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An application of graphical enumeration to PA
 Journal of Symbolic Logic
, 2003
"... For α less than ε0 let Nα be the number of occurrences of ω in the Cantor normal form of α. Further let n  denote the binary length of a natural number n, let nh denote the htimes iterated binary length of n and let inv(n) be the least h such that nh ≤ 2. We show that for any natural number h ..."
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Cited by 12 (2 self)
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For α less than ε0 let Nα be the number of occurrences of ω in the Cantor normal form of α. Further let n  denote the binary length of a natural number n, let nh denote the htimes iterated binary length of n and let inv(n) be the least h such that nh ≤ 2. We show that for any natural number h first order Peano arithmetic, PA, does not prove the following sentence: For all K there exists an M which bounds the lengths n of all strictly descending sequences 〈α0,..., αn 〉 of ordinals less than ε0 which satisfy the condition that the Norm Nαi of the ith term αi is bounded by K + i  · ih. As a supplement to this (refined Friedman style) independence result we further show that e.g. primitive recursive arithmetic, PRA, proves that for all K there is an M which bounds the length n of any strictly descending sequence 〈α0,..., αn 〉 of ordinals less than ε0 which satisfies the condition that the Norm Nαi of the ith term αi is bounded by K +i· inv(i). The proofs are based on results from proof theory and techniques from asymptotic analysis of Polyastyle enumerations. Using results from Otter and from Matouˇsek and Loebl we obtain similar characterizations for finite bad sequences of finite trees in terms of Otter’s tree constant 2.9557652856.... ∗ Research supported by a HeisenbergFellowship of the Deutsche Forschungsgemeinschaft. † The main results of this paper were obtained during the authors visit of T. Arai in
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...
Towards effective elicitation of NINAND tree causal models
, 2009
"... Abstract. To specify a Bayes net (BN), a conditional probability table (CPT), often of an effect conditioned on its n causes, needs assessed for each node. It generally has the complexity exponential on n. NoisyOR reduces the complexity to linear, but can only represent reinforcing causal interacti ..."
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Cited by 5 (3 self)
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Abstract. To specify a Bayes net (BN), a conditional probability table (CPT), often of an effect conditioned on its n causes, needs assessed for each node. It generally has the complexity exponential on n. NoisyOR reduces the complexity to linear, but can only represent reinforcing causal interactions. The nonimpeding noisyAND (NINAND) tree is the first causal model that explicitly expresses reinforcement, undermining, and their mixture. It has linear complexity, but requires elicitation of a tree topology for types of causal interactions. We study their topology space and develop two novel techniques for more effective elicitation. 1
Y.: Enumerating unlabeled and root labeled trees for causal model acquisition
, 2009
"... Abstract. To specify a Bayes net (BN), a conditional probability table (CPT), often of an effect conditioned on its n causes, needs to be assessed for each node. It generally has the complexity exponential on n. The nonimpeding noisyAND (NINAND) tree is a recently developed causal model that redu ..."
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Cited by 5 (4 self)
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Abstract. To specify a Bayes net (BN), a conditional probability table (CPT), often of an effect conditioned on its n causes, needs to be assessed for each node. It generally has the complexity exponential on n. The nonimpeding noisyAND (NINAND) tree is a recently developed causal model that reduces the complexity to linear, while modeling both reinforcing and undermining interactions among causes. Acquisition of an NINAND tree model involves elicitation of a linear number of probability parameters and a tree structure. Instead of asking the human expert to describe the structure from scratch, in this work, we develop a twostep menu selection technique that aids structure acquisition. 1
A new bijection on rooted forests
 Discrete Mathematics
, 1993
"... Erdos, P. L., A new bijection on rooted forests, Discrete Mathematics 111 (1993) 179188.
This paper extends the method due to Szekely and ErdBs (1989) on the enumeration of trees.
A bijection is introduced on certain classes of rooted forests (more exactly, on the class of semilabelled forests). Th ..."
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Cited by 4 (1 self)
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Erdos, P. L., A new bijection on rooted forests, Discrete Mathematics 111 (1993) 179188.
This paper extends the method due to Szekely and ErdBs (1989) on the enumeration of trees.
A bijection is introduced on certain classes of rooted forests (more exactly, on the class of semilabelled forests). This method yields new easy proofs for some wellknown theorems which use only elementary calculations with the sum of stirling numbers.
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.
On the Expected Number of LevelI Nodes in a
, 2004
"... A queuebased Pruferlike code is used to determine the expected number of leveli nodes in a random labeled tree on n nodes. Level1 nodes are the leaves of a given tree and leveli nodes are leaves after all nodes in levels 1 through (i1) have been deleted. More precisely, we study the expect ..."
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A queuebased Pruferlike code is used to determine the expected number of leveli nodes in a random labeled tree on n nodes. Level1 nodes are the leaves of a given tree and leveli nodes are leaves after all nodes in levels 1 through (i1) have been deleted. More precisely, we study the expected fraction f(i) of n nodes that are in levels 1 through i. Tight bounds on f(i) are obtained and used to estimate the expected radius of a random labeled tree.