Depth Of Tree Research Articles

Decision trees with minimum average depth for sorting eight elements

We prove that the minimum average depth of a decision tree for sorting 8 pairwise different elements is equal to 620160/8!. We show also that each decision tree for sorting 8 elements, which has minimum average depth (the number of such trees is approximately equal to 8.548×10326365), has also minimum depth. Both problems were considered by Knuth (1998). To obtain these results, we use tools based on extensions of dynamic programming which allow us to make sequential optimization of decision trees relative to depth and average depth, and to count the number of decision trees with minimum average depth.

Characteristics of Dried Soil Layers Under Apple Orchards of Different Ages and Their Applications in Soil Water Managements on the Loess Plateau of China

Negative soil water balance (i.e., water input < water output) can lead to soil desiccation and subsequently the occurrence of a dried soil layer (DSL). The DSLs are generally studied at a specific sampling depth (e.g., 500 cm), and the actual extent of DSLs remains unknown due to the challenge of collecting deep soil samples. To investigate the characteristics of actual DSLs under different ages of apple orchards and ascertain the optimal age of apple orchards for avoiding/controlling the formation of DSLs, soil samples were collected to a depth of 1 800 cm under apple orchards of different ages in Changwu on the Loess Plateau of China. As the ages increased, soil water content (SWC) and mean SWC in DSLs showed an overall decreasing trend, whereas while DSL thickness and the quantity of water deficit (QWD) in DSLs demonstrated an increasing trend. The DSL was the thickest (1 600 cm) under the 17-year-old orchard, the forming velocity of DSL thickness was the highest at the apple tree growth stage of 9–17 years (168 cm year−1), and the highest increasing velocity of QWD (–181 mm year−1) was also observed at this stage. The thickness of DSL was significantly correlated with growth age and root depth of apple trees (r > 0.88), whereas the QWD and mean SWC in DSLs were found to have no correlation with them. The optimal age of apple orchards for avoiding/controlling the formation of DSLs was about 9 years. This information provided pertinent references for the management of deep water resources by controlling the growth age of plants.

MFlexDT: multi flexible fuzzy decision tree for data stream classification

In many real-world applications, instances (data) arrive sequentially in the form of streams. Processing such data poses challenges to machine learning. While adhering to on-line learning strategies, in this paper we extend the Flexible Fuzzy Decision Tree (FlexDT) algorithm with multiple partitioning that makes it possible to carry out automatic on-line fuzzy data classification. The proposed method is aimed to balance accuracy and tree size in data stream mining. The objective of the classification problem is to predict the true class of each incoming instances in real time. In terms of evaluation of the method, accuracy, tree depth, and the learning time are significant factors influencing the performance. A series of experiments demonstrate that the proposed method produces optimal trees for both numeric and nominal features (variables).

A Bitmap based Data Encryption Scheme in Cloud Computing

With the development of cloud computing, the interest on spatial database outsourcing has been sharply increasing. Therefore, researches on data encryption methods for protecting location data privacy in outsourced databases have been actively performed. However, the existing data encryption schemes do not consider data distribution when generating data clusters and they use a tree-based index for processing k-nearest neighbor queries. As a result, the performance of query processing algorithm can fluctuate depending on the tree depth. To solve these problems, we propose a densityaware data encryption scheme and a query processing algorithm for database outsourcing. Our density-aware data encryption scheme uses a grid index to generate clusters and transforms the original data information into a bitmap. To provide efficient query processing, we use an algebraic coding based hash index to reduce the data retrieval time. Finally, in performance analysis, we show that the proposed scheme provides better query processing performance and guarantees the privacy of users, compared with the existing schemes.

A Bitmap based Data Encryption Scheme in Cloud Computing

With the development of cloud computing, the interest on spatial database outsourcing has been sharply increasing. Therefore, researches on data encryption methods for protecting location data privacy in outsourced databases have been actively performed. However, the existing data encryption schemes do not consider data distribution when generating data clusters and they use a tree-based index for processing k-nearest neighbor queries. As a result, the performance of query processing algorithm can fluctuate depending on the tree depth. To solve these problems, we propose a densityaware data encryption scheme and a query processing algorithm for database outsourcing. Our density-aware data encryption scheme uses a grid index to generate clusters and transforms the original data information into a bitmap. To provide efficient query processing, we use an algebraic coding based hash index to reduce the data retrieval time. Finally, in performance analysis, we show that the proposed scheme provides better query processing performance and guarantees the privacy of users, compared with the existing schemes.

The Fine Classification of Conjunctive Queries and Parameterized Logarithmic Space

We perform a fundamental investigation of the complexity of conjunctive query evaluation from the perspective of parameterized complexity. We classify sets of Boolean conjunctive queries according to the complexity of this problem. Previous work showed that a set of conjunctive queries is fixed-parameter tractable precisely when the set is equivalent to a set of queries having bounded treewidth. We present a fine classification of query sets up to parameterized logarithmic space reduction. We show that, in the bounded treewidth regime, there are three complexity degrees and that the properties that determine the degree of a query set are bounded pathwidth and bounded tree depth. We also engage in a study of the two higher degrees via logarithmic space machine characterizations and complete problems. Our work yields a significantly richer perspective on the complexity of conjunctive queries and, at the same time, suggests new avenues of research in parameterized complexity.

On Low Tree-Depth Decompositions

The theory of sparse structures usually uses tree like structures as building blocks. In the context of sparse/dense dichotomy this role is played by graphs with bounded tree depth. In this paper we survey results related to this concept and particularly explain how these graphs are used to decompose and construct more complex graphs and structures. In more technical terms we survey some of the properties and applications of low tree depth decomposition of graphs.

A robust and efficient curve skeletonization algorithm for tree-like objects using minimum cost paths

Conventional curve skeletonization algorithms using the principle of Blum's transform, often, produce unwanted spurious branches due to boundary irregularities, digital effects, and other artifacts. This paper presents a new robust and efficient curve skeletonization algorithm for three-dimensional (3-D) elongated fuzzy objects using a minimum cost path approach, which avoids spurious branches without requiring post-pruning. Starting from a root voxel, the method iteratively expands the skeleton by adding new branches in each iteration that connects the farthest quench voxel to the current skeleton using a minimum cost path. The path-cost function is formulated using a novel measure of local significance factor defined by the fuzzy distance transform field, which forces the path to stick to the centerline of an object. The algorithm terminates when dilated skeletal branches fill the entire object volume or the current farthest quench voxel fails to generate a meaningful skeletal branch. Accuracy of the algorithm has been evaluated using computer-generated phantoms with known skeletons. Performance of the method in terms of false and missing skeletal branches, as defined by human experts, has been examined using in vivo CT imaging of human intrathoracic airways. Results from both experiments have established the superiority of the new method as compared to the existing methods in terms of accuracy as well as robustness in detecting true and false skeletal branches. The new algorithm makes a significant reduction in computation complexity by enabling detection of multiple new skeletal branches in one iteration. Specifically, this algorithm reduces the number of iterations from the number of terminal tree branches to the worst case performance of tree depth. In fact, experimental results suggest that, on an average, the order of computation complexity is reduced to the logarithm of the number of terminal branches of a tree-like object.

Anomalous scaling in an age-dependent branching model.

We introduce a one-parametric family of tree growth models, in which branching probabilities decrease with branch age τ as τ(-α). Depending on the exponent α, the scaling of tree depth with tree size n displays a transition between the logarithmic scaling of random trees and an algebraic growth. At the transition (α=1) tree depth grows as (logn)(2). This anomalous scaling is in good agreement with the trend observed in evolution of biological species, thus providing a theoretical support for age-dependent speciation and associating it to the occurrence of a critical point.

How Health Managers Can Use Data Mining for Predicting Individuals’ Risks of Contracting Nosocomial Pneumonia

This article explains how managers can use a new data-mining technique for solving problems related to individual risks of contracting nosocomial pneumonia. Using the genetic algorithm, a search technique provides practitioners with an optimal choice of parameters for Gini boosting type decision tree models. Thus, managers and technicians can choose better models. These new parameters are genetically controlled: number of trees, depth of trees, trimming factor, cross-validation (to avoid overfitting), proportion of the population used, and the minimum size to split a node. This technique has been satisfactorily tested on health data.

Optimizing the Adaptive Fast Multipole Method for Fractal Sets

We have performed a detailed analysis of the fast multipole method (FMM) in the adaptive case, in which the depth of the FMM tree is nonuniform. Previous works in this area have focused mostly on special types of adaptive distributions, for example, when points accumulate on a two-dimensional manifold or accumulate around a few points in space. Instead, we considered a more general situation in which fractal sets, e.g., Cantor sets and generalizations, are used to create adaptive sets of points. Such sets are characterized by their dimension, a number between 0 and 3. We introduced a mathematical framework to define a converging sequence of octrees, and based on that, demonstrated how to increase $N \to \infty$. A new complexity analysis for the adaptive FMM is introduced. It is shown that the ${\cal{O}}(N)$ complexity is achievable for any distribution of particles, when a modified adaptive FMM is exploited. We analyzed how the FMM performs for fractal point distributions, and how optimal parameters can be picked, e.g., the criterion used to stop the subdivision of an FMM cell. A new subdividing double-threshold method is introduced, and better performance demonstrated. Parameters in the FMM are modeled as a function of particle distribution dimension, and the optimal values are obtained. A three-dimensional kernel independent black box adaptive FMM is implemented and used for all calculations.

OTSP: an optimised time synchronisation protocol for wireless sensor networks

In this work, we propose a new time synchronisation protocol for wireless sensor networks WSNs, called OTSP for Optimised Time Synchronisation Protocol. The proposed OTSP works in three phases: neighbour discovery phase NDP, centre node-localisation phase, and synchronisation phase. The purpose of neighbour discovery phase is to provide neighbours' tables to the base station BS. Using neighbours' tables, the centre node is the localised and a hierarchical tree is created. After that, the centre node will synchronise its direct neighbours and will select among them a subset of coordinators nodes. Coordinators will then propagate the synchronisation message and will also select another set of coordinators to continue the synchronisation process. This process continues until the leaf nodes are synchronised. Simulations show the superiority of OTSP, and that it outperforms TPSN, and RTSP in terms of synchronisation tree depth, energy efficiency and synchronisation error, and TSST only in term of synchronisation error.

A Hybrid Structure for Data Aggregation in Wireless Sensor Network

In recent years, wireless sensor networks have been used for various applications such as environmental monitoring, military and medical applications. A wireless sensor network uses a large number of sensor nodes that continuously collect and send data from a specific region to a base station. Data from sensors are collected from the study area in the common scenario of sensor networks. Afterward, sensed data is sent to the base station. However, neighboring sensors often lead to redundancy of data. Transmission of redundant data to the base station consumes energy and produces traffic, because process is run in a large network. Data aggregation was proposed in order to reduce redundancy in data transformation and traffic. The most popular communication protocol in this field is cluster based data aggregation. Clustering causes energy balance, but sometimes energy consumption is not efficient due to the long distance between cluster heads and base station. In another communication protocol, which is based on a tree construction, because of the short distance between the sensors, energy consumption is low. In this data aggregation approach, since each sensor node is considered as one of the vertices of a tree, the depth of tree is usually high. In this paper, an efficient hierarchical hybrid approach for data aggregation is presented. It reduces energy consumption based on clustering and minimum spanning tree. The benefit of combining clustering and tree structure is reducing the disadvantages of previous structures. The proposed method firstly employs clustering algorithm and then a minimum spanning tree is constructed based on cluster heads. Our proposed method was compared to LEACH which is a well-known data aggregation method in terms of energy consumption and the amount of energy remaining in each sensor network lifetime. Simulation results indicate that our proposed method is more efficient than LEACH algorithm considering energy consumption.

Reusing Genetic Programming for Ensemble Selection in Classification of Unbalanced Data

Classification algorithms can suffer from performance degradation when the class distribution is unbalanced. This paper develops a two-step approach to evolving ensembles using genetic programming (GP) for unbalanced data. The first step uses multiobjective (MO) GP to evolve a Pareto-approximated front of GP classifiers to form the ensemble by trading-off the minority and the majority class against each other during learning. The MO component alleviates the reliance on sampling to artificially rebalance the data. The second step, which is the focus this paper, proposes a novel ensemble selection approach using GP to automatically find/choose the best individuals for the ensemble. This new GP approach combines multiple Pareto-approximated front members into a single composite genetic program solution to represent the (optimized) ensemble. This ensemble representation has two main advantages/novelties over traditional genetic algorithm (GA) approaches. First, by limiting the depth of the composite solution trees, we use selection pressure during evolution to find small highly-cooperative groups of individuals for the ensemble. This means that ensemble sizes are not fixed a priori (as in GA), but vary depending on the strength of the base learners. Second, we compare different function set operators in the composite solution trees to explore new ways to aggregate the member outputs and thus, control how the ensemble computes its output. We show that the proposed GP approach evolves smaller more diverse ensembles compared to an established ensemble selection algorithm, while still performing as well as, or better than the established approach. The evolved GP ensembles also perform well compared to other bagging and boosting approaches, particularly on tasks with high levels of class imbalance.

Diagnosis of constant faults in read-once contact networks over finite bases

We study the depth of decision trees for diagnosis of constant 0 and 1 faults in read-once contact networks over finite bases containing only indecomposable networks. For each basis, we obtain a linear upper bound on the minimum depth of decision trees depending on the number of edges in the networks. For bases containing networks with at most 10 edges we find coefficients for linear bounds which are close to sharp.

Dendritic morphology predicts pattern recognition performance in multi-compartmental model neurons with and without active conductances.

In this paper we examine how a neuron’s dendritic morphology can affect its pattern recognition performance. We use two different algorithms to systematically explore the space of dendritic morphologies: an algorithm that generates all possible dendritic trees with 22 terminal points, and one that creates representative samples of trees with 128 terminal points. Based on these trees, we construct multi-compartmental models. To assess the performance of the resulting neuronal models, we quantify their ability to discriminate learnt and novel input patterns. We find that the dendritic morphology does have a considerable effect on pattern recognition performance and that the neuronal performance is inversely correlated with the mean depth of the dendritic tree. The results also reveal that the asymmetry index of the dendritic tree does not correlate with the performance for the full range of tree morphologies. The performance of neurons with dendritic tapering is best predicted by the mean and variance of the electrotonic distance of their synapses to the soma. All relationships found for passive neuron models also hold, even in more accentuated form, for neurons with active membranes.

Evaluating statistical learning methods for cell type classification and feature selection using RNA-seq data

Materials and methods We evaluated several statistical learning methods for classifying RNA-seq data using samples obtained from different brain regions as surrogates for single cell data. These include the ventral tegmental area (VTA, n=36), the nucleus accumbens core (AcbC, n=34) and shell (AcbS, n=30) [1]. We tested hierarchical clustering methods using different distance matrices (correlation, uncentered, abscor) and clustering algorithms. Multiscale bootstrap was used to evaluate the statistical significance of the clusters. We also tested the Bayesian Hierarchical Clustering method, which uses hypothesis testing to decide which cluster mergers increase the tree quality and calculates the optimum tree depth.

최소신장트리를 위한 크루스칼 알고리즘의 효율적인 구현

본 논문에서는 최소신장트리를 구하는 크루스칼 알고리즘의 효율적인 구현 방법을 제시한다. 제시하는 방법은 union-find 자료구조를 이용하며, 노드 집합을 나타내는 각 트리의 깊이를 줄이기 위해 union 연산시 루트까지의 경로에 있는 노드들의 위치를 최종 루트의 자식노드로 직접 이동하여 깊이를 줄이도록 하는 방법이다. 이 방법은 루트까지의 경로를 축소하고 노드의 레벨을 축소시킴으로써 트리의 깊이도 줄일 수 있다. 트리의 깊이가 줄어든다면 노드가 속하는 트리의 루트를 찾는 시간을 줄일 수 있게 되어 효율적인 방법이라 할 수 있다. 본 장에서 제안하는 방법을 그래프로 평가해보고 분석해 본 결과, 기존의 union() 방법이나 경로축소방법인 union2() 보다 트리의 깊이를 작게 유지함을 알 수 있다. In this paper, we present an efficient implementation of Kruskal's algorithm to obtain a minimum spanning tree. The proposed method utilizes the union-find data structure, reducing the depth of the tree of the node set by making the nodes in the path to root be the child node of the root of combined tree. This method can reduce the depth of the tree by shortening the path to the root and lowering the level of the node. This is an efficient method because if the tree's depth reduces, it could shorten the time of finding the root of the tree to which the node belongs. The performance of the proposed method is evaluated through the graphs generated randomly. The results showed that the proposed method outperformed the conventional method in terms of the depth of the tree.

Tree architecture varies with forest succession in evergreen broad-leaved forests in Eastern China

Vertical and temporal patterns of tree architecture and their relationship with woody plant crown exposure index varied with succession in evergreen broad-leaved forests in eastern China. Linking temporal pattern of tree architecture with changing light conditions through forest succession is important for understanding plant adaptive strategies. We determined vertical (canopy, sub-canopy, and understory species) and temporal (pioneer, mid-successional, and climax species) patterns of tree height, stem basal area, crown area and depth, leaf coverage, leaf convergence (clumped vs. dispersed leaves) and stretch direction of branches (vertical, leaned and horizontal branches) and their relationship with crown exposure index (CEI) for woody plants among three successional series in subtropical evergreen broad-leaved forests in eastern China. The series included three stages: secondary shrub (early-), young (mid-) and climax forests (late-successional stage). Tree height, crown area and depth, stem basal diameter and leaf coverage were the greatest in canopy trees, intermediate in sub-canopy trees, and the smallest in understory plants among the three successional stages; the above parameters and the proportions of dispersed leaves and leaned branches were climax > mid-successional > pioneer species. In contrast, the proportions of clumped leaves and vertical branches were pioneer > mid-successional > climax species. Between canopy and understory species, the patterns of branch stretch direction and leaf convergence were not consistent among the three successional stages. Tree height, crown area and depth, leaf coverage, and stem basal diameter were positively correlated with CEI for both vertically different species and successional species. Tree architectural traits were interactively affected by forest age and CEI (p < 0.01). In conclusion, tree architecture varies with changes in both forest vertical structure and successional status. The large variability of tree architectural traits in relation to CEI and forest age reflects a strong control of light availability that affects biomass allocation strategies of trees. Our study demonstrated that light induced shift in tree architecture may result in species coexistence through divergence of vertical space, as well as species replacement through forest succession.

Exact analysis of TTL cache networks

TTL caching models have recently regained significant research interest due to their connection to popular caching policies such as LRU. This paper advances the state-of-the-art analysis of TTL-based cache networks by developing two exact methods with orthogonal generality and computational complexity. The first method generalizes existing results for line networks under renewal requests to the broad class of caching policies whereby evictions are driven by stopping times; in addition to classical policies used in DNS and web caching, our stopping time model captures an emerging new policy implemented in SDN switches and Amazon web services. The second method further generalizes these results to feedforward networks with Markov arrival process (MAP) requests. MAPs are particularly suitable for non-line networks because they are closed not only under superposition and splitting, as known, but also under caching operations with phase-type (PH) TTL distributions. The crucial benefit of the two closure properties is that they jointly enable the first exact analysis of TTL feedforward cache networks in great generality. Moreover, numerical results highlight that existing Poisson approximations in binary-tree topologies are subject to relative errors as large as 30%, depending on the tree depth.