Set Partitioning Research Articles

Partitions, their classes, and multicolour evenness

We extend the theory of maximally even sets to determine the evenness of partitions of the chromatic universe . Interactions measure the average evenness of colour sets (partitioning sets) of . For 2-colour partitions the Clough-Douthett maximal-evenness algorithm determines maximally even partitions. But to measure the evenness of non-maximally even partitions, it is necessary to use computational methods. Moreover, for more than two colour sets there is no simple algorithm that determines maximally even partitions. Again, we rely on computational methods. We also explore collections of partitions and partition-classes (orbits under a dihedral group) and construct tables that order partition-classes according to the evenness of their partitions. We use Bell numbers, Stirling numbers of the second kind, and integer partitions to enumerate relevant combinatorial objects related to our investigation.

Development of Transmitter and Receiver With Set Partitioning 64APSK Coded Modulation Designed on Basis of Channel Capacity

In this paper, we present a prototype transmitter and receiver with set partitioning (SP)-64 amplitude and phase shift keying (APSK) coded modulation we developed for expanding satellite transmission capacity. SP-multi-level coded modulation is a transmission scheme that can improve noise immunity by properly combining a modulation scheme, bit allocation, and error-correction codes. We designed SP-64APSK coded modulation on the basis of channel capacity for expanding satellite transmission capacity. We confirmed that it has higher robustness to additive white Gaussian noise (AWGN) than DVB-S2X’s 64APSK modulation with Gray mapping through computer simulation. We also developed a prototype transmitter and receiver to implement our SP-64APSK coded modulation and evaluated its transmission performances in an AWGN channel and in a non-linear channel simulating 12-GHz-band satellite transponder characteristics. We describe the method of designing our SP-64APSK coded modulation through computer simulation, the modulation/demodulation process, and transmission performances of the prototype transmitter and receiver. We also indicate that 8K ultra-high definition television (UHDTV) with an encoded bit rate of 150 Mbps can be transmitted using the prototype within 34.5 MHz, which is the bandwidth of a single transponder for satellite broadcasting in Japan.

Resource Sharing for Cellular-Assisted D2D Communications With Imperfect CSI: A Many-to-Many Strategy

The rising demand for higher access rate and higher data rate applications will lead to the shortage of available spectrum resources. In this article, we propose a scalable framework to share resources and design a reasonable and effective resource management scheme to solve the resource allocation and power control problems of device-to-device (D2D) communication in dense scenarios. First, in order to maximize system throughput and user access rate by using many-to-many resource sharing to improve spectrum utilization. Second, considering imperfect channel state information, we propose a subcell division strategy based on location information, with the set partitioning algorithm to model intracell users as multiple many-to-many resource sharing sets satisfy the constraints. Finally, the problems of interference accumulation and low computational efficiency, a distributed power optimization algorithm based on quadratic transform is devised, which solves the multiratio fractional planning problem through the three steps of convex relaxation, fractional programming and alternating optimization. Experimental results show that the system throughput and D2D user access rate can be significantly enhanced with the proposed resource sharing framework and resource management scheme, especially in systems with a dense D2D deployment.

Improving Medical Video Coding Using Multi Scale Quincunx Lattice: From Low Bitrate to High Quality

In recent years, the healthcare sector has seen an increase in the use of medical images and videos. However, storage and transmission of this huge volume of data remain a challenging task, requiring the use of compression techniques. In this paper, the authors propose an algorithm to improve the visual quality of compressed medical video for lower bitrate without modifying the content of information such as edges and textures, this is a unique way for doctors to store and share medical data over the internet. The algorithm has not yet been sufficiently explored in medical video coding. In this study, the performances of the quincunx wavelet transform (QWT) combined with the set partitioning in hierarchical trees (SPIHT) encoder are discussed. The QWTs were chosen due to their limited number of wavelets family and reduced dilatation factor. The high efficiency of the suggested algorithm is checked against the coding standard based on the discrete cosines transform (DCT) or discrete wavelet transform (DWT). The assessment of the quality of the decoded video is based on the use of the peak signal to noise ratio (PSNR), the mean structural similarity (MSSIM) and the visual information fidelity (VIF). The results prove that the QWT+SPIHT provide competing performance where the PSNR reached 33 dB value for lower bitrate (137.408 Kbps) against previous standards.

A novel method of data compression using ROI for biomedical 2D images

In the era of modern medical imaging communal, huge volume of medical image data are being acquired and need to be transmitted and archived which necessitate the development of efficient image compression techniques on both 2D and 3D images. The compression of medical images is an essential process to support remote healthcare services. The medical diagnostics through these services require more accurate information from an image. As the property of inverse proportionality between the compression rate and quality of the image takes place in any kind of compression method, there is a need to sacrifice any one of those credentials (Quality or Compression Rate). With this context, Region of Interest (ROI) codecs are emerging and reduces this proportionality that yields more compression rate without compromising the quality. In this paper, presents an ROI based near lossless image compression method that incorporates the Set Partitioning in Hierarchical Trees (SPIHT) and Vector Quantization coding for medical images.

A new statistical test for distinguishing 2-partitions of a finite set

This paper considers a family of so-called 2-partitions of some finite set. Each of them divides the set under study into two disjoint parts. Under the assumption that two such partitions are chosen randomly, the exact probability distribution of the special cluster metric on this family is found. On this basis, a new statistical test for checking the significance of differences between 2-partitions is proposed. In addition, the distribution of the values of this metric is found for the case when both partitions are of the ledge type in ordering the set of objects in ascending order of values of some numerical indicator. This means that one of the parts of each partition, which in some sense is the main one, is a segment. The boundaries of such a segment are called normative. By comparing various estimates of the normative boundaries based on sample data, it is introduced the concept of indicative certainty of the numerical indicator. It can be regarded as the degree of confidence in this indicator as a basis for decision whether an object belongs to the main set of the ledge partition. Some application of the results to medical data processing is considered.

A note on minimum degree condition for Hamilton (a,b)-cycles in hypergraphs

Let k,a,b be positive integers with a+b=k. A k-uniform hypergraph is called an (a,b)-cycle if there is a partition (A0,B0,A1,B1,…,At−1,Bt−1) of the vertex set with |Ai|=a, |Bi|=b such that Ai∪Bi and Bi∪Ai+1 (subscripts module t) are edges for all i=0,1,…,t−1. Let H be a k-uniform n-vertex hypergraph with n≥5k and n divisible by k. By applying the concentration inequality for intersections of a uniform hypergraph with a random matching developed by Frankl and Kupavskii, we show that if there exists α∈(0,1) such that δa(H)≥(α+o(1))(n−ab) and δb(H)≥(1−α+o(1))(n−ba), then H contains a Hamilton (a,b)-cycle. As a corollary, we prove that if δℓ(H)≥(1/2+o(1))(n−ℓk−ℓ) for some ℓ≥k/2, then H contains a Hamilton (k−ℓ,ℓ)-cycle and this is asymptotically best possible.

Mahonian and Euler-Mahonian statistics for set partitions

A partition of the set [n]:={1,2,…,n} is a collection of disjoint nonempty subsets (or blocks) of [n], whose union is [n]. In this paper we consider the following rarely used representation for set partitions: given a partition of [n] with blocks B1,B2,…,Bm satisfying max⁡B1<max⁡B2<⋯<max⁡Bm, we represent it by a word w=w1w2…wn such that i∈Bwi, 1≤i≤n. We prove that the Mahonian statistics INV, MAJ, MAJd, r-MAJ, Z, DEN, MAK, MAD are all equidistributed on set partitions via this representation, and that the Euler-Mahonian statistics (des,MAJ), (mstc,INV), (exc,DEN), (des,MAK) are all equidistributed on set partitions via this representation.

The Pickup and Delivery Problem with Time Windows and Incompatibility Constraints in Cold Chain Transportation

This study investigates a new variant of the pickup and delivery problem with time windows (PDPTW) applied in cold chain transportation, which quantifies the effect of time on the quality of perishable products. Multiple commodities with incompatibility constraints are considered, where some types of products cannot be transported in a vehicle simultaneously because of their different properties and requirements for storage temperatures. The aim is to determine vehicles’ pickup and delivery routes as well as their departure times from the depot such that the travel cost and refrigeration cost of vehicles and the quality decay cost of products are minimized. We formulate this problem as a set partitioning model, which is solved exactly by a tailored branch-and-price (B&amp;P) algorithm. To tackle the asymmetry issue arising from the pricing problem of the B&amp;P framework, we develop a novel asymmetric bidirectional labeling algorithm. Benchmark instance sets based on real-world statistical data and classic PDPTW instance sets are first generated for this problem. Numerical results show that our B&amp;P algorithm can solve most instances to optimality in an acceptable time frame. Moreover, our results demonstrate that integrating the refrigeration and quality decay costs into the objective function can significantly lower the total cost of cold chain transportation activities, compared with the widely adopted objective function minimizing only the travel cost. Funding: This work was supported by the National Key R&amp;D Program of China [Grant 2018YFB1700600] and the National Natural Science Foundation of China [Grants 72101049, 71971090, 71821001]. Supplemental Material: The online supplement is available at https://doi.org/10.1287/trsc.2022.1167 .

PCB Network Analysis for Circuit Partitioning

The complexity of automatic placement and routing is proportional to the scale of the circuit. Through netlist partition algorithms, printed circuit board (PCB) circuits are divided into different submodules, and the problem scale is effectively reduced in order to obtain the optimal automatic layout and routing. In this paper, we analyze net attributes and potential patterns in netlists through visualization, and propose a heuristic PCB netlist partition approach based on net attributes and potential patterns which we discover from netlists. Our partition approach takes the netlist as input, and module partition set as output. Firstly, the modules are prepartitioned using net attributes. Further, the special patterns in circuits are discovered, and the scattered resistors, capacitors, and other components caused by prepartitioning would be allocated to initial modules by three rules—classifying, matching, and force strategy. Our method is evaluated on 11 PCB netlists which are built manually. Experimental results show that our proposed netlist partition approach significantly outperforms the state of the art on all evaluation indices, which can achieve 80–96% partition accuracy.

Hardware Implementation Of ECG Signal Compression Using SPIHT

Cardiovascular disorder is a primary cause of mortality throughout the world in both developed and underdeveloped countries. Continuous cardiac monitoring enables clinicians to identify arrhythmias and other heart conditions. Tele-cardiology introduces remote monitoring devices for tracking the cardiac activity of the patients. The large volume of Electrocardiogram (ECG) data needs to be stored, processed and transmitted by these portable health care devices. The implementation of ECG compression in hardware platform is crucial for continuous health monitoring applications. The aim of this work is to implement field programmable gate array based set partitioning in hierarchical trees-based electrocardiogram compression. Discrete wavelet transform method is employed to break up the signal into sub bands. The transformed coefficients after discrete wavelet transform are passed through dead zone quantization which rejects low magnitude values of transformed coefficients lying around zero. These quantized coefficients are then encoded by lossless set partitioning used in hierarchical trees compression approach. The introduction of dead zone quantization in the proposed technique is found to be effective and yields an increased compression ratio of 10.33 with decreased distortion value of 1.04 percent for ECG record 117 of MIT-BIH arrhythmia database.

Grayscale and color images encryption in a DRPE assisted by SPHIT, RLE, QR code and two chaos logistic maps

A non-linear encryption approach for compressed grayscale and color images is proposed in Double Random Phase Encoding (DRPE) setup using two compression algorithms: Set Partitioning in Hierarchical Trees (SPIHT) and Run-length encoding (RLE). The compressed image is then segmented and embedded in multiple Quick Response (QR) codes, which are combined again into a single image. Further, this single image is scrambled by two-coupled chaos logistic maps, at which the scrambled image is now ready for DRPE encryption. The use of two-coupled chaos logistic increases the security level of this proposed DRPE encryption from various attacks. The proposed DRPE encryption is extended to handed color images, which are segregated into three color-channels (Red, Green and Blue), where each channel is individually encrypted. For grayscale images, the proposed system will have three security key codes while for color images it has nine security key codes. The robustness and invulnerability of the proposed scheme against various attacks is demonstrated through cryptoanalysis and computer simulation results.

Even Cycle Decompositions of index 3 by a novel coloring technique

An even cycle decomposition of an Eulerian graph is a partition of the edge set into even cycles. We color the even cycles so as two cycles sharing at least one vertex receive distinct colors. If k is the minimum number of required colors in such a coloring, then the even cycle decomposition has index k . We prove that the line graph of every bridgeless cubic graph of oddness 2 has an even cycle decomposition of index 3. The same property holds for the line graphs of some infinite families of class 2 cubic graphs with arbitrary large oddness. The construction of even cycle decompositions of index 3 in the line graph of a class 2 cubic graph is alternative to the constructions that are known in the literature; that one for the line graph of a cubic graph with arbitrary large oddness is also a new contribution to the more general problem on the existence of even cycle decompositions in the line graph of a bridgeless cubic graph. The constructions are obtained by applying a novel coloring technique on the edges of the line graph.

APPROXIMATION OF POSITIONAL IMPULSE CONTROLS FOR DIFFERENTIAL INCLUSIONS

Nonlinear control systems presented as differential inclusions with positional impulse controls are investigated. By such a control we mean some abstract operator with the Dirac function concentrated at each time. Such a control ("running impulse"), as a generalized function, has no meaning and is formalized as a sequence of correcting impulse actions on the system corresponding to a directed set of partitions of the control interval. The system responds to such control by discontinuous trajectories, which form a network of so-called "Euler's broken lines." If, as a result of each such correction, the phase point of the object under study is on some given manifold (hypersurface), then a slip-type effect is introduced into the motion of the system, and then the network of "Euler's broken lines" is called an impulse-sliding mode. The paper deals with the problem of approximating impulse-sliding modes using sequences of continuous delta-like functions. The research is based on Yosida's approximation of set-valued mappings and some well-known facts for ordinary differential equations with impulses.

A web basis of invariant polynomials from noncrossing partitions

The irreducible representations of symmetric groups can be realized as certain graded pieces of invariant rings, equivalently as global sections of line bundles on partial flag varieties. There are various ways to choose useful bases of such Specht modules Sλ. Particularly powerful are web bases, which make important connections with cluster algebras and quantum link invariants. Unfortunately, web bases are only known in very special cases—essentially, only the cases λ=(d,d) and λ=(d,d,d). Building on work of B. Rhoades (2017), we construct an apparent web basis of invariant polynomials for the 2-parameter family of Specht modules with λ of the form (d,d,1ℓ). The planar diagrams that appear are noncrossing set partitions, and we thereby obtain geometric interpretations of earlier enumerative results in combinatorial dynamics.

Anti-Ramsey numbers for trees in complete multi-partite graphs

Let G be a complete multi-partite graph of order n. In this paper, we consider the anti-Ramsey number ar(G,Tq) with respect to G and the set Tq of trees with q edges, where 2≤q≤n−1. For the case q=n−1, the result has been obtained by Lu, Meier and Wang. We will extend it to q<n−1. We first show that ar(G,Tq)=ℓq(G)+1, where ℓq(G) is the maximum size of a disconnected spanning subgraph H of G with the property that any two components of H together have at most q vertices. Using this equality, we obtain the exact values of ar(G,Tq) for n−3≤q≤n−1. Moreover, for the general case when (4n−2)/5≤q≤n−1, ar(G,Tq) can be determined by a simple algorithm. In particular, the explicit expression of ar(G,Tq) is given when G has a partite set much larger than all the other partite sets.

Single Connection Amid Vertices From Two Given Sets Partitioning Vertex Set in Some Classes of Neutrosophic Graphs

Single Connection Amid Vertices From Two Given Sets Partitioning Vertex Set in Some Classes of Neutrosophic Graphs

Maximal Chains in Bond Lattices

Let $G$ be a graph with vertex set $\{1,2,\ldots,n\}$. Its bond lattice, $BL(G)$, is a sublattice of the set partition lattice. The elements of $BL(G)$ are the set partitions whose blocks induce connected subgraphs of $G$.&#x0D; In this article, we consider graphs $G$ whose bond lattice consists only of noncrossing partitions. We define a family of graphs, called triangulation graphs, with this property and show that any two produce isomorphic bond lattices. We then look at the enumeration of the maximal chains in the bond lattices of triangulation graphs. Stanley's map from maximal chains in the noncrossing partition lattice to parking functions was our motivation. We find the restriction of his map to the bond lattice of certain subgraphs of triangulation graphs. Finally, we show the number of maximal chains in the bond lattice of a triangulation graph is the number of ordered cycle decompositions.

Quadratic Clustering-Based Simplex Volume Maximization for Hyperspectral Endmember Extraction

The existence of intra-class spectral variability caused by differential scene components and illumination conditions limits the improvement of endmember extraction accuracy, as most endmember extraction algorithms directly find pixels in the hyperspectral image as endmembers. This paper develops a quadratic clustering-based simplex volume maximization (CSVM) approach to effectively alleviate spectral variability and extract endmembers. CSVM first adopts spatial clustering based on simple linear iterative clustering to obtain a set of homogeneous partitions and uses spectral purity analysis to choose pure pixels. The average of the chosen pixels in each partition is taken as a representative endmember, which reduces the effect of local-scope spectral variability. Then an improved spectral clustering based on k-means is implemented to merge homologous representative endmembers to further reduce the effect of large-scope spectral variability, and final endmember collection is determined by the simplex with maximum volume. Experimental results show that CSVM reduces the average spectral angle distance on Samson, Jasper Ridge and Cuprite datasets to below 0.02, 0.06 and 0.09, respectively, provides the root mean square errors of abundance maps on Samson and Jasper Ridge datasets below 0.25 and 0.10, and exhibits good noise robustness. By contrast, CSVM provides better results than other state-of-the-art algorithms.

Factorization of classical characters twisted by roots of unity

For a fixed integer t≥2, we consider the irreducible characters of representations of the classical groups of types A, B, C and D, namely GLtn,SO2tn+1,Sp2tn and O2tn, evaluated at elements ωkxi for 0≤k≤t−1 and 1≤i≤n, where ω is a primitive t'th root of unity. The case of GLtn was considered by D. J. Littlewood (AMS press, 1950) and independently by D. Prasad (Israel J. Math., 2016). In this article, we give a uniform approach for all cases. We also look at GLtn+1 where we specialize the elements as before and set the last variable to 1. In each case, we characterize partitions for which the character value is nonzero in terms of what we call z-asymmetric partitions, where z is an integer which depends on the group. Moreover, if the character value is nonzero, we prove that it factorizes into characters of smaller classical groups. The proof uses Cauchy-type determinant formulas for these characters and involves a careful study of the beta sets of partitions. We also give product formulas for general z-asymmetric partitions and z-asymmetric t-cores. Lastly, we show that there are infinitely many z-asymmetric t-cores for t≥z+2.