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Journal articles on the topic 'Partition tree'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Mu, Yashuang, Lidong Wang, and Xiaodong Liu. "Dynamic programming based fuzzy partition in fuzzy decision tree induction." Journal of Intelligent & Fuzzy Systems 39, no. 5 (November 19, 2020): 6757–72. http://dx.doi.org/10.3233/jifs-191497.

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Fuzzy decision trees are one of the most popular extensions of decision trees for symbolic knowledge acquisition by fuzzy representation. Among the majority of fuzzy decision trees learning methods, the number of fuzzy partitions is given in advance, that is, there are the same amount of fuzzy items utilized in each condition attribute. In this study, a dynamic programming-based partition criterion for fuzzy items is designed in the framework of fuzzy decision tree induction. The proposed criterion applies an improved dynamic programming algorithm used in scheduling problems to establish an optimal number of fuzzy items for each condition attribute. Then, based on these fuzzy partitions, a fuzzy decision tree is constructed in a top-down recursive way. A comparative analysis using several traditional decision trees verify the feasibility of the proposed dynamic programming based fuzzy partition criterion. Furthermore, under the same framework of fuzzy decision trees, the proposed fuzzy partition solution can obtain a higher classification accuracy than some cases with the same amount of fuzzy items.

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ARKIN, ESTHER M., DELIA GARIJO, ALBERTO MÁRQUEZ, JOSEPH S. B. MITCHELL, and CARLOS SEARA. "SEPARABILITY OF POINT SETS BY k-LEVEL LINEAR CLASSIFICATION TREES." International Journal of Computational Geometry & Applications 22, no. 02 (April 2012): 143–65. http://dx.doi.org/10.1142/s0218195912500021.

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Let R and B be sets of red and blue points in the plane in general position. We study the problem of computing a k-level binary space partition (BSP) tree to classify/separate R and B, such that the tree defines a linear decision at each internal node and each leaf of the tree corresponds to a (convex) cell of the partition that contains only red or only blue points. Specifically, we show that a 2-level tree can be computed, if one exists, in time O(n2). We show that a minimum-level (3 ≤ k ≤ log n) tree can be computed in time nO( log n). In the special case of axis-parallel partitions, we show that 2-level and 3-level trees can be computed in time O(n), while a minimum-level tree can be computed in time O(n5).

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3

Wood, David R. "On tree-partition-width." European Journal of Combinatorics 30, no. 5 (July 2009): 1245–53. http://dx.doi.org/10.1016/j.ejc.2008.11.010.

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Proe, M. F., and P. Millard. "Effect of N supply upon the seasonal partitioning of N and P uptake in young Sitka spruce (Piceasitchensis)." Canadian Journal of Forest Research 25, no. 10 (October 1, 1995): 1704–9. http://dx.doi.org/10.1139/x95-184.

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Seasonal relationships between N supply, tree growth, and partitioning of both N and P have been studied in young trees using 15N and 32P isotopes. Three-year-old clonal cuttings of Sitka spruce (Piceasitchensis (Bong.) Carr.) were grown for 2 years in sand irrigated with a nutrient solution containing either 1.0 mol N•m−3 (low N) or 6.0 mol N•m−3 (high N). In the first year, trees received 2-week pulses of 15N and 32P to label current nutrient uptake during either a period of rapid spring growth or shortly after bud set in summer. In the second year, trees that had been preconditioned to a low-N supply received 3-week pulses of 15N at either the low rate of application or at the high rate to simulate a single application of N fertilizer. In spring of the first year, N treatment had no effect upon tree growth. Low-N trees increased the partition of 15N uptake to roots, but the partition of 32P was not affected by N supply and was similar to the partition of 15N in the high-N treatment. At the time of the later pulse, however, growth was affected by N supply and 32P partitioning to roots increased to match the partition of 15N in the low-N treatment. During the second year, the additional 15N given to Low-N trees to simulate fertilizer application was partitioned predominantly to current shoots and roots. Results are discussed in relation to the processes of internal cycling and the partition of nutrients taken up by larger trees.

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Katrenič, Peter, and Gabriel Semanišin. "On a tree-partition problem." Electronic Notes in Discrete Mathematics 28 (March 2007): 325–30. http://dx.doi.org/10.1016/j.endm.2007.01.046.

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HURTADO, FERRAN, GIUSEPPE LIOTTA, and DAVID R. WOOD. "PROXIMITY DRAWINGS OF HIGH-DEGREE TREES." International Journal of Computational Geometry & Applications 23, no. 03 (June 2013): 213–30. http://dx.doi.org/10.1142/s0218195913500088.

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A drawing of a given (abstract) tree that is a minimum spanning tree of the vertex set is considered aesthetically pleasing. However, such a drawing can only exist if the tree has maximum degree at most 6. What can be said for trees of higher degree? We approach this question by supposing that a partition or covering of the tree by subtrees of bounded degree is given. Then we show that if the partition or covering satisfies some natural properties, then there is a drawing of the entire tree such that each of the given subtrees is drawn as a minimum spanning tree of its vertex set.

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7

Gao, Jing. "Decision Tree Generation Algorithm without Pruning." Applied Mechanics and Materials 441 (December 2013): 731–37. http://dx.doi.org/10.4028/www.scientific.net/amm.441.731.

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On the generation of decision tree based on rough set model, for the sake of classification accuracy, existing algorithms usually partition examples too specific. And it is hard to avoid the negative impact caused by few special examples on decision tree. In order to obtain this priority in traditional decision tree algorithm based on rough set, the sample is partitioned much more meticulously. Inevitably, a few exceptional samples have negative effect on decision tree. And this leads that the generated decision tree seems too large to be understood. It also reduces the ability in classifying and predicting the coming data. To settle these problems, the restrained factor is introduced in this paper. For expanding nodes in generating decision tree algorithm, besides traditional terminating condition, an additional terminating condition is involved when the restrained factor of sample is higher than a given threshold, then the node will not be expanded any more. Thus, the problem of much more meticulous partition is avoided. Furthermore, the size of decision tree generated with restrained factor involved will not seem too large to be understood.

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8

Shan, Gui Jun. "Partition Real Data in Decision Tree Using Statistical Criterion." Applied Mechanics and Materials 380-384 (August 2013): 1469–72. http://dx.doi.org/10.4028/www.scientific.net/amm.380-384.1469.

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Partition methods for real data play an extremely important role in decision tree algorithms in data mining and machine learning because the decision tree algorithms require that the values of attributes are discrete. In this paper, we propose a novel partition method for real data in decision tree using statistical criterion. This method constructs a statistical criterion to find accurate merging intervals. In addition, we present a heuristic partition algorithm to achieve a desired partition result with the aim to improve the performance of decision tree algorithms. Empirical experiments on UCI real data show that the new algorithm generates a better partition scheme that improves the classification accuracy of C4.5 decision tree than existing algorithms.

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Guttmann-Beck, Nili, and Refael Hassin. "Approximation algorithms for minimum tree partition." Discrete Applied Mathematics 87, no. 1-3 (October 1998): 117–37. http://dx.doi.org/10.1016/s0166-218x(98)00052-3.

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10

Tang, Jing, Yinghui Wang, Ningna Wang, Xiaojuan Ning, Ke Lyu, Liansheng Sui, and Zhenghao Shi. "Swaying Tree Simulation by Slicing Partition." Chinese Journal of Electronics 29, no. 5 (September 1, 2020): 826–32. http://dx.doi.org/10.1049/cje.2020.07.004.

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11

Lu, H., J. C. Woods, and M. Ghanbari. "Image segmentation by binary partition tree." Electronics Letters 42, no. 17 (2006): 966. http://dx.doi.org/10.1049/el:20061398.

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KENNY, BRIAN G., and TONY W. DIXON. "AMBIGUITY IN THE DETERMINATION OF THE FREE ENERGY ASSOCIATED WITH THE CRITICAL CIRCLE MAP." ANZIAM Journal 50, no. 2 (October 2008): 177–84. http://dx.doi.org/10.1017/s1446181108000291.

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AbstractWe consider a simple model to describe the widths of the mode-locked intervals for the critical circle map. By using two different partitions of the rational numbers based on Farey series and Farey tree levels, respectively, we calculate the free energy analytically at selected points for each partition. It emerges that the result of the calculation depends on the method of partition. An implication of this finding is that the generalized dimensions Dq are different for the two types of partition except when q=0; that is, only the Hausdorff dimension is the same in both cases.

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13

Robinson, Alan. "Partition complexes, duality and integral tree representations." Algebraic & Geometric Topology 4, no. 2 (October 22, 2004): 943–60. http://dx.doi.org/10.2140/agt.2004.4.943.

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14

Maocheng, Cai, and Yuan Xudong. "Applications of matroid partition to tree decomposition." Acta Mathematicae Applicatae Sinica 16, no. 2 (April 2000): 199–204. http://dx.doi.org/10.1007/bf02677680.

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15

Ning, Baoling, Jianzhong Li, and Shouxu Jiang. "Balanced tree partition problems with virtual nodes." Journal of Combinatorial Optimization 37, no. 4 (September 29, 2018): 1249–65. http://dx.doi.org/10.1007/s10878-018-0351-1.

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Guttmann-Beck, Nili, and Refael Hassin. "Approximation Algorithms for Min–Max Tree Partition." Journal of Algorithms 24, no. 2 (August 1997): 266–86. http://dx.doi.org/10.1006/jagm.1996.0848.

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17

Neath, Andrew A. "Polya tree distributions for statistical modeling of censored data." Journal of Applied Mathematics and Decision Sciences 7, no. 3 (January 1, 2003): 175–86. http://dx.doi.org/10.1155/s1173912603000166.

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Polya tree distributions extend the idea of the Dirichlet process as a prior for Bayesian nonparametric problems. Finite dimensional distributions are defined through conditional probabilities in P. This allows for a specification of prior information which carries greater weight where it is deemed appropriate according to the choice of a partition of the sample space. Muliere and Walker[7] construct a partition so that the posterior from right censored data is also a Polya tree. A point of contention is that the specification of the prior is partially dependent on the data. In general, the posterior from censored data will be a mixture of Polya trees. This paper will present a straightforward method for determining the mixing distribution.

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18

Budinich, Renato, and Gerlind Plonka. "A tree-based dictionary learning framework." International Journal of Wavelets, Multiresolution and Information Processing 18, no. 05 (June 18, 2020): 2050041. http://dx.doi.org/10.1142/s0219691320500411.

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We propose a new outline for adaptive dictionary learning methods for sparse encoding based on a hierarchical clustering of the training data. Through recursive application of a clustering method, the data is organized into a binary partition tree representing a multiscale structure. The dictionary atoms are defined adaptively based on the data clusters in the partition tree. This approach can be interpreted as a generalization of a discrete Haar wavelet transform. Furthermore, any prior knowledge on the wanted structure of the dictionary elements can be simply incorporated. The computational complexity of our proposed algorithm depends on the employed clustering method and on the chosen similarity measure between data points. Thanks to the multiscale properties of the partition tree, our dictionary is structured: when using Orthogonal Matching Pursuit to reconstruct patches from a natural image, dictionary atoms corresponding to nodes being closer to the root node in the tree have a tendency to be used with greater coefficients.

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19

Chen, Keh-Hsun, and Zbigniew W. Raś. "Homogeneous Information Trees." Fundamenta Informaticae 8, no. 1 (January 1, 1985): 123–49. http://dx.doi.org/10.3233/fi-1985-8110.

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By a homogeneous information tree we mean a tree with internal nodes labeled by attributes, edges by values of attributes and terminal nodes by sets of objects. Sets labeling terminal nodes form a partition of the set of objects classified by a homogeneous tree. Homogeneous information trees can be interpreted as models of expert systems, data bases with a menu or tree-structured data bases. The main problem we are dealing with concerns a minimization of a tree with respect to the storage cost. We propose a heuristic polynomial algorithm to construct an optimal tree.

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20

Bakir, Ilke, Natashia Boland, Brian Dandurand, and Alan Erera. "Sampling Scenario Set Partition Dual Bounds for Multistage Stochastic Programs." INFORMS Journal on Computing 32, no. 1 (January 2020): 145–63. http://dx.doi.org/10.1287/ijoc.2018.0885.

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We consider multistage stochastic programming problems in which the random parameters have finite support, leading to optimization over a finite scenario set. There has been recent interest in dual bounds for such problems, of two types. One, known as expected group subproblem objective (EGSO) bounds, require solution of a group subproblem, which optimizes over a subset of the scenarios, for all subsets of the scenario set that have a given cardinality. Increasing the subset cardinality in the group subproblem improves bound quality, (EGSO bounds form a hierarchy), but the number of group subproblems required to compute the bound increases very rapidly. Another is based on partitions of the scenario set into subsets. Combining the values of the group subproblems for all subsets in a partition yields a partition bound. In this paper, we consider partitions into subsets of (nearly) equal cardinality. We show that the expected value of the partition bound over all such partitions also forms a hierarchy. To make use of these bounds in practice, we propose random sampling of partitions and suggest two enhancements to the approach: sampling partitions that align with the multistage scenario tree structure and use of an auxiliary optimization problem to discover new best bounds based on the values of group subproblems already computed. We establish the effectiveness of these ideas with computational experiments on benchmark problems. Finally, we give a heuristic to save computational effort by ceasing computation of a partition partway through if it appears unpromising.

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Zhongzhi Li, and Xuegang Wang. "Spatial Clustering Algorithm Based on Hierarchical-Partition Tree." International Journal of Digital Content Technology and its Applications 4, no. 6 (September 30, 2010): 26–35. http://dx.doi.org/10.4156/jdcta.vol4.issue6.3.

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22

Cordone, Roberto, and Francesco Maffioli. "On the complexity of graph tree partition problems." Discrete Applied Mathematics 134, no. 1-3 (January 2004): 51–65. http://dx.doi.org/10.1016/s0166-218x(03)00340-8.

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23

Chen, He, Zemin Jin, Xueliang Li, and Jianhua Tu. "Heterochromatic tree partition numbers for complete bipartite graphs." Discrete Mathematics 308, no. 17 (September 2008): 3871–78. http://dx.doi.org/10.1016/j.disc.2007.07.085.

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24

Wu, Bang Ye, Hung-Lung Wang, Shih Ta Kuan, and Kun-Mao Chao. "On the uniform edge-partition of a tree." Discrete Applied Mathematics 155, no. 10 (May 2007): 1213–23. http://dx.doi.org/10.1016/j.dam.2006.10.012.

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Jin, Zemin, Shili Wen, and Shujun Zhou. "Heterochromatic tree partition problem in complete tripartite graphs." Discrete Mathematics 312, no. 4 (February 2012): 789–802. http://dx.doi.org/10.1016/j.disc.2011.11.005.

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Jin, Zemin, and Peipei Zhu. "Heterochromatic tree partition number in complete multipartite graphs." Journal of Combinatorial Optimization 28, no. 2 (October 20, 2012): 321–40. http://dx.doi.org/10.1007/s10878-012-9557-9.

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PardÀs, Montse, and Philippe Salembier. "Segmentation of video sequences for partition tree generation." Annales Des Télécommunications 52, no. 7-8 (July 1997): 388–96. http://dx.doi.org/10.1007/bf02998458.

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28

Prabha, R., and R. Kalaiyarasi. "Isometric Path Partition Problem on Tree Derived Architectures." Journal of Physics: Conference Series 1770, no. 1 (March 1, 2021): 012067. http://dx.doi.org/10.1088/1742-6596/1770/1/012067.

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Bazylevych, R. P., R. A. Melnyk, and O. G. Rybak. "Circuit Partitioning for FPGAs by the Optimal Circuit Reduction Method." VLSI Design 11, no. 3 (January 1, 2000): 237–48. http://dx.doi.org/10.1155/2000/58485.

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Mathematically the most difficult partitioning problem–packaging–is being considered. Its purpose is to minimize a number of partitions and to satisfy the constraints on the number of constituent elements and external nets. To solve the problem, the Optimal Circuit Reduction Method, suggested by R. Bazylevych is being used. The optimal reduction tree to reflect the hierarchical entrance of smaller clusters into bigger ones is being built for the first step. At the second step we select one or more tree vertices which better meet the given constraints and are the first partitions generated from. After creating every new partition we eliminate its elements from the circuit and repeat the procedure to complete all partitions. During the last stage optimization strategies to exchange some elements between the partitions are being used. Better or equivalent results among known tests confirm the effectiveness of this method.

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Zouggar, Souad Taleb, and Abdelkader Adla. "Proposal for Measuring Quality of Decision Trees Partition." International Journal of Decision Support System Technology 9, no. 4 (October 2017): 16–36. http://dx.doi.org/10.4018/ijdsst.2017100102.

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To compute a partition quality for a decision tree, we propose a new measure called NIM “New Information Measure”. The measure is simpler, provides similar performance, and sometimes outperforms the existing measures used with tree-based methods. The experimental results using the MONITDIAB application (Taleb & Atmani, 2013) and datasets from the UCI repository (Asuncion & Newman, 2007) confirm the classification capabilities of our proposal in comparison to the Shannon measure used with ID3 and C4.5 decision tree methods.

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Kumar, Sunil, Saroj Ratnoo, and Jyoti Vashishtha. "HYPER HEURISTIC EVOLUTIONARY APPROACH FOR CONSTRUCTING DECISION TREE CLASSIFIERS." Journal of Information and Communication Technology 20, Number 2 (February 21, 2021): 249–76. http://dx.doi.org/10.32890/jict2021.20.2.5.

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Decision tree models have earned a special status in predictive modeling since these are considered comprehensible for human analysis and insight. Classification and Regression Tree (CART) algorithm is one of the renowned decision tree induction algorithms to address the classification as well as regression problems. Finding optimal values for the hyper parameters of a decision tree construction algorithm is a challenging issue. While making an effective decision tree classifier with high accuracy and comprehensibility, we need to address the question of setting optimal values for its hyper parameters like the maximum size of the tree, the minimum number of instances required in a node for inducing a split, node splitting criterion and the amount of pruning. The hyper parameter setting influences the performance of the decision tree model. As researchers, we know that no single setting of hyper parameters works equally well for different datasets. A particular setting that gives an optimal decision tree for one dataset may produce a sub-optimal decision tree model for another dataset. In this paper, we present a hyper heuristic approach for tuning the hyper parameters of Recursive and Partition Trees (rpart), which is a typical implementation of CART in statistical and data analytics package R. We employ an evolutionary algorithm as hyper heuristic for tuning the hyper parameters of the decision tree classifier. The approach is named as Hyper heuristic Evolutionary Approach with Recursive and Partition Trees (HEARpart). The proposed approach is validated on 30 datasets. It is statistically proved that HEARpart performs significantly better than WEKA’s J48 algorithm in terms of error rate, F-measure, and tree size. Further, the suggested hyper heuristic algorithm constructs significantly comprehensible models as compared to WEKA’s J48, CART and other similar decision tree construction strategies. The results show that the accuracy achieved by the hyper heuristic approach is slightly less as compared to the other comparative approaches.

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DOBRINEN, NATASHA, and DAN HATHAWAY. "THE HALPERN–LÄUCHLI THEOREM AT A MEASURABLE CARDINAL." Journal of Symbolic Logic 82, no. 4 (December 2017): 1560–75. http://dx.doi.org/10.1017/jsl.2017.31.

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AbstractSeveral variants of the Halpern–Läuchli Theorem for trees of uncountable height are investigated. Forκweakly compact, we prove that the various statements are all equivalent, and hence, the strong tree version holds for one tree on any weakly compact cardinal. For any finited≥ 2, we prove the consistency of the Halpern–Läuchli Theorem ondmany normalκ-trees at a measurable cardinalκ, given the consistency of aκ+d-strong cardinal. This follows from a more general consistency result at measurableκ, which includes the possibility of infinitely many trees, assuming partition relations which hold in models of AD.

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De Felice, Francesca, Laia Humbert-Vidan, Mary Lei, Andrew King, and Teresa Guerrero Urbano. "Analyzing oropharyngeal cancer survival outcomes: a decision tree approach." British Journal of Radiology 93, no. 1111 (July 2020): 20190464. http://dx.doi.org/10.1259/bjr.20190464.

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Objectives: To analyze survival outcomes in patients with oropharygeal cancer treated with primary intensity modulated radiotherapy (IMRT) using decision tree algorithms. Methods: A total of 273 patients with newly diagnosed oropharyngeal cancer were identified between March 2010 and December 2016. The data set contained nine predictor variables and a dependent variable (overall survival (OS) status). The open-source R software was used. Survival outcomes were estimated by Kaplan–Meier method. Important explanatory variables were selected using the random forest approach. A classification tree that optimally partitioned patients with different OS rates was then built. Results: The 5 year OS for the entire population was 78.1%. The top three important variables identified were HPV status, N stage and early complete response to treatment. Patients were partitioned in five groups on the basis of these explanatory variables. Conclusion: The proposed classification tree could help to guide future research in oropharyngeal cancer field. Advances in knowledge: Decision tree method seems to be an appropriate tool to partition oropharyngeal cancer patients.

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Apandi, Tri Herdiawan, Roby Bayu Maulana, Rian Piarna, and Dwi Vernanda. "MENGANALISIS KEMUNGKINAN KETERLAMBATAN PEMBAYARAN SPP DENGAN ALGORITMA C4.5 (STUDI KASUS POLITEKNIK TEDC BANDUNG)." Jurnal Techno Nusa Mandiri 16, no. 2 (September 5, 2019): 93–98. http://dx.doi.org/10.33480/techno.v16i2.659.

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Payment of tuition as one of the sources of funds, plays an important role in the sustainability of the operations of higher education. The problem that arises is that students are not often late to make payments in a timely manner. One of the factors causing the many cases of late payment of tuition fees due to lack of policy and decisive action on the part of the campus when students are late in making payments, besides the factors of parents and students also have an influence on the delay. The purpose of this study is to classify students who are late and timely in making SPP payments using the C4.5 algorithm. From the total sample used then divided into 4 partitions, partition 1 for 90% training data and 10% testing data, partition 2 for 80% training data and 20% testing data, and partition 3 for 70% training data and 30% testing data , and partition 4 for 60% training data and 40% testing data. The classification results of the C4.5 algorithm are evaluated and validated with cross validation and confusion matrix to determine the accuracy of the C4.5 algorithm in predicting late SPP payments. Based on the comparison of the results of evaluations and validations conducted, it shows that data partition 2 has a better level of accuracy than the other partitions, which is 75%. Keywords: Data Mining, Decision Tree (C4.5), SPP.

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Aboueisha, Hassan, Victor Manuel Calo, Konrad Jopek, Mikhail Moshkov, Anna Paszyńka, Maciej Paszyński, and Marcin Skotniczny. "Element Partition Trees For H-Refined Meshes to Optimize Direct Solver Performance. Part I: Dynamic Programming." International Journal of Applied Mathematics and Computer Science 27, no. 2 (June 27, 2017): 351–65. http://dx.doi.org/10.1515/amcs-2017-0025.

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AbstractWe consider a class of two- and three-dimensional h-refined meshes generated by an adaptive finite element method. We introduce an element partition tree, which controls the execution of the multi-frontal solver algorithm over these refined grids. We propose and study algorithms with polynomial computational cost for the optimization of these element partition trees. The trees provide an ordering for the elimination of unknowns. The algorithms automatically optimize the element partition trees using extensions of dynamic programming. The construction of the trees by the dynamic programming approach is expensive. These generated trees cannot be used in practice, but rather utilized as a learning tool to propose fast heuristic algorithms. In this first part of our paper we focus on the dynamic programming approach, and draw a sketch of the heuristic algorithm. The second part will be devoted to a more detailed analysis of the heuristic algorithm extended for the case of hp-adaptive grids.

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GALANIS, ANDREAS, DANIEL ŠTEFANKOVIČ, and ERIC VIGODA. "Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models." Combinatorics, Probability and Computing 25, no. 4 (February 2, 2016): 500–559. http://dx.doi.org/10.1017/s0963548315000401.

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Recent inapproximability results of Sly (2010), together with an approximation algorithm presented by Weitz (2006), establish a beautiful picture of the computational complexity of approximating the partition function of the hard-core model. Let λc($\mathbb{T}_{\Delta}$) denote the critical activity for the hard-model on the infinite Δ-regular tree. Weitz presented anFPTASfor the partition function when λ < λc($\mathbb{T}_{\Delta}$) for graphs with constant maximum degree Δ. In contrast, Sly showed that for all Δ ⩾ 3, there exists εΔ> 0 such that (unless RP = NP) there is noFPRASfor approximating the partition function on graphs of maximum degree Δ for activities λ satisfying λc($\mathbb{T}_{\Delta}$) < λ < λc($\mathbb{T}_{\Delta}$) + εΔ.We prove that a similar phenomenon holds for the antiferromagnetic Ising model. Sinclair, Srivastava and Thurley (2014) extended Weitz's approach to the antiferromagnetic Ising model, yielding anFPTASfor the partition function for all graphs of constant maximum degree Δ when the parameters of the model lie in the uniqueness region of the infinite Δ-regular tree. We prove the complementary result for the antiferromagnetic Ising model without external field, namely, that unless RP = NP, for all Δ ⩾ 3, there is noFPRASfor approximating the partition function on graphs of maximum degree Δ when the inverse temperature lies in the non-uniqueness region of the infinite tree$\mathbb{T}_{\Delta}$. Our proof works by relating certain second moment calculations for random Δ-regular bipartite graphs to the tree recursions used to establish the critical points on the infinite tree.

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Brinkworth, Andrew R., Robert Sansom, and Matthew A. Wills. "Phylogenetic incongruence and homoplasy in the appendages and bodies of arthropods: why broad character sampling is best." Zoological Journal of the Linnean Society 187, no. 1 (May 9, 2019): 100–116. http://dx.doi.org/10.1093/zoolinnean/zlz024.

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Abstract Notwithstanding the rapidly increasing sampling density of molecular sequence data, morphological characters still make an important contribution to our understanding of the evolutionary relationships of arthropod groups. In many clades, characters relating to the number and morphological specialization of appendages are ascribed particular phylogenetic significance and may be preferentially sampled. However, previous studies have shown that partitions of morphological character matrices often imply significantly different phylogenies. Here, we ask whether a similar incongruence is observed in the appendage and non-appendage characters of arthropods. We apply tree length (incongruence length difference, ILD) and tree distance (incongruence relationship difference, IRD) tests to these partitions in an empirical sample of 53 published neontological datasets for arthropods. We find significant incongruence about one time in five: more often than expected, but markedly less often than in previous partition studies. We also find similar levels of homoplasy in limb and non-limb characters, both in terms of internal consistency and consistency relative to molecular trees. Taken together, these findings imply that sampled limb and non-limb characters are of similar phylogenetic utility and quality, and that a total evidence approach to their analysis is preferable.

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Vasconcelos, Raquel, Eudald Pujol-Buxó, Gustavo A. Llorente, Ahmed Saeed, and Salvador Carranza. "Micro-Hotspots for Conservation: An Umbrella Tree Species for the Unique Socotran Reptile Fauna." Forests 11, no. 3 (March 21, 2020): 353. http://dx.doi.org/10.3390/f11030353.

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Umbrella species are defined as species that can be rare and sensitive to human disturbance, whose protection may confer the protection of other co-occurring species. The dragon’s blood tree Dracaena cinnabari Balf.f. was already considered an umbrella species on Socotra Island (Indic Ocean, Yemen) due to its ecological importance for some native biota. We studied the reptile community living on D. cinnabari from Socotra Island. We sampled reptiles on trees across most D. cinnabari populations and applied co-occurrence and network partition analyses to check if the presence of reptiles on D. cinnabari populations was random or structured. Regardless of its patched and scarce actual distribution, we report the use of this tree as a habitat by more than half of the reptile community (12 endemic reptiles). Co-occurrence and network partition analyses demonstrate that this community is structured across the distribution of dragon’s blood trees, reflecting complex allopatric, vicariant, and biotic interaction processes. Hence, these trees act as micro-hotspots for reptiles, that is, as areas where endemic and rare species that are under threat at the landscape scale co-occur. This Socotra endemic tree is currently threatened by overgrazing, overmaturity, and climate change. Its protection and declaration as an umbrella species are expected to benefit the reptile community and to protect evolutionary processes that are partially driven by the ecological links between reptiles and this tree. To our knowledge, no tree species has been proposed as an umbrella species for island vertebrate endemics so far, highlighting the ecological uniqueness of Socotra Island.

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Dekking, F. M., and L. E. Meester. "An almost sure result for path lengths in binary search trees." Advances in Applied Probability 35, no. 02 (June 2003): 363–76. http://dx.doi.org/10.1017/s0001867800012295.

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This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variableZ. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜mof finite binary trees with at mostmnodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector timesZ.

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40

Dekking, F. M., and L. E. Meester. "An almost sure result for path lengths in binary search trees." Advances in Applied Probability 35, no. 2 (June 2003): 363–76. http://dx.doi.org/10.1239/aap/1051201652.

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This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variable Z. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜m of finite binary trees with at most m nodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector times Z.

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41

Park, Jong In, Young Po Lee, and Seok Ho Yoon. "Partition-Based Hybrid MIMO Decoding Schemes with Combined Depth- and Breath-First Search." Applied Mechanics and Materials 284-287 (January 2013): 2652–56. http://dx.doi.org/10.4028/www.scientific.net/amm.284-287.2652.

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In this paper, we propose a novel maximum likelihood (ML) decoding scheme based on the combination of depth- and breadth-first search methods on a partitioned tree for multiple input multiple output systems. The proposed scheme first partitions the searching tree into several stages, each of which is then searched by a depth- or breadth-first search method, possibly exploiting the advantages of both the depth- and breadth-first search methods in an organized way. Numerical results indicate that, when the depth- and breadth-first search algorithms are adopted appropriately, the proposed scheme exhibits substantially lower computational complexity than conventional ML decoders while maintaining the ML bit error performance.

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42

ITO, TAKEHIRO, XIAO ZHOU, and TAKAO NISHIZEKI. "PARTITIONING TREES OF SUPPLY AND DEMAND." International Journal of Foundations of Computer Science 16, no. 04 (August 2005): 803–27. http://dx.doi.org/10.1142/s0129054105003303.

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Assume that a tree T has a number ns of "supply vertices" and all the other vertices are "demand vertices." Each supply vertex is assigned a positive number called a supply, while each demand vertex is assigned a positive number called a demand. One wishes to partition T into exactly ns subtrees by deleting edges from T so that each subtree contains exactly one supply vertex whose supply is no less than the sum of demands of all demand vertices in the subtree. The "partition problem" is a decision problem to ask whether T has such a partition. The "maximum partition problem" is an optimization version of the partition problem. In this paper, we give three algorithms for the problems. The first is a linear-time algorithm for the partition problem. The second is a pseudo-polynomial-time algorithm for the maximum partition problem. The third is a fully polynomial-time approximation scheme (FPTAS) for the maximum partition problem.

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Silva, Jorge F., and Shrikanth Narayanan. "Complexity-Regularized Tree-Structured Partition for Mutual Information Estimation." IEEE Transactions on Information Theory 58, no. 3 (March 2012): 1940–52. http://dx.doi.org/10.1109/tit.2011.2177771.

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Shiono, Yasunori, Tadaaki Kirishima, Yoshinori Ueda, and Kensei Tsuchida. "Drawing Algorithm for Fuzzy Graphs Using the Partition Tree." Journal of Advanced Computational Intelligence and Intelligent Informatics 16, no. 5 (July 20, 2012): 641–52. http://dx.doi.org/10.20965/jaciii.2012.p0641.

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Fuzzy graphs have been used frequently and effectively as a method for sociogram analysis. A fuzzy graph has the fundamental characteristic of being able to express a variety of relationships between nodes. The drawing of fuzzy graphs has been studied in computer-aided analysis systems with human interfaces and methods using genetic algorithms. However, computer-aided analysis systems with human interfaces do not provide for automatic drawing, while methods using genetic algorithms have the defect of requiring too much execution time for finding a locally optimum solution. To overcome these defects, we propose an algorithm for drawing intelligible and comprehensive fuzzy graphs using a partition tree. This method automatically draws the fuzzy graphwith nodes arranged on the intersections of a latticed space. Since nodes are optimally arranged on the latticed intersections and put together at a nearby position in accordance with the transition of clusters according to cluster levels in the partition tree, drawing the algorithm makes fuzzy relations easier to understand through fuzzy graph representation. Moreover, fuzzy graphs can be drawn faster than by conventional methods. This paper describes the algorithm and its verification by introducing a system implementing the method for displaying fuzzy graphs. Moreover, we have carried out a case study in which a questionnaire has been administered to students, allowing us to analyze human relations quantitatively using a method based on fuzzy theory. Human relations are represented as fuzzy graphs by our algorithm and analyzed using the fuzzy graph.

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Jian-Qiang, HAO, GONG Yun-Zhan, Tan Li, and Duan Da-Gao. "Apply Partition Tree to Compute Canonical Labelings of Graphs." International Journal of Grid and Distributed Computing 9, no. 5 (May 31, 2016): 241–64. http://dx.doi.org/10.14257/ijgdc.2016.9.5.21.

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Cheng, Qing, Zhong Liu, Jincai Huang, and Guangquan Cheng. "Community detection in hypernetwork via Density-Ordered Tree partition." Applied Mathematics and Computation 276 (March 2016): 384–93. http://dx.doi.org/10.1016/j.amc.2015.12.039.

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Cordone, Roberto. "A subexponential algorithm for the coloured tree partition problem." Discrete Applied Mathematics 155, no. 10 (May 2007): 1326–35. http://dx.doi.org/10.1016/j.dam.2007.02.001.

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Wang, Qing Ren. "A flexible tree design in an edit-partition scheme." Pattern Recognition Letters 5, no. 4 (April 1987): 261–65. http://dx.doi.org/10.1016/0167-8655(87)90055-9.

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Wißmann, Thorsten, Hans-Peter Deifel, Stefan Milius, and Lutz Schröder. "From generic partition refinement to weighted tree automata minimization." Formal Aspects of Computing 33, no. 4-5 (March 23, 2021): 695–727. http://dx.doi.org/10.1007/s00165-020-00526-z.

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AbstractPartition refinement is a method for minimizing automata and transition systems of various types. Recently, we have developed a partition refinement algorithm that is generic in the transition type of the given system and matches the run time of the best known algorithms for many concrete types of systems, e.g. deterministic automata as well as ordinary, weighted, and probabilistic (labelled) transition systems. Genericity is achieved by modelling transition types as functors on sets, and systems as coalgebras. In the present work, we refine the run time analysis of our algorithm to cover additional instances, notably weighted automata and, more generally, weighted tree automata. For weights in a cancellative monoid we match, and for non-cancellative monoids such as (the additive monoid of) the tropical semiring even substantially improve, the asymptotic run time of the best known algorithms. We have implemented our algorithm in a generic tool that is easily instantiated to concrete system types by implementing a simple refinement interface. Moreover, the algorithm and the tool are modular, and partition refiners for new types of systems are obtained easily by composing pre-implemented basic functors. Experiments show that even for complex system types, the tool is able to handle systems with millions of transitions.

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ZHAO, YAN, YIYU YAO, and JINGTAO YAO. "LEVEL-WISE CONSTRUCTION OF DECISION TREES FOR CLASSIFICATION." International Journal of Software Engineering and Knowledge Engineering 16, no. 01 (February 2006): 103–26. http://dx.doi.org/10.1142/s0218194006002690.

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A partition-based framework is presented for a formal study of classification problems. An information table is used as a knowledge representation, in which all basic notions are precisely defined by using a language known as the decision logic language. Solutions to, and solution space of, classification problems are formulated in terms of partitions. Algorithms for finding solutions are modelled as searching in a space of partitions under the refinement order relation. We focus on a particular type of solutions called conjunctively definable partitions. Two level-wise methods for decision tree construction are investigated, which are related to two different strategies: local optimization and global optimization. They are not in competition with, but are complementary to each other. Experimental results are reported to evaluate the two methods.

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