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On path entropy functions for rooted trees

A. Meir, J.W. Moon
1995 Discrete Mathematics  
Ifu is a terminal node of a rooted tree T. with n terminal nodes, let h(u) = ~f (d(v)) where the sum is over all interior nodes v in the path from the root of T. to u, d(v) is the out-degree of v, and  ...  cost function f corresponding to the interior node v; and the path entropy function h(T.)= •h(u), where the sum is over all terminal nodes u of T., is a measure of the complexity of T.. showed that if  ...  Hoede have considered the problem of determining the structure of trees T, that minimize h(T,,) for certain cost functions f(k) in: "On an Optimality Property of Ternary Trees", Information and Control  ... 
doi:10.1016/0012-365x(94)00214-4 fatcat:sw4mbkffebbu7ovm3jo37dqya4

Optimal Organizational Hierarchies: Source Coding: Disaster Relief [article]

G. Rama Murthy
2011 arXiv   pre-print
In this effort two novel concepts : prefix free path, graph entropy are introduced. Some results of graph entropy are provided. Also some results on Kraft inequality are discussed.  ...  ulticasting is an important communication paradigm for enabling the dissemination of information selectively.  ...  , assign him a path of small length from root node in the D-ary tree.  ... 
arXiv:1111.6149v1 fatcat:tpwuflflkncpnoyigvfriimere

Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs [article]

Gil Cohen
2015 arXiv   pre-print
In the language of theoretical computer science, our work resolves the problem of explicitly constructing two-source dispersers for polylogarithmic entropy.  ...  In his 1947 paper that inaugurated the probabilistic method, Erdős proved the existence of 2n-Ramsey graphs on n vertices.  ...  Acknowledgement I wish to thank Ran Raz and Avi Wigderson for their warm encouragement. On a personal note, it is uncustomary to acknowledge one's partner in life in mathematical papers.  ... 
arXiv:1506.04428v1 fatcat:bnla4jwtuje55c62rdinrcboqm

Informational Divergence and Entropy Rate on Rooted Trees with Probabilities [article]

Georg Böcherer, Rana Ali Amjad
2013 arXiv   pre-print
Rooted trees with probabilities are used to analyze properties of a variable length code. A bound is derived on the difference between the entropy rates of the code and a memoryless source.  ...  The bound is used to derive converses for exact random number generation, resolution coding, and distribution matching.  ...  TABLE I CHAIN I RULES ON ROOTED TREES WITH PROBABILITIES. function increment un-normalized normalized . I, we display various instances of the LANSIT.  ... 
arXiv:1310.2882v1 fatcat:nvol37wjtvdepgbxri5wdbnf7a

Entropy Bounds for Hierarchical Molecular Networks

Matthias Dehmer, Stephan Borgert, Frank Emmert-Streib, Enrico Scalas
2008 PLoS ONE  
In addition to the derivation of the entropy bounds, we provide a numerical analysis for two special graph classes, rooted trees and generalized trees, and demonstrate hereby not only the computational  ...  In this paper we derive entropy bounds for hierarchical networks.  ...  Acknowledgments We would like to thank the anonymous referee as well as Danail Bonchev and Claus Grupen for fruitful discussions and valuable comments. Author Contributions  ... 
doi:10.1371/journal.pone.0003079 pmid:18769487 pmcid:PMC2518648 fatcat:m2nair4zrffdbg3nmro3o7kvkm

A survey of recent results in (generalized) graph entropies [article]

Xueliang Li, Meiqin Wei
2015 arXiv   pre-print
indices and generalized graph entropies, complexity for calculation of graph entropies.  ...  Various applications of graph entropies together with some open problems and conjectures are also presented for further research.  ...  [33] studied bounds for the entropies of rooted trees and so-called generalized trees. Here we give the entropy bounds of rooted trees first. Theorem 2.15 [33] Let T be a rooted tree.  ... 
arXiv:1505.04658v2 fatcat:7ixzutkuvbbyrmejeese7linha

Decision Trees, Protocols, and the Fourier Entropy-Influence Conjecture [article]

Andrew Wan and John Wright and Chenggang Wu
2013 arXiv   pre-print
This conjecture is one of the major open problems in the analysis of Boolean functions, and settling it would have several interesting consequences.  ...  Here, H[f^2] denotes the Shannon entropy of f's spectral distribution, and Inf[f] is the total influence of f.  ...  a bound on the entropy of X using a prefix-free protocol for Y.  ... 
arXiv:1312.3003v1 fatcat:6xw5ii2xgvef3mizwjnqbulaxe

Page 473 of Journal of Chemometrics Vol. 5, Issue 5 [page]

1991 Journal of Chemometrics  
Certainty factors are equal to the product of logistic values along the classification path (from the root to a leaf of the tree).  ...  Crisp classification trees often use a depth first search, for which a single path leads to the category designation at the leaf of the tree.  ... 

Optimal pruning with applications to tree-structured source coding and modeling

P.A. Chou, T. Lookabaugh, R.M. Gray
1989 IEEE Transactions on Information Theory  
A concentration on the first of these and a detailed analysis of variable-rate tree-structured vector quantization are provided.  ...  These include variable-rate and minimum-entropy tree-structured vector quantization, minimum expected cost decision trees, variable-order Markov modeling, optimum bit allocation, and computer graphics  ...  The vector variable U maintains a running account of U( .) for the tree as it is pruned. The first inner "whle" loop searches a path beginning at the root for the node with the minimum X ( t ) .  ... 
doi:10.1109/18.32124 fatcat:rbvjz2ydrfcazmgpy6gmkhnbfm

Rooted Trees with Probabilities Revisited [article]

Georg Böcherer
2013 arXiv   pre-print
Using the differential LANSIT, Pinsker's inequality is formulated for rooted trees with probabilities, with an application to the approximation of product distributions.  ...  Rooted trees with probabilities are convenient to represent a class of random processes with memory.  ...  / 34 Path Length Lemma [3, Lemma 2.1] Function w (j) := length of path to node j. For each j ∈ N \ 0: ∆w (j) = 1. w (0) = 0.  ... 
arXiv:1302.0753v1 fatcat:faliosr6tbftfabr7opdb2aeei

Best basis for joint representation: The median of marginal best bases for low cost information exchanges in distributed signal representation

Abdourrahmane M. Atto, Kavé Salamatian, Philippe Bolon
2014 Information Sciences  
The paper provides algorithms for computing this median basis with respect to standard tree structured libraries of bases such as wavelet packet bases or cosine trees.  ...  a marginal best basis), the paper proposes the median basis computed on a set of best marginal bases for joint representation or fusion of distributed/dependent signals.  ...  Acknowledgements The authors address special thanks to Professor Muriel M EDARD for insightful discussions on distributive coding and compression.  ... 
doi:10.1016/j.ins.2014.06.040 fatcat:nr7ppxb5pbggnf5yhwy4y2ejia

Pebbling, Entropy, and Branching Program Size Lower Bounds

Balagopal Komarath, Jayalal Sarma
2015 ACM Transactions on Computation Theory  
We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced in [4].  ...  Our main technique is the entropy method introduced by Jukna and Zak[6] originally in the context of proving lower bounds for read-once branching programs.  ...  A computation path on input x is a directed path from the root (the start state) and each edge in the path is consistent with x. At least one such computation must end in a final state.  ... 
doi:10.1145/2751320 fatcat:v7vms43mdfhwzd7dny3aczsele

Computing quadratic entropy in evolutionary trees

Drago Bokal, Matt DeVos, Sandi Klavžar, Aki Mimoto, Arne Ø. Mooers
2011 Computers and Mathematics with Applications  
Furthermore, on ultrametric phylogenetic trees, the maximum of quadratic entropy is a measure of pairwise evolutionary distinctness in conservation biology, introduced by S. Pavoine.  ...  This fact allows us to deduce some efficient algorithms for computing the quadratic entropy in the case of given tip weights, which may be useful for community biodiversity measures.  ...  In a rooted tree, every vertex v ∈ V (T ) has a unique incident edge e v that lies on the shortest path connecting v with r. The component of T − e v containing v is the subtree T v rooted at v.  ... 
doi:10.1016/j.camwa.2011.09.030 fatcat:nrydqmwpjbgjhc2k2xkyvgxzly

INFORMATION-THEORETIC CONCEPTS FOR THE ANALYSIS OF COMPLEX NETWORKS

Matthias Dehmer
2008 Applied Artificial Intelligence  
The main contribution of this article is a method for determining the structural information content of graphs that is based on a tree decomposition.  ...  ., entropy-based approaches are powerful tools to investigate complex systems on a graph or sequence-based level (Bonchev and Rouvray 2005; Bonchev 1983; Claussen 2007a; Kullback 1959; Kullback and Leibler  ...  The level of a vertex t in a rooted tree T is simply the length of the path from r to t. The path with the largest path length from the root to a leaf is denoted as h.  ... 
doi:10.1080/08839510802164101 fatcat:h6e2vwy2bbbqdc7ls3fqp2t2ka

Fast Computation of Entropic Profiles for the Detection of Conservation in Genomes [chapter]

Matteo Comin, Morris Antonello
2013 Lecture Notes in Computer Science  
We show that Fast EP is suitable for large genomes and for the discovery of motifs with unbounded length.  ...  In this paper we discuss and improve the function Entropic Profile, introduced by Vinga and Almeida in [23] .  ...  Figure 2 shows an example of suffix tree for the string s = T CGGCGGCAAC. We can observe that each suffix of the string s is present in the tree as a labeled path from the root to a leaf.  ... 
doi:10.1007/978-3-642-39159-0_25 fatcat:bhehvqtrbfe2hiwjsvkknpuvsm
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