通过纯有理运算确定矩阵的Jordan块结构

本文讨论如何通过有限步有理运算求得给定矩阵的Jordan块结构(JBS),因为有理运算可以通过符号计算精确实现.与此对照,迄今为止用数值计算求矩阵的JBS与理论结果相距甚远.证明是构造性的,分两大部分:1)确定矩阵A的不变因子,2)根据A的不变因子确定初等因子结构.为求得A的不变因子,我们提出一种新的Las Vegas算法.它是一种概率型算法,这种算法允许失败,但是当且仅当求得正确答案时才停止运算;     

Defect location clustering schemes

Algorithms for clustering n objects typically require O(n²) operations. This report presents a special approach for a certain class of data that requires O(n) operations and O(n) storage. Such data commonly occur when a microscopic signal structure is imposed on a medium with potential for macroscopic defects, and the signal elements are then checked sequentially for error. The algorithm can be used to cluster other classes of data in O(n log n) operations. An application to videodisc defect consolidation is presented.     

Regular Splitting and Potential Reduction Method for Solving Quadratic Programming Problem with Box Constraints

A regular splitting and potential reduction method is presented for solving a quadratic programming problem with box constraints (QPB) in this paper. A general algorithm is designed to solve the QPB problem and generate a sequence of iterative points. We show that the number of iterations to generate an e-minimum solution or an e-KKT solution by the algorithm is bounded by O( nlog(1 )), and the total running time is bounded by O(n2(n logn log1/ε)(n/εlog1/ε logn)) arithmetic operations.     

对称r-循环矩阵的快速算法和并行算法

借助于快速付立叶变换(FFT),给出了一种判断对称r-循环线性系统是否有解的快速算法,并且在有解的情况下求出其解,该算法的计算复杂度为O(nlogn),且具有很好的并行性,若使用n台处理机并行处理该算法则只需要O(logn)步.当r=0时,对称r-循环矩阵变成一个上三角型Hankel矩阵,我们也给出了此类矩阵求逆的一种算法.最后将该算法推广到线性同余系统,其运算量仅为O(nlogn).     

FAST DENSE MATRIX METHOD FOR THE SOLUTION OF INTEGRAL EQUATIONS OF THE SECOND KIND

We present a fast algorithm based on polynomial interpolation to approximate matrices arising from the discretization of second-kind integral equations where the kernel function is either smooth, non-oscillatory and possessing only a finite number of singularities or a product of such function with a highly oscillatory coefficient function. Contrast to wavelet-like approximations, ourapproximation matrix is not sparse. However, the approximation can be construced in O(n) operations and requires O(n) storage, where n is the number of quadrature points used in the discretization. Moreover, the matrix-vector multiplication cost is of order O(nlogn). Thus our scheme is well suitable for conjugate gradient type methods. Our numerical results indicate that the algorithm is very accurate and stable for high degree polynomial interpolation.     

Topologically sweeping visibility complexes via pseudotriangulations

This paper describes a new algorithm for constructing the set of free bitangents of a collection ofn disjoint convex obstacles of constant complexity. The algorithm runs in timeO(n logn + k), where,k is the output size, and uses,O(n) space. While earlier algorithms achieve the same optimal running time, this is the first optimal algorithm that uses only linear space. The visibility graph or the visibility complex can be computed in the same time and space. The only complicated data structure used by the algorithm is a splittable queue, which can be implemented easily using red-black trees. The algorithm is conceptually very simple, and should therefore be easy to implement and quite fast in practice. The algorithm relies on greedy pseudotriangulations, which are subgraphs of the visibility graph with many nice combinatorial properties. These properties, and thus the correctness of the algorithm, are partially derived from properties of a certain partial order on the faces of the visibility complex. A preliminary version of this work appeared in theProceedings of the 11th Annual ACM Symposium on Computational Geometry, Vancouver, June 1995, pages 248–257.     

An O(n 3 L) potential reduction algorithm for linear programming

We describe a primal-dual potential function for linear programming: $$\phi (x,s) = \rho \ln (x^T s) - \sum\limits_{j = 1}^n {\ln (x_j s_j )} $$ whereρ? n, x is the primal variable, ands is the dual-slack variable. As a result, we develop an interior point algorithm seeking reductions in the potential function with \(\rho = n + \sqrt n \) . Neither tracing the central path nor using the projective transformation, the algorithm converges to the optimal solution set in \(O(\sqrt n L)\) iterations and uses O(n ³ L) total arithmetic operations. We also suggest a practical approach to implementing the algorithm.     

二次规划的直接椭球算法

本文对凸二次规划问题，给出了一个直接椭球算法，并证明了算法的复杂度为Ｏ（ｎ４Ｌ）．     

A machine learning approach to process improvement in a telecommunications company

Redesigning and improving business processes to better serve customer needs has become a priority in service industries as they scramble to become more competitive. This paper describes an approach to process improvement that is being developed collaboratively by applied researchers at US WEST, a major telecommunications company, and the University of Colorado. Motivated by the need to streamline and to add more quantitative power to traditional quality improvement processes, the new approach uses an artificial intelligence (AI) statistical tree growing method that uses customer survey data to identify operations areas where improvements are expected to affect customers most. This AI/statistical method also identifies realistic quantitative targets for improvement and suggests specific strategies (recommended combinations of actions) that are predicted to have high impact. This research, funded in part by the Colorado Advanced Software Institute (CASI) in an effort to stimulate profitable innovations, has resulted in a practical methodology that has been used successfully at US WEST to help set process improvement priorities and to guide resource allocation decisions throughout the company.     

关于矩阵乘法的一个改进算法的时间复杂度

两个ｎ阶非负整数方阵相乘,常规算法的时间复杂度为Ｏ（ｎ^３）,文献［１］提出一个“运算次数”为Ｏ（ｎ²）的“最佳”算法,文献［２］对此算法做了进一步研究,提出三种改进策略．本文根据算法分析理论,得出改进后的算法的时间复杂度仍不低于Ｏ（ｎ^３ｌｏｇｎ）,因而其阶仍高于常规算法的运算量的阶．     

Modular Counting of Rational Points over Finite Fields

Let F_q be the finite field of q elements, where q = p^h. Let f(x) be a polynomial over F_q in n variables with m nonzero terms. Let N(f) denote the number of solutions of f(x) = 0 with coordinates in F_q. In this paper we give a deterministic algorithm which computes the reduction of N(f) modulo p^b in O(n(8m)p^(h+b)p) bit operations. This is singly exponential in each of the parameters {h, b, p}, answering affirmatively an open problem proposed by Gopalan, Guruswami, and Lipton.     

An efficient label setting/correcting shortest path algorithm

We design a new label shortest path algorithm by applying the concept of a pseudo permanent label. This approach allows an algorithm to partition the set of nodes into two new sets: pseudo permanently labeled nodes and its complementary set. From this point of view, this new label method can be considered as a label setting method. Moreover, at least one node becomes permanently labeled when some nodes which belong to the set of pseudo permanently labeled nodes are scanned in each iteration of the algorithm. In the case of networks with non-negative length arcs it is easy to prove that this node has the minimum distance label among the non-pseudo permanently labeled nodes. On the other hand, it is not known during the computation which pseudo permanently labeled nodes are permanently labeled. Therefore, all distance labels are temporary and the algorithm becomes a label correcting method. Nevertheless, the proposed algorithm exhibits some nice features, such as: (1) the time bound for the running of the algorithm for a network with n nodes and m arcs is O(nm); (2) the number of node scan operations in the algorithm is less than the number of these operations in the previous label correcting algorithms as is observed in the computational experience; (3) the algorithm incorporates two new rules which allow easy detection of a negative cycle in the network; (4) the algorithm is quite simple and very easy to implement, and does not require sophisticated data structures; (5) the algorithm exhibits flexibility in the order in which the new pseudo permanently labeled nodes are scanned. The above features are possible through the application of the pseudo permanent label concept.     

THE EXISTENCE OF OVERLARGE SETS OF IDEMPOTENT QUASIGROUPS

A idempotent quasigroup (Q, o) of order n is equivalent to an n(n-1)×3 partial orthogonal array in which all of rows consist of 3 distinct elements. Let X be a (n+1)-set. Denote by T(n+1) the set of (n+1)n(n-1) ordered triples of X with the property that the 3 coordinates of each ordered triple are distinct. An overlarge set of idempotent quasigroups of order n is a partition of T(n+1) into n+1 n(n-1)×3 partial orthogonal arrays A_x, x∈X based on X\{x}. This article gives an almost complete solution of overlarge sets of idempotent quasigroups.     

A low-memory algorithm for finding short product representations in finite groups

We describe a space-efficient algorithm for solving a generalization of the subset sum problem in a finite group G, using a Pollard-ρ approach. Given an element z and a sequence of elements S, our algorithm attempts to find a subsequence of S whose product in G is equal to z. For a random sequence S of length d log₂ n, where n = #G and d ≥ 2 is a constant, we find that its expected running time is O(?n log n){O(\sqrt{n}\,{\rm log}\,n)} group operations (we give a rigorous proof for d > 4), and it only needs to store O(1) group elements. We consider applications to class groups of imaginary quadratic fields, and to finding isogenies between elliptic curves over a finite field.     

关于矩阵乘法的一个算法的时间复杂度

两个n阶非负整数方阵相乘,常规算法的时间复杂度为O(n³),文献[1]提出一个“运算次数”为O(n²)的“最佳”算法,本文根据算法分析理论得出此算法的时间复杂度不低于O(n³log₂ⁿ),因而比常规算法的运算量还大.     

Patch-adaptive multilevel iteration

Themultilevel adaptive iteration is an attempt to improve both the robustness and efficiency of iterative sparse system solvers. Unlike in most other iterative methods, the order of processing and sequence of operations is not determined a priori. The method consists of a relaxation scheme with an active set strategy and can be viewed as an efficient implementation of the Gauß-Southwell relaxation. With this strategy, computational work is focused on where it can efficiently improve the solution quality. To obtain full efficiency, the algorithm must be used on a multilevel structure. This algorithm is then closely related to multigrid or multilevel preconditioning algorithms, and can be shown to have asymptotically optimal convergence. In this paper the focus is on a variant that uses data structures with a locally uniform grid refinement. The resulting grid system consists of a collection of patches where each patch is a uniform rectangular grid and where adaptive refinement is accomplished by arranging the patches flexibly in space. This construction permits improved implementations that better exploit high performance computer designs. This will be demonstrated by numerical examples.     

Nearest Neighbor Queries in Metric Spaces

Given a set S of n sites (points), and a distance measure d , the nearest neighbor searching problem is to build a data structure so that given a query point q , the site nearest to q can be found quickly. This paper gives data structures for this problem when the sites and queries are in a metric space. One data structure, D(S) , uses a divide-and-conquer recursion. The other data structure, M(S,Q) , is somewhat like a skiplist. Both are simple and implementable. The data structures are analyzed when the metric space obeys a certain sphere-packing bound, and when the sites and query points are random and have distributions with an exchangeability property. This property implies, for example, that query point q is a random element of . Under these conditions, the preprocessing and space bounds for the algorithms are close to linear in n . They depend also on the sphere-packing bound, and on the logarithm of the distance ratio of S , the ratio of the distance between the farthest pair of points in S to the distance between the closest pair. The data structure M(S,Q) requires as input data an additional set Q , taken to be representative of the query points. The resource bounds of M(S,Q) have a dependence on the distance ratio of S Q . While M(S,Q) can return wrong answers, its failure probability can be bounded, and is decreasing in a parameter K . Here K≤ |Q|/n is chosen when building M(S,Q) . The expected query time for M(S,Q) is O(Klog n)log , and the resource bounds increase linearly in K . The data structure D(S) has expected O( log n) ^O(1) query time, for fixed distance ratio. The preprocessing algorithm for M(S,Q) can be used to solve the all nearest neighbor problem for S in O(n(log n) ² (log ϒ(S)) ² ) expected time. Received September 17, 1996, and in revised form November 1, 1998.     

Hierarchical partition—a new optimal pivoting algorithm

A new pivotal strategy for reducing storage and arithmetic operations in computing the inverse of a matrix is outlined. This algorithm uses recursive deletion and partition to generate an efficient structure. The optimal selection of rows and columns to be deleted (spikes) is greatly facilitated through the use of certain exclusion criteria. The algorithm produces significantly fewer fill-ins and requires significantly fewer operations than the alternative P3 and P4 algorithms for the product form of the inverse (PFI). It also produces satisfactory structures for problems for which P3 and P4 would generate zero pivots. Three variants of HP algorithm specifically tailored for the elimination form of the inverse (EFI) and for greater speed are also presented. The algorithm has been implemented on a CDC 6400 computer. Performance comparison on six sample problems is given.     

构建最小k 重控制集的概率算法

点集D ⊆ V (G) 称为图G 的k 重控制集, 如果D 满足V (G) - D 中任意结点在D 中至少有k 个邻居. 在无线网络中, 最小k 重控制集(MkDS) 用以构建健壮的虚拟骨干网. 构建虚拟骨干网是无线网络中最基本也是最重要的问题. 在本文中, 我们提出一种快速的分布式概率算法来构建k重控制集. 我们构建的k 重控制集的期望大小不超过最优解的O(k²) 倍. 算法的运行时间复杂度为O((Δ logΔ+log log n)n),其中Δ = max{|D(p)|}, D(p) 是以p 为中心半径为1 的圆盘中的结点, 最大值的比较范围是给定集合中所有的p 点.     

Binary operations and lattice structure for a model of matching with contracts

We consider a restricted model of many-to-one matching with contracts and we order the set of stable allocations according both to the unanimous-for-doctors partial ordering and Blair’s partial ordering for hospitals. We define two binary operations to calculate the least upper bound and greatest lower bound for each pair of elements of this set in a simple way. By using these operations, we show that the set of stable allocations has dual lattice structures, thus reflecting an expected counterposition of interests between both sides of the market.