Lattice translates of a polytope and the Frobenius problem

For a non-negative integer $p$, one of the generalized Frobenius numbers, which is called the $p$-Frobenius number, is the largest integer that is represented at most in $p$ ways as a linear combination with nonnegative integer coefficients of a given set of positive integers whose greatest common divisor is one. The famous so-called Frobenius number proposed by Frobenius is reduced to the $0$-Frobenius number when $p=0$. The explicit formula for the Frobenius number with two variables was found in the 19th century, but a formula with more than two variables is very difficult to find, and closed formulas of Frobenius numbers have been found only in special cases such as geometric, Thabit, Mersenne, and so on. The case of $p>0$ was even more difficult, and not a single formula was known. However, most recently, we have finally succeeded in giving the $p$-Frobenius numbers as closed-form expressions of the triangular number triplet ([8]), repunits ([9]), Fibonacci triplet ([12]) and Jacobsthal triplet ([11]). In this paper, we give closed-form expressions of the $p$-Frobenius number for the finite sequence $\{a b^n-c\}_n$, where $a$, $b$ and $c$ are integers with $a\ge 1$, $b\ge 2$ and $c\ne 0$. This sequence includes the cases for geometric, Thabit and Mersenne as well as their variations.

Let p = (p1,…,pn,) be a vector of positive integers whose greatest common divisor is unity. The Frobenius problem is to find the largest integer f* which cannot be written as a nonnegative integral combination of the pi. In this note we relate the Frobenius problem to the topic of maximal lattice free bodies and describe an algorithm for n = 3.

Consider positive integral solutions to the equation a0x0 + … + anxn = t. In the so called unbounded subset sum problem, the objective is to decide whether such a solution exists, whereas in the Frobenius problem, the objective is to compute the largest t such that there is no such solution. In this paper we study the algorithmic complexity of the unbounded subset sum, the Frobenius problem and a generalization of both problems. More precisely, we study pseudo-polynomial time algorithms with a running time that depends on the smallest number a0 or respectively the largest number an. For the parameter a0, we show that all considered problems are subquadratically equivalent to (min, +)-convolution, a fundamental algorithmic problem from the area of fine-grained complexity. By this equivalence, we obtain hardness results for the considered problems (based on the assumption that an algorithm with a subquadratic running time for (min, +)-convolution does not exist) as well as algorithms with improved running time. The proof for the equivalence makes use of structural properties of solutions, a technique that was developed in the area of integer programming. In case of the complexity of the problems parameterized by an, we present improved algorithms. For example we give a quasi linear time algorithm for the Frobenius problem as well as a hardness result based on the strong exponential time hypothesis.

In this correspondence, a new viewpoint is proposed for estimating an image from its distorted versions in presence of noise without the a priori knowledge of the distortion functions. In z-domain, the desired image can be regarded as the greatest common polynomial divisor among the distorted versions. With the assumption that the distortion filters are finite impulse response (FIR) and relatively coprime, in the absence of noise, this becomes a problem of taking the greatest common divisor (GCD) of two or more two-dimensional (2-D) polynomials. Exact GCD is not desirable because even extremely small variations due to quantization error or additive noise can destroy the integrity of the polynomial system and lead to a trivial solution. Our approach to this blind deconvolution approximation problem introduces a new robust interpolative 2-D GCD method based on a one-dimensional (1-D) Sylvester-type GCD algorithm. Experimental results with both synthetically blurred images and real motion-blurred pictures show that it is computationally efficient and moderately noise robust.

We present new methods for computing the greatest common right divisor of polynomial matrices. These methods involve the recently studied generalized Sylvester and generalized Bezoutian resultant matrices, which require no polynomial operations. They can provide a row proper greatest common right divisor, test for coprimeness and calculate dual dynamical indices. The generalized resultant matrices are developments of the scalar Sylvester and Bezoutian resultants and many of the familiar properties of these latter matrices are demonstrated to have analogs with the properties of the generalized resultant matrices for matrix polynomials.

A matrix Euclidean algorithm induced by state space realization

Differential privacy is a definition giving a strong privacy guarantee even in the presence of auxiliary information. In this work we pursue the application of geometric techniques for achieving differential privacy, a highly promising line of work initiated by Hardt and Talwar [26], focusing on the problem of marginal release. Here, a database is a collection of the data of n individuals, each characterized by d binary attributes. A k-way marginal query is specified by a subset S of k attributes, together with a |S|-dimensional binary vector β specifying their values. The true answer to this query is a count of the number of people in the database whose attribute vector restricted to S agrees with β. Information theoretically, the error complexity of marginal queries -- how wrong do the answers have to be in order to preserve differential privacy -- is well-understood: the perquery additive error is known to be at least Ω(min{√n,dk/2}) and at most O(√nd1/4,dk/2). However, no polynomial time algorithm with error complexity as low as the information theoretic upper bound is known for small n. We present a polynomial time algorithm that matches the best known information-theoretic bounds when k = 2; more generally, by reducing to the case k = 2, for any distribution on marginal queries, our algorithm achieves average error at most O(√nd[k/2]/4), an improvement over previous work on when k is small and when error o(n) is desirable. Using private boosting we are also able to give nearly matching worst-case error bounds. Our algorithms are based on the geometric techniques of Nikolov, Talwar, and Zhang [29], wherein a vector of sufficiently noisy answers is projected onto a particular convex body. We reduce projection, which is expensive, to a simple geometric question: given (a succinct representation of) a convex body K, find a containing convex body L that one can efficiently optimize over, while keeping the Gaussian width of L small. This reduction is achieved by a careful use of the Frank-Wolfe algorithm.

SummaryThe implementation of cryptography algorithms is constantly observing threat from multiple techniques in breaking secure system. The current trend in breaking the secure system of cryptography is by power analysis technique. In order to break the secure key, the energy consumed in the digital circuit during computation is measured. This technique is commonly known as differential power analysis, explored by the hackers to break the secure systems. To circumvent these type of attacks, it is necessary to explore different designs, which dissipate less energy. Ideally, reversible circuits dissipate zero energy. We present a new reversible architecture for greatest common divisor (GCD) computation using modified Binary GCD algorithm. We present the generalized design methodology of reversible GCD computation unit. We compare the proposed GCD computation design with the existing design. The proposed reversible GCD architecture takes less number of iterations compared with the existing GCD architecture in the literature. The proposed design outperforms the existing GCD design in terms of Quantum Cost, Gate Count, and Ancilla Inputs. Copyright © 2016 John Wiley & Sons, Ltd.

We give a probabilistic algorithm for computing the greatest common divisor (GCD) of two polynomials over an algebraic number field. We can compute the GCD using O(llog5(l)) expected binary operations where l is the size of the GCD given by standard estimations. Since we require time Ω(l) just to write down the GCD, the algorithm is close to optimal.

A faster FPTAS for knapsack problem with cardinality constraint

Under the word RAM model, we design three data structures that can be constructed in O(n √{lg n}) time over n points in an n × n grid. The first data structure is an O(n lg^ε n)-word structure supporting orthogonal range reporting in O(lg lg n+k) time, where k denotes output size and ε is an arbitrarily small constant. The second is an O(n lg lg n)-word structure supporting orthogonal range successor in O(lg lg n) time, while the third is an O(n lg^ε n)-word structure supporting sorted range reporting in O(lg lg n+k) time. The query times of these data structures are optimal when the space costs must be within O(n polylog n) words. Their exact space bounds match those of the best known results achieving the same query times, and the O(n √{lg n}) construction time beats the previous bounds on preprocessing. Previously, among 2d range search structures, only the orthogonal range counting structure of Chan and Pǎtraşcu (SODA 2010) and the linear space, O(lg^ε n) query time structure for orthogonal range successor by Belazzougui and Puglisi (SODA 2016) can be built in the same O(n √{lg n}) time. Hence our work is the first that achieve the same preprocessing time for optimal orthogonal range reporting and range successor. We also apply our results to improve the construction time of text indexes.

The study of the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD) explores the fundamental field of number theory. In addition to being essential to theoretical number theory, these mathematical ideas have extensive applications in computer science, algebra, and real-world settings like network routing and scheduling. Using techniques like the Euclidean algorithm for GCD calculation and prime factorization for LCM, the study starts by examining the fundamental concepts and notations of GCD and LCM. Then the author goes further into the detail about how these ideas are used to polynomial equations, highlighting how they may be used to solve challenging issues and simplify algebraic expressions. The studys practical applications, which demonstrate how GCD and LCM optimize scheduling tasks and boost network routing efficiency, are presented in the conclusion. The results of this study highlight the broad impact and adaptability of LCM and GCD in theoretical and applied mathematics, providing valuable insights for further research in these domains.

We propose an algorithm to compute a greatest common divisor (GCD) of univariate polynomials with large integer coefficients on Graphics Processing Units (GPUs). At the highest level, our algorithm relies on modular techniques to decompose the problem into subproblems that can be solved separately. Next, we employ resultant-based or matrix algebra methods to compute a GCD of each modular image in parallel. Our approach exhibits block structure to distribute the computation of a single modular GCD over several thread blocks, and thus to remove any hardware limitations on the maximal size of polynomials that can be handled. To "combine" all modular results, we have adopted Mixed-Radix Conversion (MRC) algorithm running on the GPU. Our approach shows a significant speed-up over host-based GCD algorithm from Maple 13.

The objective of this is to show the application of Euclid's Lemma to calculate the Greatest Common Divisor in Elementary Education and evaluate the receptivity that the students had the method. We studied and wrote about the division in integers, divisibility and properties involved, the theorem euclidian division, the greatest common divisor, Euclid's Lemma, the euclidean algorithm, the Fundamental Theorem of Arithmetic and also on the Diophantine Equations they are an interesting application of the Greatest Common Divisor. We use a short course on Greatest Common Divisor in two elementary school classes and present some results obtained.

This paper presents a preliminary report on a new algorithm for computing the Greatest Common Divisor (GCD) of two multivariate polynomials over the integers. The algorithm is strongly influenced by the method used for factoring multivariate polynomials over the integers. It uses an extension of the Hensel lemma approach originally suggested by Zassenhaus for factoring univariate polynomials over the integers. We point out that the cost of the Modular GCD algorithm applied to sparse multivariate polynomials grows at least exponentially in the number of variables appearing in the GCD. This growth is largely independent of the number of terms in the GCD. The new algorithm, called the EZ (Extended Zassenhaus) GCD Algorithm, appears to have a computing bound which in most cases is a polynomial function of the number of terms in the original polynomials and the sum of the degrees of the variables in them. Especially difficult cases for the EZ GCD Algorithm are described. Applications of the algorithm to the computation of contents and square-free decompositions of polynomials are indicated.