两类整数分解算法的分析与改进

给出了整数分解的两种算法,试除法和Pollard算法.根据素数分布的规律,通过减少试除次数提高了试除法运算效率,使得其性能显著提高;对Pollard算法进行分析后,变换随机序列产生式并重启算法使算法运行更稳定有效.给出了这两类改进算法的运行时间对比表,结果表明,改进的试除法在分解32位内小整数效果更佳而改进的Pollard算法在分解32位以上大整数有明显的优化.     

Non-negative matrix factorization: Ill-posedness and a geometric algorithm

Non-negative matrix factorization (NMF) has been proposed as a mathematical tool for identifying the components of a dataset. However, popular NMF algorithms tend to operate slowly and do not always identify the components which are most representative of the data. In this paper, an alternative algorithm for performing NMF is developed using the geometry of the problem. The computational costs of the algorithm are explored, and it is shown to successfully identify the components of a simulated dataset. The development of the geometric algorithm framework illustrates the ill-posedness of the NMF problem and suggests that NMF is not sufficiently constrained to be applied successfully outside of a particular class of problems.     

二值分解压缩和Consensus算法

奇异值分解(SVD)是一种流行的用于高维数据压缩的方法,二值分解是奇异值分解的一种简化形式.实现二值分解的主要算法有两种:迭代启发式算法和贪婪算法.但这两种算法都不是很理想的算法:迭代启发式算法在很多情况下不能保证收敛性,贪婪算法不满足大型数值矩阵分解的需要.采用了一种新的算法来实现二值分解:Consensus的算法.Consensus算法可在渐进多项式时间内找到一般图中的极大二分团.对于某些二分图,该算法的复杂度是多项式时间的.实验结果表明,当迭代启发式算法不起作用时,Consensus算法是一种很好的求解二值分解的方法.该算法远比贪婪算法的效率高,且具有稳定收敛性.     

A factorization algorithm to compute Pfaffians

We describe an explicit algorithm to factorize an even antisymmetric N² matrix into triangular and trivial factors. The construction resembles the Crout algorithm for LU factorization. It allows for a straight forward computation of Pfaffians (including their signs) at the cost of N³/3 flops.     

Accelerated image factorization based on improved NMF algorithm

Non-negative matrix factorization (NMF) is widely used in feature extraction and dimension reduction fields. Essentially, it is an optimization problem to determine two non-negative low rank matrices \(W_{m \times k}\) and \(H_{k \times n}\) for a given matrix \(A_{m \times n}\), satisfying \(A_{m \times n} \approx W_{m \times k}H_{k \times n}\). In this paper, a novel approach to improve the image decomposing and reconstruction effects by introducing the Singular Value Decomposing (SVD)-based initialization scheme of factor matrices W and H, and another measure called choosing rule to determine the optimum value of factor rank k, are proposed. The input image is first decomposed using SVD to get its singular values and corresponding eigenvectors. Then, the number of main components as the rank value k is extracted. Then, the singular values and corresponding eigenvectors are used to initialize W and H based on selected rank k. Finally, convergent results are obtained using multiplicative and additive update rules. However, iterative NMF algorithms’ convergence is very slow on most platforms limiting its practicality. To this end, a parallel implementation frame of described improved NMF algorithm using CUDA, a tool for algorithms parallelization on massively parallel processors, i.e., many-core graphics processors, is presented. Experimental results show that our approach can get better decomposing effect than traditional NMF implementations and dramatic accelerate rate comparing to serial schemes as well as existing distributed-system implementations.     

The expected linearity of a simple equivalence algorithm

The average time needed to form unions of disjoint equivalence classes, using an algorithm suggested by Aho, Hopcroft, and Ullman, is shown to be linear in the total number of elements, thereby establishing a conjecture of Yao. The analytic methods used to prove this result are of interest in themselves, as they are based on extensions of Stepanov's approach to the study of random graphs. Several refinements of Yao's analyses of related algorithms are also presented.     

矩阵的Doolittle递归分解算法及符号程序设计

将矩阵An×n的Doolittle分解推广到Am×n上，并在常规的迭代算法上加以创新，给出了递归的分解算法．在实现算法的过程中，对数据进行了巧妙处理，使中间数据及最终计算结果都具有分数形式，提高了结果的精确度，而且更符合人们阅读的习惯．经过运行测试，算法设计合理，程序运行高效准确．程序是对MathSoft公司的交互式的数学文字软件Mathcad的矩阵分解的数值计算扩充到符号运算．     

A Toeplitz algorithm for polynomial J-spectral factorization

A block Toeplitz algorithm is proposed to perform the J-spectral factorization of a para-Hermitian polynomial matrix. The input matrix can be singular or indefinite, and it can have zeros along the imaginary axis. The key assumption is that the finite zeros of the input polynomial matrix are given as input data. The algorithm is based on numerically reliable operations only, namely computation of the null-spaces of related block Toeplitz matrices, polynomial matrix factor extraction and linear polynomial matrix equations solving.     

Population size versus runtime of a simple evolutionary algorithm

Evolutionary algorithms (EAs) find numerous applications, and practical knowledge on EAs is immense. In practice, sophisticated population-based EAs employing selection, mutation and crossover are applied. In contrast, theoretical analysis of EAs often concentrates on very simple algorithms such as the (1+1) EA, where the population size equals 1. In this paper, the question is addressed whether the use of a population by itself can be advantageous. A population-based EA that neither makes use of crossover nor any diversity-maintaining operator is investigated on an example function. It is shown that an increase of the population size by a constant factor decreases the expected runtime from exponential to polynomial. Thereby, the best gap known so far is improved from superpolynomial vs. polynomial to exponential vs. polynomial. Moreover, it is proved that the exponential and polynomial runtime bounds occur with a probability exponentially close to one if the population size is a constant (resp., a small polynomial). Finally, a second example function, where only a small population leads to a polynomial runtime, and a hierarchy result on the appropriate population size are presented. The analyses show formally how the population size can lead to different attractors in the search space.     

A note on Gao’s algorithm for polynomial factorization

Shuhong Gao (2003) [6] has proposed an efficient algorithm to factor a bivariate polynomial f over a field F. This algorithm is based on a simple partial differential equation and depends on a crucial fact: the dimension of the polynomial solution space G associated with this differential equation is equal to the number r of absolutely irreducible factors of f. However, this holds only when the characteristic of F is either zero or sufficiently large in terms of the degree of f. In this paper we characterize a vector subspace of G for which the dimension is r, regardless of the characteristic of F, and the properties of Gao’s construction hold. Moreover, we identify a second vector subspace of G that leads to an analogous theory for the rational factorization of f.     

A generalized Schur-type algorithm for the joint factorization of a structured matrix and its inverse

A Schur-type algorithm is presented for the simultaneous triangular factorization of a given (non-degenerate) structured matrix and its inverse. The algorithm takes the displacement generator of a Hermitian, strongly regular matrixR as an input, and computes the displacement generator of the inverse matrixR ⁻¹ as an output. From these generators we can directly deduce theLD ⁻¹ L ^* (lower-diagonal-upper) decomposition ofR, and theUD ⁻¹ U ^* (upper-diagonallower) decomposition ofR ⁻¹. The computational complexity of the algorithm isO(rn ²) operations, wheren andr denote the size and the displacement rank ofR, respectively. Moreover, this method is especially suited for parallel (systolic array) implementations: usingn processors the algorithm can be carried out inO(n) steps.     

Analysis of a parallel volume rendering system based on theshear-warp factorization

This paper presents a parallel volume rendering algorithm that can render a 256×256×225 voxel medical data set at over 15 Hz and a 512×512×334 voxel data set at over 7 Hz on a 32-processor Silicon Graphics Challenge. The algorithm achieves these results by minimizing each of the three components of execution time: computation time, synchronization time, and data communication time. Computation time is low because the parallel algorithm is based on the recently-reported shear-warp serial volume rendering algorithm which is over five times faster than previous serial algorithms. The algorithm uses run-length encoding to exploit coherence and an efficient volume traversal to reduce overhead. Synchronization time is minimized by using dynamic load balancing and a task partition that minimizes synchronization events. Data communication costs are low because the algorithm is implemented for shared-memory multiprocessors, a class of machines with hardware support for low-latency fine-grain communication and hardware caching to hide latency. We draw two conclusions from our implementation. First, we find that on shared-memory architectures data redistribution and communication costs do not dominate rendering time. Second, we find that cache locality requirements impose a limit on parallelism in volume rendering algorithms. Specifically, our results indicate that shared-memory machines with hundreds of processors would be useful only for rendering very large data sets     

Tumble, a fast simple iteration algorithm for Fibonacci

A faster linear iteration process for Fibonacci (and Lucas) numbers is given. Algorithms are derived for individual numbers and for sequence generation. A general algorithm for generalized Fibonacci numbers is derived from these.     

A simple Voronoi diagram algorithm for a reconfigurable mesh

In this paper, we introduce a simple and efficient algorithm for computing the Voronoi Diagram for n planar points on a reconfigurable mesh of size O(n)×O(n). The algorithm has a worst case running of O(log n log log n) time. The algorithm exploits the O(1) communication diameter of the reconfigurable mesh model to implement efficient load balancing     

Numerical stability of a convex hull algorithm for simple polygons

A numerically stable and optimalO(n)-time implementation of an algorithm for finding the convex hull of a simple polygon is presented. Stability is understood in the sense of a backward error analysis. A concept of the condition number of simple polygons and its impact on the performance of the algorithm is discussed. It is shown that if the condition number does not exceed (1+O())/(3), then, in floating-point arithmetic with the unit roundoff, the algorithm produces the vertices of a convex hull for slightly perturbed input points. The relative perturbation does not exceed 3(1+O()).J. W. Jaromczyk was partially supported by a grant from the Center for Robotics and Manufacturing Systems at the University of Kentucky and G. W. Wasilkowski was partially supported by the National Science Foundation under Grants CCR-89-05371 and CCR-91-14042.     

A simple contour matching algorithm

Two similarity measures between strings are proposed. This correspondence also describes an experiment performed to illustrate the inadequacy of a commonly used measure.