size, internal path length and certain generalisations thereof. The corresponding counting parameters are labelled by an integer k> 1. We show the existence of a joint limit distribution for these ..."
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Cited by 5 (1 self)
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We analyse qfunctional equations arising from treelike combinatorial structures, which are counted by size, internal path length and certain generalisations thereof. The corresponding counting parameters are labelled by an integer k> 1. We show the existence of a joint limit distribution for these parameters in the limit of infinite size, if the size generating function has a square root as dominant singularity. The limit distribution coincides with that of integrals of (k − 1)th powers of the standard Brownian excursion. Our method yields a recursion for the moments of the joint distribution and admits an extension to other types of singularities. 1
Pseudoknot RNA structures with arclenght � 4
 J. Comp. bio
, 2008
"... Abstract. In this paper we study knoncrossing RNA structures with arclength ≥ 3, i.e. RNA molecules in which for any i, the nucleotides labeled i and i + j (j = 1,2) cannot form a bond and in which there are at most k − 1 mutually crossing arcs. Let Sk,3(n) denote their number. Based on a novel fu ..."
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Cited by 5 (3 self)
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Abstract. In this paper we study knoncrossing RNA structures with arclength ≥ 3, i.e. RNA molecules in which for any i, the nucleotides labeled i and i + j (j = 1,2) cannot form a bond and in which there are at most k − 1 mutually crossing arcs. Let Sk,3(n) denote their number. Based on a novel functional equation for the generating function P n≥0 Sk,3(n)z n, we derive for arbitrary k ≥ 3 exponential growth factors and for k = 3 the subexponential factor. Our main result is the derivation of the formula S3,3(n) ∼ 6.11170·4! n(n−1)...(n−4) 4.54920n. 1.
A Hybrid of Darboux’s Method and Singularity Analysis in Combinatorial Asymptotics, in "The Electronic
 n o 1, June 2006, R103, http://www.combinatorics.org/Volume_13/PDF/v13i1r103.pdf. Algo 11
"... Abstract. A “hybrid method”, dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux’s method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy su ..."
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Cited by 5 (1 self)
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Abstract. A “hybrid method”, dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux’s method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions—this, even in the case when the unit circle is a natural boundary. A prime application is to coefficients of several types of infinite product generating functions, for which full asymptotic expansions (involving periodic fluctuations at higher orders) can be derived. Examples relative to permutations, trees, and polynomials over finite fields are treated in this way.
RNA pseudoknot structures with arclength ≥ 3 and stacklength ≥ σ. submitted
"... Abstract. In this paper we enumerate knoncrossing RNA pseudoknot structures with given minimum arc and stacklength. That is, we study the numbers of RNA pseudoknot structures with arclength ≥ 3, stacklength ≥ σ and in which there are at most k − 1 mutually crossing bonds, denoted by T [3] k,σ ( ..."
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Cited by 3 (0 self)
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Abstract. In this paper we enumerate knoncrossing RNA pseudoknot structures with given minimum arc and stacklength. That is, we study the numbers of RNA pseudoknot structures with arclength ≥ 3, stacklength ≥ σ and in which there are at most k − 1 mutually crossing bonds, denoted by T [3] k,σ (n). In particular we prove that the numbers of 3, 4 and 5noncrossing RNA structures with arclength ≥ 3 and stacklength ≥ 2 satisfy T [3] 3,2 (n) ∼ K3 n−52.5723n,
DESTRUCTION OF VERY SIMPLE TREES
"... Abstract. We consider the total cost of cutting down a random rooted tree chosen from a family of socalled very simple trees (which include ordered trees, dary trees, and Cayley trees); these form a subfamily of simply generated trees. At each stage of the process an edge is chose at random from t ..."
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Cited by 2 (1 self)
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Abstract. We consider the total cost of cutting down a random rooted tree chosen from a family of socalled very simple trees (which include ordered trees, dary trees, and Cayley trees); these form a subfamily of simply generated trees. At each stage of the process an edge is chose at random from the tree and cut, separating the tree into two components. In the onesided variant of the process the component not containing the root is discarded, whereas in the twosided variant both components are kept. The process ends when no edges remain for cutting. The cost of cutting an edge from a tree of size n is assumed to be n α. Using singularity analysis and the method of moments, we derive the limiting distribution of the total cost accrued in both variants of this process. A salient feature of the limiting distributions obtained (after normalizing in a familyspecific manner) is that they only depend on α. 1.
ISOLATING NODES IN RECURSIVE TREES
"... Abstract. We consider the number of random cuts that are necessary to isolate the node with label λ, 1 ≤ λ ≤ n, in a random recursive tree of size n. At each stage of the edgeremoval procedure considered an edge is chosen at random from the tree and cut, separating the tree into two subtrees. The p ..."
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Cited by 2 (1 self)
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Abstract. We consider the number of random cuts that are necessary to isolate the node with label λ, 1 ≤ λ ≤ n, in a random recursive tree of size n. At each stage of the edgeremoval procedure considered an edge is chosen at random from the tree and cut, separating the tree into two subtrees. The procedure is then continued with the subtree containing the specified label λ, whereas the other subtree is discarded. The procedure stops when the node with label λ is isolated. Using a recursive approach we are able to give asymptotic expansions for all ordinary moments of the random variable Xn,λ, which counts the number of random cuts required to isolate the vertex with label λ in a random sizen recursive tree, for small labels, i. e., λ = l, and large labels, i. e., λ = n + 1 − l, with l ≥ 1 fixed and n → ∞. Moreover, we can characterize the limiting distribution of a scaled variant of Xn,λ, for the instance of large labels. 1.
Central and local limit theorems for RNA structures
 J. Theor. Biol
, 2007
"... Abstract. A knoncrossing RNA pseudoknot structure is a graph over {1,..., n} without 1arcs, i.e. arcs of the form (i, i + 1) and in which there exists no kset of mutually intersecting arcs. In particular, RNA secondary structures are 2noncrossing RNA structures. In this paper we prove a central ..."
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Cited by 2 (1 self)
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Abstract. A knoncrossing RNA pseudoknot structure is a graph over {1,..., n} without 1arcs, i.e. arcs of the form (i, i + 1) and in which there exists no kset of mutually intersecting arcs. In particular, RNA secondary structures are 2noncrossing RNA structures. In this paper we prove a central and a local limit theorem for the distribution of the numbers of 3noncrossing RNA structures over n nucleotides with exactly h bonds. We will build on the results of [?] and [?], where the generating function of knoncrossing RNA pseudoknot structures and the asymptotics for its coefficients have been derived. The results of this paper explain the findings on the numbers of arcs of RNA secondary structures obtained by molecular folding algorithms and predict the distributions for knoncrossing RNA folding algorithms which are currently being developed. 1.
A REPERTOIRE FOR ADDITIVE FUNCTIONALS OF UNIFORMLY DISTRIBUTED mARY SEARCH TREES
"... Abstract. Using recent results on singularity analysis for Hadamard products of generating functions, we obtain the limiting distributions for additive functionals on mary search trees on n keys with toll sequence (i) nα with α ≥ 0 (α = 0 and α = 1 correspond roughly to the space requirement and to ..."
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Cited by 2 (2 self)
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Abstract. Using recent results on singularity analysis for Hadamard products of generating functions, we obtain the limiting distributions for additive functionals on mary search trees on n keys with toll sequence (i) nα with α ≥ 0 (α = 0 and α = 1 correspond roughly to the space requirement and total path length, respectively); (ii) ln ` n ´, which corresponds to the som−1 called shape functional; and (iii) 1n=m−1, which corresponds to the number of leaves. 1.
Additive functionals on random search trees
 Department of Mathematical Sciences, The Johns Hopkins University
, 2003
"... Search trees are fundamental data structures in computer science. We study functionals on random search trees that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the socalled shape functional fall under t ..."
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Cited by 1 (1 self)
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Search trees are fundamental data structures in computer science. We study functionals on random search trees that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the socalled shape functional fall under this framework. Our goal is to derive asymptotics of moments and identify limiting distributions of these functionals under two commonly studied probability models—the random permutation model and the uniform model. For the random permutation model, our approach is based on establishing transfer theorems that link the order of growth of the input into a particular (deterministic) recurrence to the order of growth of the output. For the uniform model, our approach is based on the complexanalytic tool of singularity analysis. To facilitate a systematic analysis of these additive functionals we extend singularity analysis, a class of methods by which one can translate on a termbyterm basis an asymptotic expansion of a functional around its dominant singularity into a corresponding expansion for the Taylor coefficients of the function. The most important extension is the determination of how singularities are composed under the operation of Hadamard product of analytic power series. The transfer theorems derived are used in conjunction with the method of moments to establish limit laws for mary search trees under the random permutation model. For the uniform model on binary search trees, the extended singularity analysis toolkit is employed to establish the asymptotic behavior of the moments of a wide class of functionals. These asymptotics are used, again in conjunction with the method of moments, to derive limit laws.