Query-Efficient Algorithms for Polynomial Interpolation over Composites

Systolic algorithm for polynomial interpolation and related problems

Matching and packing problems have formed an important class of NP-hard problems. There have been a number of recently developed techniques for parameterized algorithms for these problems, including greedy localization, color-coding plus dynamic programming, and randomized divide-and-conquer. In this paper, we provide further theoretical study on the structures of these problems, and develop improved algorithmic methods that combine existing and new techniques to obtain improved algorithms for matching and packing problems. For the 3-set packing problem, we present a deterministic algorithm of time O *(4.613k), which significantly improves the previous best deterministic algorithm of time O *(12.83k). For the 3-d matching problem, we develop a new randomized algorithm of running time O *(2.323k) and a new deterministic algorithm of running time O *(2.773k). Our randomized algorithm improves the previous best randomized algorithm of running time O *(2.523k), and our deterministic algorithm significantly improves the previous best deterministic algorithm of running time O *(12.83k). Our results also imply improved algorithms for various triangle packing problems in graphs.KeywordsTime ComplexityDynamic ProgrammingMatch ProblemPacking ProblemDynamic Programming AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A directed graph D is semicomplete if for every pair x, y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph D=(V,A) and a pair of natural numbers k and ell , we are to decide whether there is a subset X of V of size k such that the largest strongly connected component in D-X has at most ell vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for ell =1. We study the parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: k, ell ,ell +k and n-ell . In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time O^*(2^{16k}) but not in time O^*(2^{o(k)}) unless the Exponential Time Hypothesis (ETH) fails. The upper bound O^*(2^{16k}) implies the upper bound O^*(2^{16(n-ell )}) for the parameter n-ell . We complement the latter by showing that there is no algorithm of time complexity O^*(2^{o({n-ell })}) unless ETH fails. Finally, we improve (in dependency on ell ) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter ell +k on general digraphs from O^*(2^{O(kell log (kell ))}) to O^*(2^{O(klog (kell ))}). Note that Drange, Dregi and van ’t Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time O^*(2^{o(klog ell )}) unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity O^*(2^{o(klog k)}).

Algebraic coding theory is one of the areas that routinely gives rise to computational problems involving various structured matrices, such as Hankel, Vandermonde, Cauchy matrices, and certain generalizations thereof. Their structure has often been used to derive efficient algorithms; however, the use of the structure was pattern-specific, without applying a unified technique. In contrast, in several other areas, where structured matrices are also widely encountered, the concept of displacement rank was found to be useful to derive efficient algorithms in a unified manner (i.e., not depending on a particular pattern of structure). The latter technique allows one to compress, in a unified way, different types of n × n structured matrices to only αn parameters. This typically leads to computational savings (in many applications the number α, called the displacement rank, is a small fixed constant).In this paper we demonstrate the power of the displacement structure approach by deriving in a unified way efficient algorithms for a number of decoding problems. We accelerate the Sudan's list decoding algorithm for Reed-Solomon codes, its generalization to algebraic-geometric codes by Shokrollahi and Wasserman, and the improvement of Guruswami and Sudan in the case of Reed-Solomon codes. In particular, we notice that matrices that occur in the context of list decoding have low displacement rank, and use this fact to derive algorithms that use O(n2 l) and O(n7/3 l) operations over the base field for list decoding of Reed-Solomon codes and algebraic-geometric codes from certain plane curves, respectively. Here l denotes the length of the list; assuming that l is constant, this gives algorithms of running time O(n2) and O(n7/3), which is the same as the running time of conventional decoding algorithms. We also present efficient parallel algorithms for the above tasks.To the best of our knowledge this is the first application of the concept of displacement rank to the unified derivation of several decoding algorithms; the technique can be useful in finding efficient and fast methods for solving other decoding problems.

This article provides an in-depth exploration of Big-Oh and small-oh notations, shedding light on their practical implications in the analysis of algorithm complexity. Big-Oh notation offers a valuable tool for estimating an upper bound on the growth rate of an algorithm's running time, whereas small-oh notation delineates a lower limit on this growth rate. The piece delves into a comprehensive examination of various complexity classes that emerge through the application of Big-Oh notation, underscoring the significance of small-oh notation as it complements and enriches complexity analysis. In the realm of programming and computer science, the employment of these notations holds paramount importance. They empower developers and researchers to make informed decisions regarding algorithm selection and optimization. It is crucial to recognize that while complexity analysis is a vital facet of effective programming, ongoing research endeavors may yield more refined methodologies and approaches within this domain. By understanding and harnessing the power of Big-Oh and small-oh notations, professionals can effectively evaluate algorithm efficiency and scalability. This knowledge equips them with the ability to design and implement algorithms that meet specific performance criteria, which is pivotal in the ever-evolving landscape of technology and computation. As pushing the boundaries of what is possible in the field of algorithm design is being continued, these notations remain invaluable tools for navigating the complex terrain of algorithmic analysis and optimization. By embracing Big-Oh and small-oh notations, professionals can finely assess algorithmic efficiency, ensuring they meet performance criteria in the evolving technological landscape. These notations remain indispensable for algorithmic analysis.

We show how to solve a number of problems in numerical linear algebra, such as least squares regression, lp-regression for any p ≥ 1, low rank approximation, and kernel regression, in time T(A)poly(log(nd)), where for a given input matrix A ∈ Rn×d, T(A) is the time needed to compute A · y for an arbitrary vector y ∈ Rd. Since T(A) ≤ O(nnz(A)), where nnz(A) denotes the number of non-zero entries of A, the time is no worse, up to polylogarithmic factors, as all of the recent advances for such problems that run in input-sparsity time. However, for many applications, T(A) can be much smaller than nnz(A), yielding significantly sublinear time algorithms. For example, in the overconstrained (1+ε)-approximate polynomial interpolation problem, A is a Vandermonde matrix and T(A) = O(n log n); in this case our running time is n · poly (log n) + poly (d/ε) and we recover the results of Avron, Sindhwani, and Woodruff (2013) as a special case. For overconstrained autoregression, which is a common problem arising in dynamical systems, T(A) = O(n log n), and we immediately obtain n· poly (log n) + poly(d/ε) time. For kernel autoregression, we significantly improve the running time of prior algorithms for general kernels. For the important case of autoregression with the polynomial kernel and arbitrary target vector b ∈ Rn, we obtain even faster algorithms. Our algorithms show that, perhaps surprisingly, most of these optimization problems do not require much more time than that of a polylogarithmic number of matrix-vector multiplications.

The success of data mining tasks depends on the selection of an accurate classification algorithm. To evaluate the efficacy of a classification algorithm of interest, classification accuracy and algorithm run time are used. Trialing with several algorithms can raise the cost of the data mining operation. In this paper, information gain that is feature selection algorithm is used to improve the classification accuracy and reduce run time of classification technique. The paper uses classification techniques such as part rule, sequential minimal optimization, naïve Bayes, and random forest for a medical dataset. Dataset used in the paper is regular about physical activity, which is vital to maintain or even improve a person’s health. In this paper, the procedure has been built based on record the accuracy and run time of the classification algorithms before and after using the information_gain algorithm. As a result, the accuracy after the information_gain algorithm was the highest for the part rule algorithm and the best run time to implement the naïve Bayes algorithm.KeywordsPhysical activityFeature selectionInformation gainPart ruleClassification

We show that there is a set of pointsp 1,p 2,...,p n such that any arithmetic circuit of depthd for polynomial evaluation (or interpolation) at these points has size $$\Omega \left( {\frac{{n\log n}}{{\log (2 + d/\log n}}} \right).$$ Moreover, for circuits of sub-logarithmic depthd, we obtain a lower bound of Ω(dn 1+1/d ) on its size.

A parameterized algorithm for the hyperplane-cover problem

The paper is devoted to the problem of multivariate polynomial interpolation and its application to quantum secret sharing. We show that using quantum Fourier transform one can produce the protocol for quantum secret sharing distribution.

Unisolvency for multivariate polynomial interpolation in Coatmèlec configurations of nodes

In this paper we deal with association rule mining in the context of a complex, interactive and iterative knowledge discovery process. After a general introduction covering the basics of association rule mining and of the knowledge discovery process in databases we draw the attention to the problematic aspects concerning the integration of both. Actually, we come to the conclusion that with regard to human involvement and interactivity the current situation is far from being satisfying. In our paper we tackle this problem on three sides: First of all there is the algorithmic complexity. Although today’s algorithms efficiently prune the immense search space the achieved run times do not allow true interactivity. Nevertheless we present a rule caching schema that significantly reduces the number of mining runs. This schema helps to gain interactivity even in the presence of extreme run times of the mining algorithms. Second, today the mining data is typically stored in a relational database management system. We present an efficient integration with modern database systems which is one of the key factors in practical mining applications. Third, interesting rules must be picked from the set of generated rules. This might be quite costly because the generated rule sets normally are quite large whereas the percentage of useful rules is typically only a very small fraction. We enhance the traditional association rule mining framework in order to cope with this situation.KeywordsAssociation RuleMining AlgorithmFrequent ItemsetsAssociation Rule MiningGeneralize Association RuleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let be a class of probability distributions over the discrete domain [n] = {1, …, n}. We show that if satisfies a rather general condition – essentially, that each distribution in can be well-approximated by a variable-width histogram with few bins – then there is a highly efficient (both in terms of running time and sample complexity) algorithm that can learn any mixture of k unknown distributions from . We analyze several natural types of distributions over [n], including log-concave, monotone hazard rate and unimodal distributions, and show that they have the required structural property of being well-approximated by a histogram with few bins. Applying our general algorithm, we obtain near-optimally efficient algorithms for all these mixture learning problems as described below. More precisely, Log-concave distributions: We learn any mixture of k log-concave distributions over [n] using k · Õ(1/ε4) samples (independent of n) and running in time Õ(k log(n)/ε4) bit-operations (note that reading a single sample from [n] takes Θ(log n) bit operations). For the special case k = 1 we give an efficient algorithm using Õ(1/ε3) samples; this generalizes the main result of [DDS12b] from the class of Poisson Binomial distributions to the much broader class of all log-concave distributions. Our upper bounds are not far from optimal since any algorithm for this learning problem requires Ω(k/ε5/2) samples. Monotone hazard rate (MHR) distributions: We learn any mixture of k MHR distributions over [n] using O(k log(n/ε)/ε4) samples and running in time Õ(k log (n)/ε4) bit-operations. Any algorithm for this learning problem must use Ω(k log(n)/ε3) samples. Unimodal distributions: We give an algorithm that learns any mixture of k unimodal distributions over [n] using O(k log(n)/ε4) samples and running in time Õ(k log2(n)/ε4) bit-operations. Any algorithm for this problem must use Ω(k log(n)/ε3) samples.

Problem statement. Currently, the main efforts in the development of the general theory of extrapolation and its practical applications are aimed at predicting processes, moments of time and fields. At the same time, as a result of increased processing quality requirements, it is necessary to take into account the effect of various kinds of interference and noise. Therefore, most of the existing approaches to solving extrapolation problems rely to some extent on analytical methods. At the same time, the formulation of tasks mainly concerns processes and sequences, the changes of which are largely determined by random components. Due to the fact that the mathematical description of objects and the environment (or their probabilistic characteristics) is often unknown in advance, and it is not possible to determine them, an adaptive approach is typical for solving such problems. With an adaptive approach, some suboptimal solution is sought, obtained, for example, by using maximum likelihood or minimum average risk criteria. It is from this point of view that the problems of filtering, interpolation and extrapolation are often considered. The main focus is on the application of the basic principles of the theory of systems and adaptive Kalman filtering. It is assumed that the Gaussian approximation, a linear model of the system, as well as linear processing of input sequences are used. The optimal joint assessment of the parameters of the state model leads to a nonlinear filter structure, the implementation of which is associated with high requirements for performance and computer memory. Therefore, preference is usually given to suboptimal methods, on the basis of which simplified algorithms are formed that do not require large amounts of computer memory. Goal – development of adaptive algorithms for the functioning of multidimensional linear and nonlinear extrapolators implemented in the time and private domain by digital signal processing (DSP) methods, as well as analysis of the quality characteristics of their functioning and application features. Algorithms for the functioning of multidimensional extrapolators are proposed using examples of the development of adaptive algorithms using the method of linear prediction "forward" and "backward" and their ombinations. Algorithms for the operation of nonlinear "forward" extrapolators based on Kolmogorov-Gabor series in the time and frequency domains have been developed. Adaptive algorithms of nonlinear extrapolators based on the use of partitioning methods for calculating convolution are proposed. It is shown that algorithms for the functioning of nonlinear extrapolators based on the use of partitioning methods for calculating convolution are more efficient than linear shift-invariant systems and at the same time require less memory. The features of creating adaptive filtering, interpolation and extrapolation algorithms using neurocomputers built on the basis of existing microchips of domestic and foreign production are considered.

Let G = (V, E) be an unweighted undirected graph with n vertices and m edges. Dor, Halperin, and Zwick [FOCS 1996, SICOMP 2000] presented an Õ(n2)-time algorithm that computes estimated distances with a multiplicative approximation of 3. Berman and Kasiviswanathan [WADS 2007] improved the approximation of Dor et al. and presented an Õ(n2)-time algorithm that produces for every u, v ϵ V an estimate (u, v) such that: dG(u, v) ≤ (u, v) ≤ 2dG(u, v) + 1. We refer to such an approximation as an (α, β)-approximation, where a is the multiplicative approximation and β is the additive approximation. A prerequisite for an O(n2−ε)-time algorithm, where ε ϵ (0, 1), is a data structure that uses O(n2−δ) space, for some δ ≥ ε, and answers queries in constant time. An O(n2−ε)-time (3, 0)-approximation algorithm became plausible after Thorup and Zwick [STOC 2001, JACM 2005] presented their approximate distance oracles, and in particular an O(n1.5)-space data structure that reports a (3, 0)-approximate distance in O(1) time. Indeed, using Thorup and Zwick distance oracles together with more ideas, Baswana, Gaur, Sen, and Upadhyay [ICALP 2008] improved the running time of Dor et al., and obtained an O(n2−ε) time algorithm, at the cost of introducing also an additive approximation. They presented an algorithm that in Õ(m + n23/12) expected running time constructs an O(n1.5)-space data structure, that in O(1) time reports a (3, 14)-approximate distance. An O(n2−ε)-time (2, 1)-approximation algorithm became plausible only after Pǎtraşcu and Roditty [FOCS 2010, SICOMP 2014] presented an O(n5/3)-space data structure that reports (2, 1)-approximate distances in O(1) time. However, only few years ago, Sommer [ICALP 2016] obtained an Õ(n2) time algorithm that computes a (2, 1)-distance oracle with Õ(n5/3) space. This leads to the following natural question of whether Ω(n2) time is a lower bound for any (3−α, β)-approximation, where α ϵ (0, 1), and β is constant. In this paper we show that this is not the case by presenting an algorithm that for every ε ϵ (0, 1/2) computes in Õ(m) + n2−Ω(ε) time an -space data structure that in O(1/ε) time reports, for every u, v ϵ V, an estimate (u, v) such that: Our result improves, simultaneously, the running time and the multiplicative approximation of the Õ(n2)-time (3, 0)-approximation algorithm of Dor et al. at the cost of introducing also an additive approximation.