Integer Arrays Research Articles

A REFINED CLASSIFICATION OF THE SUBSET SUM PROBLEM IN POLYNOMIAL TIME COMPLEXITY

The problem considered is the selection of at least one subset from a set (array) of distinct positive integers, such that the sum of the subset's elements exactly matches a given target sum (target certificate). According to R. M. Karp, this problem belongs to the class of NP-complete problems. Diophantine equations and an auxiliary problem, which facilitates the solution of the original problem and has independent scientific interest, have been introduced. A novel method has been developed, which includes proven lemmas and theorems. These results enable the development of efficient and straightforward algorithms for solving the subset sum problem. The time and space complexity for selecting the required subsets do not exceed the square of the length of the original set. An analytical framework has been proposed for managing indices within the original set. These algorithms are applicable to solving problems related to the independent set of cardinality k and the k-vertex cover problem. Additionally, we present examples to confirm claimed results. It should be noted that the time complexity of sorting an array of integers is proportional to the square of the array's size, and this problem belongs to class P. Therefore, based on the newly developed method, it can be inferred that the subset sum problem, originally classified as NP-complete within the NP class, also belongs to P.

MARS Plus: An Improved Molecular Design Tool for Complex Compounds Involving Ionic, Stereo, and Cis-Trans Isomeric Structures.

MARS (Molecular Assembling and Representation Suite) (Hsu et al. J. Chem. Inf. Model. 2019, 59, 3703-3713) is a toolbox for the molecular design of organic molecules. MARS uses integer arrays to represent the elements and connectivity between elements of a molecule. It provides a collection of operations to manipulate the elemental composition and connectivity of a molecule (or a pair of molecules), enabling the creation of novel chemical compounds. In this work, the original MARS is extended to handle complex molecular structures, including geometric (cis-trans) isomers, stereo isomers, cyclic compounds, and ionic species. The extended version of MARS, referred to as MARS+, has a more comprehensive coverage of the chemical space and therefore can explore molecules with a greater chemical and physical diversity. Compared to other molecular design tools, MARS+ is designed to perform all possible manipulations on a given molecule or a pair of molecules. Molecular structure manipulation can be conducted in either a controlled or a random fashion. Furthermore, every structure manipulation has a counterpart so that the operation can be reversed. Nearly any possible chemical structure can be generated with MARS+ via a combination of molecular operations. The capabilities of MARS+ are examined by the design of new ionic liquids (ILs). The results show that MARS+ is a useful tool for computer-aided molecular design (CAMD) and molecular structure enumeration.

VisClust: A visual clustering algorithm based on orthogonal projections

We present a novel clustering algorithm, visClust, that is based on lower dimensional data representations and visual interpretation. Thereto, we design a transformation that allows the data to be represented by a binary integer array enabling the use of image processing methods to select a partition. Qualitative and quantitative analyses measured in accuracy and an adjusted Rand-Index show that the algorithm performs well while requiring low runtime as well and RAM. We compare the results to 6 state-of-the-art algorithms with available code, confirming the quality of visClust by superior performance in most experiments. Moreover, the algorithm asks for just one obligatory input parameter while allowing optimization via optional parameters. The code is made available on GitHub and straightforward to use.

HEaaN.MLIR: An Optimizing Compiler for Fast Ring-Based Homomorphic Encryption

Homomorphic encryption (HE) is an encryption scheme that provides arithmetic operations on the encrypted data without doing decryption. For Ring-based HE, an encryption scheme that uses arithmetic operations on a polynomial ring as building blocks, performance improvement of unit HE operations has been achieved by two kinds of efforts. The first one is through accelerating the building blocks, polynomial operations. However, it does not facilitate optimizations across polynomial operations such as fusing two polynomial operations. The second one is implementing highly optimized HE operations in an amalgamated manner. The written codes have superior performance, but they are hard to maintain. To resolve these challenges, we propose HEaaN.MLIR, a compiler that performs optimizations across polynomial operations. Also, we propose Poly and ModArith, compiler intermediate representations (IRs) for integer polynomial arithmetic and modulus arithmetic on integer arrays. HEaaN.MLIR has compiler optimizations that are motivated by manual optimizations that HE developers do. These include optimizing modular arithmetic operations, fusing loops, and vectorizing integer arithmetic instructions. HEaaN.MLIR can parse a program consisting of the Poly and ModArith instructions and generate a high-performance, multithreaded machine code for a CPU. Our experiment shows that the compiled operations outperform heavily optimized open-source and commercial HE libraries by up to 3.06x in a single thread and 4.55x in multiple threads.

Springer Numbers and Arnold Families Revisited

For the calculation of Springer numbers (of root systems) of type $$B_n$$ and $$D_n$$ , Arnold introduced a signed analogue of alternating permutations, called $$\beta _n$$ -snakes, and derived recurrence relations for enumerating the $$\beta _n$$ -snakes starting with k. The results are presented in the form of double triangular arrays ( $$v_{n,k}$$ ) of integers, $$1\le |k|\le n$$ . An Arnold family is a sequence of sets of such objects as $$\beta _n$$ -snakes that are counted by $$(v_{n,k})$$ . As a refinement of Arnold’s result, we give analogous arrays of polynomials, defined by recurrence, for the calculation of the polynomials associated with successive derivatives of $$\tan x$$ and $$\sec x$$ , established by Hoffman. Moreover, we provide some new Arnold families of combinatorial objects that realize the polynomial arrays, which are signed variants of André permutations and Simsun permutations.

MacMahon’s statistics on higher-dimensional partitions

Abstract We study some combinatorial properties of higher-dimensional partitions which generalize plane partitions. We present a natural bijection between d-dimensional partitions and d-dimensional arrays of nonnegative integers. This bijection has a number of important applications. We introduce a statistic on d-dimensional partitions, called the corner-hook volume, whose generating function has the formula of MacMahon’s conjecture. We obtain multivariable formulas whose specializations give analogues of various formulas known for plane partitions. We also introduce higher-dimensional analogues of dual stable Grothendieck polynomials which are quasisymmetric functions and whose specializations enumerate higher-dimensional partitions of a given shape. Finally, we show probabilistic connections with a directed last passage percolation model in $\mathbb {Z}^d$ .

Machine learning and IoT-based garbage detection system for smart cities

Today, detecting waste, collecting it, processing it, and getting rid of it are among the most significant environmental issues in developing and undeveloped counties. It has been observed that a large amount of garbage remains strewn on the roadside. This study presented a garbage detection technology such as machine learning and gadgets connected to the Internet of Things (IoT), such as an IP-enabled CCTV camera, to take pictures and send them to the city’s main server. The input images are transformed into a two-dimension array of integers using Python modules and divided into the garbage and no garbage classes. There is an 80:20 split between the training and testing datasets from the input dataset. Preprocessed images are then utilised as inputs for a wide range of machine learning and neural network models for classification; these include K-Nearest Neighbour (KNN), Logistic Regression (LR), Naive Bayes (NB), and Support Vector Machine (SVM). The test data sets are applied, and a confusion matrix is formed for all models to analyse the efficiency and performance of the trained models. Results from the confusion matrix are contrasted with those from the area under the Receiver characteristics operating curve (AUC). As a result, the ConvNet model is best suited for classifying garbage or no garbage present in open space, and the LR model proposed best suits the garbage detection problem. The proposed models are best suitable for improving the efficiency of existing garbage identification systems and developing a new system for smart cities.

String inference from longest-common-prefix array

The suffix array, perhaps the most important data structure in modern string processing, is often augmented with the longest common prefix (LCP) array which stores the lengths of the longest common prefixes for lexicographically adjacent suffixes of a string. Together the two arrays are roughly equivalent to the suffix tree with the LCP array representing the tree shape.In order to better understand the combinatorics of LCP arrays, we consider the problem of inferring a string from an LCP array, i.e., determining whether a given array of integers is a valid LCP array, and if it is, reconstructing some string or all strings with that LCP array. There are recent studies of inferring a string from a suffix tree shape but using significantly more information (in the form of suffix links) than is available in the LCP array.We provide two main results. (1) We describe two algorithms for inferring strings from an LCP array when we allow a generalized form of LCP array defined for a multiset of cyclic strings: a linear time algorithm for binary alphabet and a general algorithm with polynomial time complexity for a constant alphabet size. (2) We prove that determining whether a given integer array is a valid LCP array is NP-complete when we require more restricted forms of LCP array defined for a single cyclic or non-cyclic string or a multiset of non-cyclic strings. The result holds whether or not the alphabet is restricted to be binary. In combination, the two results show that the generalized form of LCP array for a multiset of cyclic strings is fundamentally different from the other more restricted forms.

A scalable algorithm for many-body dissipative particle dynamics using multiple general purpose graphic processing units

We present a novel algorithm for the many-body Dissipative Particle Dynamics (DPD) forces calculation which allows to efficiently scale the DL_MESO software package on Multiple General Purpose Graphic Processing Units. Together with the extension to 64-bit integer arrays and addition of hard surface boundary conditions, the proposed algorithm allows to simulate very large complex mesoscale systems up to 14 billion beads. The implementation takes advantages of the CUDA language stream features to overlap the exchange of particle positions and local densities and the computation of the short range forces. We tested a water drop between two plates system using tree of the main European supercomputers: Piz Daint, Marconi and JUWELS. Results shows an improvement on the speedup compared to a naive implementation up to 1.5x when using 1024 GPUs.

Green Path Project Develops Using Image Processing

Abstract: The Green Path project develops Image Processing which includes changing the nature of an image in the order to improve its pictorial information for human interpretation, for autonomous machine perception. In digital image processing, image is converted to an array of small integers, called pixels, representing a physical quantity such as scene radiance, stored in a digital memory, and processed by computer. Interest in digital image processing methods is in two principal applications areas: first one is improvement of pictorial information for human interpretation; and second one is processing of image data for storage, transmission, and representation for image or machine. Edge detection is one of the most difficult tasks in image processing hence it is a problem of fundamental importance in image processing method. In this paper we find out different steps of digital image processing. Like, a high-speed filter implementation is presented. Then Filter solves the purpose of removing the impulse noise from the image and reducing distortion in the image. The Image Processing is very important in the project. Then indicate a specific spot required for pointing to show the main spot for empty places. So, we required to divide the image pixelization points and coloring into different parts. And last one spotted the spot which is very important for project. Image processing is a method to perform operations on an image to extract information from it or enhance it. Digital image processing has a broad range of applications such as image restoration, medical imaging, remote sensing, image segmentation, etc. Every process requires a different technique. Keywords: Image Processing, scene radiance

Learning deep neural networks' architectures using differential evolution. Case study: Medical imaging processing

The COVID-19 pandemic has changed the way we practice medicine. Cancer patient and obstetric care landscapes have been distorted. Delaying cancer diagnosis or maternal-fetal monitoring increased the number of preventable deaths or pregnancy complications. One solution is using Artificial Intelligence to help the medical personnel establish the diagnosis in a faster and more accurate manner. Deep learning is the state-of-the-art solution for image classification. Researchers manually design the structure of fix deep learning neural networks structures and afterwards verify their performance. The goal of this paper is to propose a potential method for learning deep network architectures automatically. As the number of networks architectures increases exponentially with the number of convolutional layers in the network, we propose a differential evolution algorithm to traverse the search space. At first, we propose a way to encode the network structure as a candidate solution of fixed-length integer array, followed by the initialization of differential evolution method. A set of random individuals is generated, followed by mutation, recombination, and selection. At each generation the individuals with the poorest loss values are eliminated and replaced with more competitive individuals. The model has been tested on three cancer datasets containing MRI scans and histopathological images and two maternal-fetal screening ultrasound images. The novel proposed method has been compared and statistically benchmarked to four state-of-the-art deep learning networks: VGG16, ResNet50, Inception V3, and DenseNet169. The experimental results showed that the model is competitive to other state-of-the-art models, obtaining accuracies between 78.73% and 99.50% depending on the dataset it had been applied on.

USING ARDUINO IN STUDYING THE SECTION "ALGORITHMIZATION AND PROGRAMMING" OF THE SCHOOL INFORMATICS COURSE AS A PREPARATION FOR LEARNING THE BASICS OF ENGINEERING

The article discusses the possibility of preparing schoolchildren for the study of the basics of engineering in the framework of training in the section "Algorithmization and programming" of the informatics course based on the Arduino platform. Such training can be organized by including in the material under study practical work on the creation of electronic controlled devices: a weather station, a "smart" night light, a traffic light, a music box, etc. A simple assembly system allows using the Arduino kit for 40-minute lessons: 10 to 20 minutes depending on the complexity of the project. Also, in the classroom, it is proposed to use Tinkercad, a free 3D modeling program that allows you to assemble an electronic device online, write a code, test it, make sure it works, and download a sketch code for further use. The article presents a methodology for carrying out work in the 10th grade to create a "music box" device in Arduino. The purpose of this work is to generalize and systematize the topic "One-dimensional arrays of integers". The practical part of the lesson involves working with light, sound, control, and rotation of the figure. Thus, the concept of "array", algorithmic structures "loop" and "branching" are repeated. This work can also be carried out as an extracurricular event, for example, within the framework of a subject week.

On odd ranks of odd Durfee symbols

An odd Durfee symbol of [Formula: see text] is an array of positive odd integers and a subscript [Formula: see text], [Formula: see text] such that [Formula: see text], [Formula: see text], and [Formula: see text]. Andrews defined the odd rank of an odd Durfee symbol as [Formula: see text]. Let [Formula: see text] be the number of odd Durfee symbols of [Formula: see text] with odd rank congruent to [Formula: see text] modulo [Formula: see text]. We decompose the generating function of [Formula: see text] into modular and mock modular parts. Specifically, we derive some special cases of the generating functions of [Formula: see text] for [Formula: see text]. Some generating functions are related to classical mock theta functions.

Practical trade‐offs for the prefix‐sum problem

AbstractGiven an integer array A, the prefix‐sum problem is to answer sum(i) queries that return the sum of the elements in A[0..i], knowing that the integers in A can be changed. It is a classic problem in data structure design with a wide range of applications in computing from coding to databases. In this work, we propose and compare practical solutions to this problem, showing that new trade‐offs between the performance of queries and updates can be achieved on modern hardware.

A Novel Block Cipher Based on Randomly Shuffled Key Strings

Information security has immense importance in today’s world of digital communication. Sharing information between intended parties while keeping it secret from any outsider is a challenging task. Cryptography provides the security of information by keeping the information secret among only the communicating parties. In this paper, a new cryptographic algorithm is propounded, which is a block cipher algorithm. A positive integer, n, two $$n\times 256$$ tables of ASCII characters, and a string, Mapper, are used as shared private keys in the proposed method. Every row of the key tables and Mapper contains a random permutation of all the ASCII characters. In the beginning of encryption of a block, an Exclusive Or (XOR) operation is done between the block and a random array of integers. Then, a 3-round substitution operation is done using the key tables and Mapper. We found that the encryption and the decryption speed of the proposed algorithm is faster than the encryption–decryption speed of Advanced Encryption Standard (AES) and Data Encryption Standard (DES). The strength of the proposed cipher was analyzed in the light of some popular attacks, and we found that the system is strong enough against those attacks. We hope that if integrated with the modern cryptography protocols, this method will enhance modern digital communication security.

A parallel solution to finding nodal neighbors in generic meshes

In this paper we specifically present a parallel solution to finding the one-ring neighboring nodes and elements for each vertex in generic meshes. The finding of nodal neighbors is computationally straightforward but expensive for large meshes. To improve the efficiency, the parallelism is adopted by utilizing the modern Graphics Processing Unit (GPU). The presented parallel solution is heavily dependent on the parallel sorting, scan, and reduction. Our parallel solution is efficient and easy to implement, but requires the allocation of large device memory.•Our parallel solution can generate the speedups of approximately 55 and 90 over the serial solution when finding the neighboring nodes and elements, respectively.•It is easy to implement due to the reason it does not need to perform the mesh-coloring before finding neighbors•There are no complex data structures, only integer arrays are needed, which makes our parallel solution very effective.

Acceleration of large integer arrays sorting using ranges of values and frequencies of elements

Acceleration of large integer arrays sorting using ranges of values and frequencies of elements

Stepping Stones to Inductive Synthesis of Low-Level Looping Programs

Inductive program synthesis, from input/output examples, can provide an opportunity to automatically create programs from scratch without presupposing the algorithmic form of the solution. For induction of general programs with loops (as opposed to loop-free programs, or synthesis for domain-specific languages), the state of the art is at the level of introductory programming assignments. Most problems that require algorithmic subtlety, such as fast sorting, have remained out of reach without the benefit of significant problem-specific background knowledge. A key challenge is to identify cues that are available to guide search towards correct looping programs. We present MAKESPEARE, a simple delayed-acceptance hillclimbing method that synthesizes low-level looping programs from input/output examples. During search, delayed acceptance bypasses small gains to identify significantly-improved stepping stone programs that tend to generalize and enable further progress. The method performs well on a set of established benchmarks, and succeeds on the previously unsolved “Collatz Numbers” program synthesis problem. Additional benchmarks include the problem of rapidly sorting integer arrays, in which we observe the emergence of comb sort (a Shell sort variant that is empirically fast). MAKESPEARE has also synthesized a record-setting program on one of the puzzles from the TIS100 assembly language programming game.

Chain code compression with modified interpolative coding

A lossless chain code compression algorithm consisting of Burrows-Wheeler Transform, Move-To-Front Transform, and a modified interpolative coding is presented in this paper. The interpolative coding divides a strictly increasing array of integers and encodes the middle element recursively, where the required number of bits is obtained from the difference of the array’s border values. In this paper, it is shown that, for some intervals, Golomb coding is more efficient than the interpolative coding. This fact was used to construct a hybrid approach that yields slightly better compression results than the state-of-the-art while compressing Three OrThogonal chain code, Freeman chain code in four and eight directions, and Vertex Chain Code.

Counting with Borel’s triangle

Borel’s triangle is an array of integers closely related to the classical Catalan numbers. In this paper we study combinatorial statistics counted by Borel’s triangle. We present various combinatorial interpretations of Borel’s triangle in terms of lattice paths, binary trees, and pattern avoiding permutations and matchings, and derive a functional equation that is useful in analyzing the involved structures.