Set Partitioning Research Articles

Distributed Feature Selection for Power System Dynamic Security Region Based on Grid-Partition and Fuzzy-Rough Sets

In order to satisfy the requirements of modern online security assessment of power systems with continuously increasing complexity in terms of structure and scale, it is desirable to develop a power system dynamic security region (DSR) analysis. However, data-driven methods suffer from expensive model training costs and overfitting when determining DSR boundaries with high-dimensional grid features. Given this problem, a distributed feature selection method based on grid partition and fuzzy-rough sets is proposed in this paper. The method first employs the Louvain algorithm to partition the power grid and divide the original feature set so that high-dimensional features can be allocated to multiple computational units for distributed screening. At this point, the connections between features of different computational units are minimized to a relatively low level, thereby avoiding large errors in the distributed results. Then, an incremental search algorithm based on the fuzzy-rough set theory (FRST) is used for feature selection at each computational unit, which can effectively take into account the intrinsic connections between features. Finally, the results of all computational units are integrated in the coordination unit to complete the overall feature selection. The experimental results based on the IEEE-39 bus system show that the proposed method can help simplify the power system DSR analysis with high-dimensional features by screening the critical features. And compared with other commonly used filter methods, it has higher screening accuracy and lower time costs.

The finite sequences and the partitions whose members are finite of a set

Abstract In this paper, we investigate relationships between $|\text{{seq}}(A)|$ and $|\text{{Part}}_{\text{{fin}}}(A)|$ in the absence of the Axiom of Choice, where $\text{{seq}}(A)$ is the set of finite sequences of elements in a set $A$ and $\text{{Part}}_{\text{{fin}}}(A)$ is the set of partitions of $A$ whose members are finite. We show that $|\text{{seq}}(A)|<|\text{{Part}}_{\text{{fin}}}(A)|$ if $A$ is Dedekind-infinite and the condition cannot be removed. Moreover, this relationship holds for an arbitrary infinite set $A$ if we restrict $\text{{seq}}(A)$ to the set of finite sequences with a bounded length.

GraphLambda: Fusion Graph Neural Networks for Binding Affinity Prediction.

Predicting the binding affinity of protein-ligand complexes is crucial for computer-aided drug discovery (CADD) and the identification of potential drug candidates. The deep learning-based scoring functions have emerged as promising predictors of binding constants. Building on recent advancements in graph neural networks, we present graphLambda for protein-ligand binding affinity prediction, which utilizes graph convolutional, attention, and isomorphism blocks to enhance the predictive capabilities. The graphLambda model exhibits superior performance across CASF16 and CSAR HiQ NRC benchmarks and demonstrates robustness with respect to different types of train-validation set partitions. The development of graphLambda underscores the potential of graph neural networks in advancing binding affinity prediction models, contributing to more effective CADD methodologies.

A New Algorithm Framework for the Influence Maximization Problem Using Graph Clustering

Given a social network modelled by a graph, the goal of the influence maximization problem is to find k vertices that maximize the number of active vertices through a process of diffusion. For this diffusion, the linear threshold model is considered. A new algorithm, called ClusterGreedy, is proposed to solve the influence maximization problem. The ClusterGreedy algorithm creates a partition of the original set of nodes into small subsets (the clusters), applies the SimpleGreedy algorithm to the subgraphs induced by each subset of nodes, and obtains the seed set from a combination of the seed set of each cluster by solving an integer linear program. This algorithm is further improved by exploring the submodularity property of the diffusion function. Experimental results show that the ClusterGreedy algorithm provides, on average, higher influence spread and lower running times than the SimpleGreedy algorithm on Watts–Strogatz random graphs.

Formula omitted]-packing coloring of cubic Halin graphs

Given a non-decreasing sequence S=(s1,s2,…,sk) of positive integers, an S-packing coloring of a graph G is a partition of the vertex set of G into k subsets {V1,V2,…,Vk} such that for each 1≤i≤k, the distance between any two distinct vertices u and v in Vi is at least si+1. In this paper, we study the problem of S-packing coloring of cubic Halin graphs, and we prove that every cubic Halin graph is (1,1,2,3)-packing colorable. In addition, we prove that such graphs are (1,2,2,2,2,2)-packing colorable.

Compression of Medical images using SPIHT Algorithm for Telemedicine Application

Image compression plays a pivotal role in the medical field for the storage and transfer of DICOM images. This research work focuses on the compression of medical images using Set Partitioning in Hierarchy Trees (SPIHT) algorithm. The CT/MR images are used as input, the images are subjected to filtering by a median filter. The CT images in general are corrupted by Gaussian noise and MR images are corrupted by rician noise. The SPIHT algorithm comprises of following phases; transformation into wavelet domain, refinement pass and sorting pass. The Haar wavelet transform is employed and the wavelet coefficients are subjected to sorting and refinement pass. The Haar wavelet transform generates LL, HL, HL and HH sub-bands. In the sorting pass, the coefficients are classified into significant and insignificant. The refinement pass creates the precision bits for the significant coefficients. The main characteristic of the SPIHT algorithm is that it does not use an entropy coder. The reconstructed image in the decoding stage was validated by performance metrics. The SPIHT algorithm generates proficient results, when compared with the classical algorithms like wavelet and embedded zero tree wavelet (EZW) algorithms.

Block mapping and dual-matrix-based watermarking for image authentication with self-recovery capability.

Numerous image authentication techniques have been devised to address the potential security issue of malicious tampering with image content since digital images can be easily duplicated, modified, transformed and diffused via the Internet transmission. However, the existing works still remain many shortcomings in terms of the recovery incapability and detection accuracy with extensive tampering. To improve the performance of tamper detection and image recovery, we present a block mapping and dual-matrix-based watermarking scheme for image authentication with self-recovery capability in this paper. The to-be-embedded watermark information is composed of the authentication data and recovery data. The Authentication Feature Composition Calculation algorithm is proposed to generate the authentication data for image tamper detection and localization. Furthermore, the recovery data for tampered region recovery is comprised of self-recovery bits and mapped-recovery bits. The Set Partition in Hierarchical Trees encoding algorithm is applied to obtain the self-recovery bits, whereas the Rehashing Model-based Block Mapping algorithm is proposed to obtain the mapped-recovery bits for retrieving the damaged codes caused by tampering. Subsequently, the watermark information is embedded into the original image as digital watermarking with the guidance of a dual-matrix. The experimental results demonstrate that comparing with other state-of-the-art works, our proposed scheme not only improves the performance in recovery, but also extends the limitation of tampering rate up to 90%. Furthermore, it obtains a desirable image quality above 40 dB, large watermark payload up to 3.169 bpp, and the effective resistance to malicious attack, such as copy-move and collage attacks.

A novel edge connectivity based on edge partition for hypercube and folded hypercube

Edge connectivity is often used to capture edge fault tolerance. As for most common networks, their edge connectivity is exactly equal to their minimum degree. Edge matroidal connectivity and conditional edge matroidal connectivity are two new graph edge connectivity parameters that can be defined when a partition of the edge set is given, for example when the network is built with different kinds of edges having different expected faultiness. And those two edge connectivity parameters can measure edge fault tolerance more than the traditional definition. The edge matroidal connectivity is based on the associated edge partition and the other is a generalization called conditional edge matroidal connectivity. We analyze these new parameters on hypercubes and folded hypercubes which are well studied networks. In this study, we consider their standard dimensional partition of the edges. This study leads to more structural insights about edge connectivity and yields many interesting questions.

LIBS combined with SG-SPXY spectral data pre-processing for cement raw meal composition analysis.

Rapid testing of cement raw meal plays a crucial role in the cement production process, so there is an urgent need for a fast and accurate testing method. In this paper, a method based on the Savitzky-Golay (SG) smoothing and sample set partitioning based on joint x-y distance (SPXY) spectral data pre-processing is proposed to improve the accuracy of the laser-induced breakdown spectroscopy (LIBS) technique for quantitative analysis of cement raw meal components. Firstly, the spectral data is denoised by SG smoothing, which effectively reduces the noise and baseline variations in the spectra. Then, the denoised data is divided into sample sets by combining the SPXY sample division method, which improves the efficiency of data analysis. Finally, the delineated data set is modeled for quantitative analysis by a back-propagation (BP) neural network. Compared to the modeling effect of the four oxide contents of CaO, S i O 2, A l 2 O 3, and F e 2 O 3 in the Hold-Out method, the correlation coefficient (R) was improved by 26%, 10%, 17%, and 4%, respectively. The root mean square error (RMSE) was reduced by 47%, 33%, 43%, and 21%, respectively. The mean absolute percentage error (MAPE) was reduced by 63%, 60%, 36%, and 51%, respectively. The results show that there is a significant improvement in the model effect, which can effectively improve the accuracy of quantitative analysis of cement raw meal composition by LIBS. This is of great significance for the real-time detection of cement raw meal composition analysis.

An Embedding of the Skein Action on Set Partitions into the Skein Action on Matchings.

Rhoades defined a skein action of the symmetric group on noncrossing set partitions which generalized an action of the symmetric group on matchings. The $\mathfrak{S}_n$-action on matchings is made possible via the Ptolemy relation, while the action on set partitions is defined in terms of a set of skein relations that generalize the Ptolemy relation. The skein action on noncrossing set partitions has seen applications to coinvariant theory and coordinate rings of partial flag varieties. In this paper, we will show how Rhoades' $\mathfrak{S}_n$-module can be embedded into the $\mathfrak{S}_n$-module generated by matchings, thereby explaining how Rhoades' generalized skein relations all arise from the Ptolemy relation.

A Comparison of Integer Partitions Based on Smallest Part

For positive integers $n, L$ and $s$, consider the following two sets that both contain partitions of $n$ with the difference between the largest and smallest parts bounded by $L$: the first set contains partitions with smallest part $s$, while the second set contains partitions with smallest part at least $s+1$. Let $G_{L,s}(q)$ be the generating series whose coefficient of $q^n$ is difference between the sizes of the above two sets of partitions. This generating series was introduced by Berkovich and Uncu (2019). Previous results concentrated on the nonnegativity of $G_{L,s}(q)$ in the cases $s=1$ and $s=2$. In the present paper, we show the eventual positivity of $G_{L,s}(q)$ for general $s$ and also find a precise nonnegativity result for the case $s=3$.

A State Table SPHIT Approach for Modified Curvelet-based Medical Image Compression

Medical imaging plays a significant role in clinical practice. Storing and transferring a large volume of images can be complex and inefficient. This paper presents the development of a new compression technique that combines the fast discrete curvelet transform (FDCvT) with state table set partitioning in the hierarchical trees (STS) encoding scheme. The curvelet transform is an extension of the wavelet transform algorithm that represents data based on scale and position. Initially, the medical image was decomposed using the FDCvT algorithm. The FDCvT algorithm creates symmetrical values for the detail coefficients, and these coefficients are modified to improve the efficiency of the algorithm. The curvelet coefficients are then encoded using the STS and differential pulse-code modulation (DPCM). The greatest amount of energy is contained in the coarse coefficients, which are encoded using the DPCM method. The finest and modified detail coefficients are encoded using the STS method. A variety of medical modalities, including computed tomography (CT), positron emission tomography (PET), and magnetic resonance imaging (MRI), are used to verify the performance of the proposed technique. Various quality metrics, including peak signal-to-noise ratio (PSNR), compression ratio (CR), and structural similarity index (SSIM), are used to evaluate the compression results. Additionally, the computation time for the encoding (ET) and decoding (DT) processes is measured. The experimental results showed that the PET image obtained higher values of the PSNR and CR. The CT image provides high quality for the reconstructed image, with an SSIM value of 0.96 and the fastest ET of 0.13 seconds. The MRI image has the shortest DT, which is 0.23 seconds.

On degree conditions of semi-balanced k-partite Hamiltonian graphs

For two integers [Formula: see text] and [Formula: see text] with [Formula: see text], a [Formula: see text]-partite graph is said to be a semi-balanced[Formula: see text]-partite graph if each partite set has either [Formula: see text] or [Formula: see text] vertices. We give some degree conditions for semi-balanced [Formula: see text]-partite graphs to be Hamiltonian and almost pancyclic, respectively. Furthermore, using these results, we obtain the degree conditions for a balanced [Formula: see text]-partite graph to be vertex bipancyclic if [Formula: see text] is even, and vertex [Formula: see text]-pancyclic if [Formula: see text] is odd.

Moments of permutation statistics and central limit theorems

We show that if a permutation statistic can be written as a linear combination of bivincular patterns, then its moments can be expressed as a linear combination of factorials with constant coefficients. This generalizes a result of Zeilberger. We use an approach of Chern, Diaconis, Kane and Rhoades, previously applied on set partitions and matchings. In addition, we give a new proof of the central limit theorem (CLT) for the number of occurrences of classical patterns, which uses a lemma of Burstein and Hästö. We give a simple interpretation of this lemma and an analogous lemma that would imply the CLT for the number of occurrences of any vincular pattern. Furthermore, we obtain explicit formulas for the moments of the descents and the minimal descents statistics. The latter is used to give a new direct proof of the fact that we do not necessarily have asymptotic normality of the number of pattern occurrences in the case of bivincular patterns. Closed forms for some of the higher moments of several popular statistics on permutations are also obtained.

Combining spectral and texture feature of UAV image with plant height to improve LAI estimation of winter wheat at jointing stage.

Leaf area index (LAI) is a critical physiological and biochemical parameter that profoundly affects vegetation growth. Accurately estimating the LAI for winter wheat during jointing stage is particularly important for monitoring wheat growth status and optimizing variable fertilization decisions. Recently, unmanned aerial vehicle (UAV) data and machine/depth learning methods are widely used in crop growth parameter estimation. In traditional methods, vegetation indices (VI) and texture are usually to estimate LAI. Plant Height (PH) unlike them, contains information about the vertical structure of plants, which should be consider. Taking Xixingdian Township, Cangzhou City, Hebei Province, China as the research area in this paper, and four machine learning algorithms, namely, support vector machine(SVM), back propagation neural network (BPNN), random forest (RF), extreme gradient boosting (XGBoost), and two deep learning algorithms, namely, convolutional neural network (CNN) and long short-term memory neural network (LSTM), were applied to estimate LAI of winter wheat at jointing stage by integrating the spectral and texture features as well as the plant height information from UAV multispectral images. Initially, Digital Surface Model (DSM) and Digital Orthophoto Map (DOM) were generated. Subsequently, the PH, VI and texture features were extracted, and the texture indices (TI) was further constructed. The measured LAI on the ground were collected for the same period and calculated its Pearson correlation coefficient with PH, VI and TI to pick the feature variables with high correlation. The VI, TI, PH and fusion were considered as the independent features, and the sample set partitioning based on joint x-y distance (SPXY) method was used to divide the calibration set and validation set of samples. The ability of different inputs and algorithms to estimate winter wheat LAI were evaluated. The results showed that (1) The addition of PH as a feature variable significantly improved the accuracy of the LAI estimation, indicating that wheat plant height played a vital role as a supplementary parameter for LAI inversion modeling based on traditional indices; (2) The combination of texture features, including normalized difference texture indices (NDTI), difference texture indices (DTI), and ratio texture indices (RTI), substantially improved the correlation between texture features and LAI; Furthermore, multi-feature combinations of VI, TI, and PH exhibited superior capability in estimating LAI for winter wheat; (3) Six regression algorithms have achieved high accuracy in estimating LAI, among which the XGBoost algorithm estimated winter wheat LAI with the highest overall accuracy and best results, achieving the highest R2 (R2=0.88), the lowest RMSE (RMSE=0.69), and an RPD greater than 2 (RPD=2.54). This study provided compelling evidence that utilizing XGBoost and integrating spectral, texture, and plant height information extracted from UAV data can accurately monitor LAI during the jointing stage of winter wheat. The research results will provide a new perspective for accurate monitoring of crop parameters through remote sensing.

Monotonic Entropies

We introduce an axiomatization of entropy that generates, as special cases, novel entropy types. These entropies generalize Shannon’s entropy and allow the introduction of entropy for partitions of sets of objects located in metric spaces, and for partitions of sets of vertices in undirected graphs. Corresponding metrics on the sets of partitions are introduced. Also, we hint to applications of these metrics for evaluation of clustering quality.

A new approach for counting patterns in language theory, matchings, and set partitions.

A new approach for counting patterns in language theory, matchings, and set partitions.

Minimizing Carbon Emission in Hybrid Flow Shop Scheduling: A Comparative Analysis of Flow-based and Set Partitioning Formulations

This article addresses the Hybrid Flow Shop Green Scheduling Problem with Sequence and Machine Dependent Setup Times (HFSGSP-SMDST) in the context of Industry 4.0. The objective is to minimize carbon emission while optimizing scheduling. Two mathematical formulations, a flow-based positional formulation (FB) and a set partitioning positional formulation (HCG), are proposed to provide tighter lower bounds. The FB formulation is solved using a commercial solver, while the HCG algorithm incorporates an Iterated Local Search (ILS) to generate initial solutions and an upper bound for the problem. Computational experiments demonstrate that HCG achieves better lower bounds, although it requires more time compared to FB. The proposed algorithms offer valuable solutions for carbon emission optimization and scheduling in Industry 4.0, with a trade-off between lower bounds and convergence speed.

Radio number of $ 2- $ super subdivision for path related graphs

<abstract><p>We studied radio labelings of graphs in response to the Channel Assignment Problem (CAP). In a graph $ G, $ the radio labeling is a mapping $ \varpi:V(G) \rightarrow \{0, 1, 2, ..., \}, $ such as $ |\varpi(\mu')-\varpi(\mu'')|\geq diam(G)+1-d(\mu', \mu''). $ The label of $ \mu $ for under $ \varpi $ is defined by the integer $ \varpi(\mu), $ and the span under is defined by $ span(\varpi) = max \{|\varpi(\mu')-\varpi(\mu'')|: \mu', \mu'' \in V(G)\}. $ $ rn(G) = min_{\varpi} span(\varpi) $ is defined as the radio number of $ G $ when the minimum over all radio labeling $ \varpi $ of $ G $ is taken. $ G $ is said to be optimal if its radio labeling is $ span(\varpi) = rn(G). $ A graph H is said to be an $ m $ super subdivision if $ G $ is replaced by the complete bipartite graph $ K_{m, m} $ with $ m = 2 $ in such a way that the end vertices of the edge are merged with any two vertices of the same partite set $ X $ or $ Y $ of $ K_{m, m} $ after removal of the edge of $ G $. Up to this point, many lower and upper bounds of $ rn(G) $ have been found for several kinds of graph families. This work presents a comprehensive analysis of the radio number $ rn(G) $ for a graph $ G $, with particular emphasis on the $ m $ super subdivision of a path $ P_{n} $ with $ n (n \geq 3) $ vertices, along with a complete bipartite graph $ K_{m, m} $ consisting of $ m $ v/ertices, where $ m = 2 $.</p></abstract>

Enhancing multi-view ensemble learning with zig-zag pattern-based feature set partitioning

Enhancing multi-view ensemble learning with zig-zag pattern-based feature set partitioning