Set Partitioning Research Articles

An Aggregation Metric Based on Partitioning and Consensus for Asymmetric Distributions in Likert Scale Responses

A questionnaire is a basic tool for collecting information in survey research. Often, these questions are measured using a Likert scale. With multiple items on the same broad object, these codes could be summed or averaged to give an indication of each respondent’s overall positive or negative orientation towards that object. This is the basis for Likert scales. Aggregation methods have been widely used in different research areas. Most of them are mathematical methods, such as the arithmetic mean, the weighted arithmetic mean, or the OWA (Ordered Weighted Averaging) operator. The usual presentation of Likert scale derived data are Mean. This paper presents a new approach to compute an aggregate value that represents Likert scale responses as a histogram adequate to treat better than Mean with asymmetric distributions. This method generates a set of partitions using an approach based on successive division. After every division, each partition is evaluated using a consensus measure and the one with the best value is then selected. Once the process of division has finished, the aggregate value is computed using the resulting partitions. Promising results have been obtained. Experiments show that our method is appropriate for distributions with large asymmetry and is not far from the behavior of the arithmetic mean for symmetric distributions. Overall, the article sheds light on the need to consider other presentations of Likert scale derived data beyond Mean more suitable for asymmetric distributions.

Faà di Bruno Hopf algebras

This is a short review on the Faà di Bruno formulas, implementing composition of real-analytic functions, and a Hopf algebra associated to such formulas. This structure provides, among several other things, a short proof of the Lie-Scheffers theorem, and relates the Lagrange inversion formulas with antipodes. It is also the maximal commutative Hopf subalgebra of the one used by Connes and Moscovici to study diffeomorphisms in a noncommutative geometry setting. The link of Faà di Bruno formulas with the theory of set partitions is developed in some detail.

Some results on the multipartite Ramsey numbers mj(C3,Cm,n1K2,n2K2,…,niK2)

The graph Kj×t is a graph which is complete and multipartite which includes j partite sets and t vertices in each partite set. The multipartite Ramsey number (M-R-number) mj(G1,G2,…,Gn) is the smallest integer t for the mentioned graphs G1,G2,…,Gn, in a way which for each n-edge-coloring (G1,G2,…,Gn) of the edges of Kj×t, Gi contains a monochromatic copy of Gi for at least one i. The size of M-R-number mj(nK2,C7) for j≥2, n≤6, the M-R-number mj(nK2,C7) for j=2,3,4, n≥2, the M-R-number mj(nK2,C7) for each j≥5, n≥2, the M-R-number mj(C3,C3,n1K2,n2K2,…,niK2) for j≤6, and i,ni≥1, and the size of M-R-number mj(C3,C3,nK2) for j≥2 and n≥1 have been calculated in various articles hitherto. We acquire some bounds of M-R-number mj(C3,C3,n1K2,n2K2,…,niK2) in this essay in which i,j≥2, and ni≥1, also the size of M-R-number m4(C3,C4,nK2) for each n≥1 is computed in this paper.

LayerLock: Layer-Wise Collision-Free Multi-Robot Additive Manufacturing Using Topologically Interlocked Space-Filling Shapes

We present LayerLock, an approach for synchronous multi-robot additive manufacturing (cooperative 3D printing or C3DP). Our approach is based on Delaunay Lofts, a class of topologically interlocked shapes that are generated by stacking layers of Voronoi partitions of a set of moving Voronoi sites based on wallpaper symmetries. Our approach is based on two key insights. First, each layer of a Delaunay Loft is simply a tessellation of convex polygons allowing for easy division of cells for collision-free simultaneous material deposition. Second, the unique transition of Voronoi cells along the layers naturally leads to topological interlocking, thereby providing better energy absorption ability compensating for the loss of structural strength due to segmented printing. In this work, we constrain our current investigation to a two-robot system and develop the LayerLock algorithm consisting of three steps: (1) a distance-based division of the Voronoi cells at each layer of the Delaunay Loft, (2) a moving-front strategy for determining the sequence of cells for each robot, and (3) print path generation based on the cell sequence, which allows synchronous collaboration. We evaluate our algorithm for a range of geometric parameters such as part orientation and cell resolution. We also demonstrate it practically using a two-robot cooperative 3D printing platform.

Method for accelerating the joining of distributed datasets by a given criterion

Introduction: rapidly growing volumes of information pose new challenges to modern data analysis technologies. Currently, based on cost and performance considerations, data processing is usually performed in cluster systems. One of the most common related operations in analytics is the joins of datasets. Join is an extremely expensive operation that is difficult to scale and increase efficiency in distributed databases or systems based on the MapReduce paradigm. Despite the fact that a lot of effort has been put into improving the performance of this operation, often the proposed methods either require fundamental changes in the MapReduce structure, or are aimed at reducing the overhead of the operation, such as balancing the load on the network. Objective: to develop an algorithm to accelerate the integration of data sets in distributed systems. Results: a review of the Apache Spark architecture and the features of distributed computing based on MapReduce is performed, typical methods for combining datasets are analyzed, the main recommendations for optimizing the operation of combining data are presented, an algorithm that allows you to speed up the special case of combining implemented in Apache Spark is presented. This algorithm uses the methods of partitioning and partial transfer of sets to the computing nodes of the cluster, in such a way as to take advantage of the merge and broadcast associations. The experimental data presented demonstrate that the method is all the more effective the larger the volume of input data. So, for 2Tb compressed data, acceleration up to ~37% was obtained in comparison with standard Spark SQL.

Decomposition of zero divisor graph into cycles and stars

For a graph G and a subgraph H of G, an H-decomposition of G is a partition of the edge set of G into subsets Ei,1⩽i⩽k, such that each Ei induces a graph isomorphic to H. A graph Γ(ℝ) is said to be non-zero zero divisor graph of commutative ring ℝ with identity if u,v∈V(Γ(ℝ)) and (u,v)∈E(Γ(ℝ)) if and only if uv=0. It is prove that complete decomposible into cycle of length 4 of an H-decomposition of the zero divisor graph Γ(ℝ) where H is any simple connected graph. In particular, we give a complete solution to the problem in the case Zp×Zp×Zp×,…,×Zp (n times). For any positive integer n>2, there exists a decomposition of Γ(ℝ) into cycle and stars in a commutative ring ℝ. We show that the obvious the graph Γ(ℝ) is decomposition into cycle and stars. Overall, the proposed of the graph Γ(ℝ) has significantly improved the decomposing to algebraic structure which can be useful for networking. In this paper we investigate the concept of Γ(ℝ) is decomposition into cycles and stars as a commutative rings R=Zp×Zp,Zp×Zp×Zp,Zp×Zp×Zp×Zp and Zp×Zp×Zp×,…,×Zp with p is a prime number. It is prove that the zero divisor graph Γ(R) is complete decomposible into cycle of length 4 and star. In particular, we give a complete solution to the problem in the case Zp×Zp×Zp×,…,×Zp (n times). For any positive integer n>2, there exists a decomposition of Γ(R) into cycle and stars in a commutative ring ℝ. We show that the obvious the graph Γ(R) is decomposition into cycle and stars. Overall, the proposed of the graph Γ(R) has significantly improved the decomposing to algebraic structure which can be useful for networking area.

Detection and analysis of sweet potato defects based on hyperspectral imaging technology

In order to identify the defective sweet potato quickly, this study used hyperspectral imaging technology to classify healthy, frostbitten and diseased sweet potatoes. The training set and prediction set were divided according to the ratio of 3:1 by sample set partitioning based on joint x-y distances (SPXY) algorithm, and then the standard normal variable (SNV) pretreatment was carried out for the spectral data. In order to improve the running speed of the model and eliminate redundant variables, Monte Carlo uninformative variables elimination (MCUVE), Random frog (RF) algorithm and successive projections algorithm (SPA) were used to extract characteristic wavelengths. Finally, partial least square discrimination analysis (PLS-DA) and linear discrimination analysis (LDA) were used to establish classification models. The final results showed that the classification performance of the SPA-LDA model was the best, and the total accuracy of the prediction set reached 99.52%. In summary, hyperspectral imaging technology can accomplish the accurate detection of sweet potato defects, and provides feasible ideas for the automatic classification of sweet potatoes.

Gini index on generalized r-partitions

Abstract The Gini index of a set partition π of size n is defined as 1 − δ ( π ) n 2 , $1-\frac{\delta (\pi )}{{{n}^{2}}},$ where δ(π) is the sum of the squares of the block cardinalities of π. In this paper, we study the distribution of the δ statistic on various kinds of set partitions in which the first r elements are required to lie in distinct blocks. In particular, we derive the generating function for the distribution of δ on a generalized class of r-partitions wherein contents-ordered blocks are allowed and elements meeting certain restrictions may be colored. As a consequence, we obtain simple explicit formulas for the average δ value, equivalently for the average Gini index, in all r-partitions, r-permutations and r-Lah distributions of a given size. Finally, combinatorial proofs can be found for these formulas in the case r = 0 corresponding to the Gini index on classical set partitions, permutations and Lah distributions.

Collaboration graph for feature set partitioning in data classification

The curse of dimensionality of features in data classification is still an open issue. An approach to solve this problem is to partition features into several sub-sets of features hence the data classification task for every subset is performed. Then, an ensemble of these classifications are reported as the result of the classification problem. However, the feature set partitioning into sub-sets of features is still an area of research interest. Thus, in this paper, an innovative framework is proposed in which, first, a collaboration measure between each two features is defined and measured. Then, the collaboration graph, consisted of features as nodes and measured collaborations as edges’ weights, is generated according to the collaboration measures calculated. After that, a community detection method is used to find the graph communities. The communities are considered as the feature subsets and a base classifier is trained for each subset based on the corresponding training data of the subsets. Then, the ensemble classifier is created by a combination of base classifiers according to the AdaBoost Aggreagation. The simulation results of the proposed approach over the real and synthetic datasets indicate that the proposed approach considerably increases the classification accuracy in comparison to previous methods.

Hyperspectral Inversion of Soil Carbon and Nutrient Contents in the Yellow River Delta Wetland

Hyperspectral inversion techniques can facilitate soil quality monitoring and evaluation. In this study, the Yellow River Delta Wetland Nature Reserve was used as the study area. By measuring and analyzing soil samples under different vegetation types and collecting soil reflectance spectra, the relationships between vegetation types, soil depth, and the changes in soil total carbon (TC), total nitrogen (TN), and total phosphorus (TP) contents were assessed. The spectral data set was changed by spectral first derivative processing and division of the sample set according to vegetation type. The correlation between soil carbon, nitrogen, and phosphorus contents, and soil spectra was also analyzed, sensitive bands were selected, and the partial least-squares (PLS) method, support vector machine (SVM) method, and random forest (RF) model were used to establish the inversion model based on the characteristic bands. The optimal combination of spectral transformation, sample set partitioning, and inversion model was explored. The results showed significant differences (p < 0.05) in soil TC, TN, and TP contents under reed and saline alkali poncho vegetation, but not between soil element contents under different stratifications of the same plant species. The first derivative reflectance had higher correlation coefficients with soil TC, TN, and TP contents compared with the original reflectance, while the sensitive bands and quantities of the three elements differed. The division of the sample sets according to vegetation type and the first derivative treatment can improve the prediction accuracy of the model. The best combination of sample set plus FD plus RF for TC, TN, and TP in reed soil and sample set plus FD plus SVM for TC, TN, and TP in saline alkali pine soil provides technical support to further improve the prediction accuracy of TC, TN, and TP in wetland soil.

Multivariate calibration strategy in simultaneous determination of temperature properties of petroleum diesel by near infrared spectrometry

It is essential to control the quality of diesel products so that they comply with relevant fuel specifications, however, the quality assessments rely upon conventional wet chemical analyses that are costly and time consuming. Rapid, simultaneous quality measurement enabling immediate online optimisation for process control and blending offers tremendous cost savings by minimising product quality give-away, shipment demurrage, tank inventory, and laboratory analysis. In this study, the use of near infrared spectroscopy and chemometrics demonstrates a straightforward workflow for simultaneous determination of the petroleum diesel’s boiling point at 95% recovery (T95), flash point (FP), cloud point (CP), and cetane index (CI) calibration development. It involved appropriate spectral region selection, calibration/validation set partition, data pre-processing, regression modelling and validation. Based on the calibration and validation results, the supervised learning models that are obtained from a combination region of 4000–4800 cm−1 on a randomly selected calibration set managed to deliver promising predictive performance in terms of coefficient of determination for prediction (r2P/T95 ≥ 0.94, r2P/FP ≥ 0.89, r2P/CP ≥ 0.89, r2P/CI ≥ 0.993), root mean square error of prediction (RMSEP (T95) ≤ 5.2°C, RMSEP (FP) ≤ 2.0°C, RMSEP (CP) ≤ 2.4°C, RMSEP (CI) ≤ 0.3), and ratio of performance deviation (RPD (T95) ≥ 3.7, RPD (FP) ≥ 3.0, RPD (CP) ≥ 2.9, RPD (CI) ≥ 11). Regardless of principal component regression or partial least square regression on either the multiplicative scattering corrected spectra or Savitzky-Golay second derivative spectra, the developed models met respective ASTM reproducibility requirements, and were considered adequate for immediate quality assessment of diesel.

An optimal partition problem for the localization of eigenfunctions

We study the minimizers of a functional on the set of partitions of a domain $\Omega \subset R^n$ into $N$ subsets $W_j$ of locally finite perimeter in $\Omega$, whose main term is $\sum_{j=1^N} \int_{\Omega \cap \partial W_j} a(x) dH^{n--1}(x)$. Here the positive bounded function $a$ may for instance be related to the Landscape function of some Schr{\o}dinger operator. We prove the existence of minimizers through the equivalence with a weak formulation, and the local Ahlfors regularity and uniform rectifiability of the boundaries $\Omega \cap \partial W_j$.

Robust and Fault-Tolerant Fog Design and Dimensioning for Reliable Operation

Internet of Things (IoT) applications depend on reliable external storage and processing such as cloud data centers. In response to high latency from cloud, fog computing has been introduced as a network of microdata centers closer to IoT devices that provides a geo-distributed low-latency response. Current contributions regarding design and dimensioning of fog infrastructures are developed to service a static set of IoT traffic and a reliable fog network. However, these designs are not fault tolerant. This article explores the implementation of reliable and fault-tolerant fog infrastructures via dynamically available fog nodes—standby nodes that activate when a nearby fog node fails. We formulate the design and dimensioning of dynamically available nodes as a set partitioning problem, which is solved via a mixed-integer linear program (MILP). This MILP formulation proves to be intractable; we therefore introduce a column generation approach to increase scalability with little loss to optimal design and dimensioning cost. Compared to other benchmark heuristic methods, our column generation approach yields reduced cost, with proportional solution time.

An Experimental Study of Fast Greedy Algorithms for Fair Allocation Problems

This paper is concerned with two salient allocationproblems in fair division of indivisible goods, aiming atmaximizing egalitarian and Nash product social welfare.These problems are computationally NP-hard, meaning thatachieving polynomial time algorithms is impossible, unlessP = NP. Approximation algorithms, which return near-optimalsolution with a theoretical guarantee, have been widely usedfor tackling the problems. However, most of them are often ofhigh computational complexity or not easy to implement. It istherefore of great interest to explore fast greedy methods thatcan quickly produce a good solution. This paper presents anempirical study of the performance of several such methods.Interestingly, the obtained results show that fair allocationproblems can be practically approximated by greedy algorithms.Keywords: Fair allocation, exact algorithm, greedy algorithm,mixed-integer linear program, NP-hard.I. INTRODUCTIONIn this paper, we study the fair allocation problem, whichhas shown its growing interest during last decades, with awide range of real-world applications [1]. In short, this is acombinatorial optimization problem which asks to allocate???? discrete items amongst a set of ???? agents (or players)so as to meet a certain notion of fairness. It is assumedthat every item is “indivisible” and “non-sharable”, thatis, i) it cannot be broken in pieces before allocating toagents, and ii) it cannot be shared by two or more agents.For example, laptops and cell-phones are indivisible itemswhich agents might not want to share with others. Anallocation of items to agents is simply a partition of thewhole set of items into ???? disjoint subsets. There are up to???????? such partitions, making the solution space large enoughso that an exhaustive search for an optimal solution isimpossible.It now remains to define what a fair allocation is, aconcept that is of independent interest in the field ofEconomic and Social Choice Theory [2, 3]. In general, thereare many different ways of defining fairness, depending onparticular applications. The most common way is to eitheruse a so-called Collective Utility Function (CUF), which isa function for aggregating individual agents’ utilities in afair manner, or to follow an orthogonal method relying ondetermining the fair share of agents. Since we are focusingon the first method in this paper, we refer the reader tothe paper [4] and the references therein for more details ofthe second method. Suppose that every agent evaluates thevalue of items through a utility function, which maps eachsubset of items to a numerical value representing the utilityof the agent for the subset. Then, one can define a maxmin fair allocation to be the one that maximizes the 
 
 

Variable neighbourhood search for the integrated runway sequencing, taxiway scheduling, and gate reassignment problem

With the rapid development of air transportation, the surface management of large airports has become more complicated. This paper proposes an integrated scheduling model of arrival aircraft (ISAA), in which complex constraints related to runway restrictions, gate assignment and taxiway conflicts are incorporated. With the aim of reducing the total passenger runway delay, passenger taxi delay, and passenger swap cost, the model is built as a set partition model. We propose various neighbourhood search (VNS) algorithm for solving the problem. A variable neighbourhood descent (VND) as local search is integrated into the general variable neighbourhood framework. This method is compared with a sequential method that solves runway sequencing, gate reassignment and taxiway scheduling problem separately. Numerical results based on Nanjing Lukou Internation Airport indicate that, although the swap cost for gate reassignment is slightly increased, the whole performance is significantly improved.

Minimizing user inconvenience and operational costs in a dial-a-flight problem for flying safaris

We study a Dial-a-Flight Problem faced by one of the major safari airline companies in Tanzania. Given a set of daily passenger requests and a fleet of heterogeneous airplanes, the problem requires to determine the best set of itineraries to transport the passengers from their origins to the requested destinations within specific time windows, while satisfying a number of operational constraints. The aim is to minimize user inconvenience, measured by delays with respect to the pre-defined time windows and by the number of intermediate stops in the itineraries, and operational cost. The problem is complicated by the high number of daily requests in peak touristic periods, and by the fact that refueling is possible only at a limited number of airstrips. We solve the problem by means of an adaptive large neighborhood search, which we enrich with local search operators and a set partitioning model. Extensive computational tests on real-world instances prove the effectiveness of the proposed algorithm, which can improve the solutions found by the company both in terms of operational cost and user inconvenience, in reasonable computational time.

Field programmable gate array implementation of an adaptive filtering based noise reduction and enhanced compression technique for healthcare applications

AbstractNowadays, the development of e‐health monitoring devices is getting more demand for remote health applications. In addition, cardiovascular diseases is considered the most chronic disease in recent years. So, this requires a fast diagnosis and therapy to clear this issue and requires an information transformation from the patient to the distant hospital. But, it faces many challenges like more energy consumption, data security, storage, and transmission. In addition, the electrocardiogram (ECG) data must be processed in real‐time, and the information must have high predictability and lossless compression. So, in this article, an enhanced set partitioning in hierarchical tree decoder with deep belief neural network is developed for ECG compression. Due to an inappropriate channel in telemedicine applications, the ECG signals transmitted through a wireless network are affected by noise. So, fast normalized least mean square (FNLMS) algorithm based adaptive filter is deliberated to eliminate the unwanted noise from the ECG signal. Also, the structure of the FNLMs based adaptive filter is modified by developing a systolic structure to increase the speed of filtering process. Moreover, the parallel and pipelined architecture is provided for the proposed compression unit to reduce the processing time and the area overhead as compared to the existing methods. A publicly available database called the MIT‐BIH database and real‐time ECG data records are used for simulation purposes. The proposed design was implemented on the Xilinx platform and the MATLAB tool. The performance of the proposed design is estimated by the parameters like compression ratio, signal to noise ratio, root mean square error, and percentage root mean square difference (PRD), PRD normalized, and quality‐score. It is compared with existing designs to show the efficiency of our proposed design.

Finite-state parameter space maps for pruning partitions in modularity-based community detection

Partitioning networks into communities of densely connected nodes is an important tool used widely across different applications, with numerous methods and software packages available for community detection. Modularity-based methods require parameters to be selected (or assume defaults) to control the resolution and, in multilayer networks, interlayer coupling. Meanwhile, most useful algorithms are heuristics yielding different near-optimal results upon repeated runs (even at the same parameters). To address these difficulties, we combine recent developments into a simple-to-use framework for pruning a set of partitions to a subset that are self-consistent by an equivalence with the objective function for inference of a degree-corrected planted partition stochastic block model (SBM). Importantly, this combined framework reduces some of the problems associated with the stochasticity that is inherent in the use of heuristics for optimizing modularity. In our examples, the pruning typically highlights only a small number of partitions that are fixed points of the corresponding map on the set of somewhere-optimal partitions in the parameter space. We also derive resolution parameter upper bounds for fitting a constrained SBM of K blocks and demonstrate that these bounds hold in practice, further guiding parameter space regions to consider. With publicly available code (http://github.com/ragibson/ModularityPruning), our pruning procedure provides a new baseline for using modularity-based community detection in practice.

A deep learning approach to cardiovascular disease classification using empirical mode decomposition for ECG feature extraction

Deep learning has achieved promising results on a broad spectrum of tasks using an end-to-end approach, and domain-specific knowledge can be used to supplement it by either constraining the solution-space, or to transform data such that relevant signals are more influential during the training process. This is especially important in the context of continuous recordings or wearable-based monitoring, where signal quality may be constrained by hardware, such as a limited number of available electrodes, or by external perturbations. In this study, a global feature extraction method for multi-periodic signals is proposed, which effectively classifies multi-cycle normal and abnormal heartbeats. The abnormal heartbeats are classed as the Left Bundle Branch Block beat (LBBB), Right Bundle Branch Block beat (RBBB), and Paced beat (P). The signal analysis process of the electrocardiogram (ECG) recordings consist of three main stages: (I) segmentation of the ECG data and partitioning of the data set; (II) generation of an overall feature map representing the “heartbeat condition” based on Empirical Mode Decomposition (EMD); and (III) a classification stage for determining the patient’s heartbeat condition. The extracted feature image is used to classify the heartbeat condition using a two-dimensional Convolutional Neural Network (CNN). This method is applied to the publicly available MIT-BIH arrhythmia database. Experimental results show that the reconstructed ECG features outperform use of the raw ECG signals. Under self-test conditions of 3 min of ECG signal, the Total Classification Accuracy (TCA) was approximately 99.01%, while the TCA without the proposed method was approximately 90.69%.

Perfect 3-colorings of Cubic Graphs of Order 8

Perfect coloring is a generalization of the notion of completely regular codes, given by Delsarte. A perfect $m$-coloring of a graph $G$ with $m$ colors is a partition of the vertex set of $G$ into m parts $A_1$, $\dots$, $A_m$ such that, for all $ i,j\in \lbrace 1,\cdots ,m\rbrace $, every vertex of $A_i$ is adjacent to the same number of vertices, namely, $a_{ij}$ vertices, of $A_j$ . The matrix $A=(a_{ij})_{i,j\in \lbrace 1,\cdots ,m\rbrace }$ is called the parameter matrix. We study the perfect 3-colorings (also known as the equitable partitions into three parts) of the cubic graphs of order $8$. In particular, we classify all the realizable parameter matrices of perfect 3-colorings for the cubic graphs of order 8.