pinecone/core-2020-05-10-deduplication huggingface dataset:
Partial Fraction Decomposition in C(z) and Simultaneous Newton Iteration for Factorization in C[z]

AbstractThe subject of this paper is fast numerical algorithms for factoring univariate polynomials with complex coefficients and for computing partial fraction decompositions (PFDs) of rational functions in C(z). Numerically stable and computationally feasible versions of PFD are specified first for the special case of rational functions with all singularities in the unit disk (the “bounded case”) and then for rational functions with arbitrarily distributed singularities. Two major algorithms for computing PFDs are presented: The first one is an extension of the “splitting circle method” by A. Schönhage (“The Fundamental Theorem of Algebra in Terms of Computational Complexity,” Technical Report, Univ. Tübingen, 1982) for factoring polynomials in C[z] to an algorithm for PFD. The second algorithm is a Newton iteration for simultaneously improving the accuracy of all factors in an approximate factorization of a polynomial resp. all partial fractions of an approximate PFD of a rational function. Algorithmically useful starting value conditions for the Newton algorithm are provided. Three subalgorithms are of independent interest. They compute the product of a sequence of polynomials, the sum of a sequence of rational functions, and the modular representation of a polynomial. All algorithms are described in great detail, and numerous technical auxiliaries are provided which are also useful for the design and analysis of other algorithms in computational complex analysis. Combining the splitting circle method with simultaneous Newton iteration yields favourable time bounds (measured in bit operations) for PFD, polynomial factoring, and root calculation. In particular, the time bounds for computing high accuracy PFDs, high accuracy factorizations, and high accuracy approximations for zeros of squarefree polynomials are linear in the output size (and hence optimal) up to logarithmic factors
The Exponent of Discrepancy Is at Least 1.0669

AbstractLetP⊂[0,1]dbe ann-point set and letw:P→[0,∞) be a weight function withw(P)=∑z∈Pw(z)=1. TheL2-discrepancy of the weighted set (P,w) is defined as theL2-average ofD(x)=vol(Bx)−w(P∩Bx) overx∈[0,1]d, where vol(Bx) is the volume of thed-dimensional intervalBx=∏dk=1[0,xk). The exponent of discrepancyp* is defined as the infimum of numberspsuch that for all dimensionsd⩾1 and allε>0 there exists a weighted set of at mostKε−ppoints in [0,1]dwithL2-discrepancy at mostε, whereK=K(p) is a suitable number independent ofεandd. Wasilkowski and Woźniakowski proved thatp*⩽1.4779, by combining known bounds for the error of numerical integration and using their relation toL2-discrepancy. In this note we observe that a careful treatment of a classical lower- bound proof of Roth yieldsp*⩾1.04882, and by a slight modification of the proof we getp*⩾1.0669. Determiningp* exactly seems to be quite a difficult problem
Scrambling Sobol' and Niederreiter–Xing Points

AbstractHybrids of equidistribution and Monte Carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter. In particular, randomized (0,m,s)-nets in basebproduce unbiased estimates of the integral, have a variance that tends to zero faster than 1/nfor any square integrable integrand and have a variance that for finitenis never more thane≐2.718 times as large as the Monte Carlo variance. Lower bounds thaneare known for special cases. Some very important (t,m,s)-nets havet>0. The widely used Sobol' sequences are of this form, as are some recent and very promising nets due to Niederreiter and Xing. Much less is known about randomized versions of these nets, especially ins>1 dimensions. This paper shows that scrambled (t,m,s)-nets enjoy the same properties as scrambled (0,m,s)-nets, except the sampling variance is guaranteed only to be belowbt[(b+1)/(b−1)]stimes the Monte Carlo variance for a least-favorable integrand and finiten
Feasible Real Random Access Machines

AbstractWe present a modified real RAM model which is equipped with the usual discrete and real-valued arithmetic operations and with a finite precision test <kwhich allows comparisons of real numbers only up to a variable uncertainty 1/(k+1). Furthermore, ourfeasible RAMhas an extended semantics which allows approximative computations. Using a logarithmic complexity measure we prove that all functions computable on a RAM in time O(t) can be computed on a Turing machine in time O(t2·log(t)·loglog(t)). Vice versa all functions computable on a Turing machine in time O(t) are computable on a RAM in time O(t). Thus, our real RAM model does not only express exactly the computational power of Turing machines on real numbers (in the sense of Grzegorczyk), but it also yields a high-level tool for realistic time complexity estimations on real numbers
On theL2-Discrepancy for Anchored Boxes

AbstractTheL2-discrepancy for anchored axis-parallel boxes has been used in several recent computational studies, mostly related to numerical integration, as a measure of the quality of uniform distribution of a given point set. We point out that if the number of points is not large enough in terms of the dimension (e.g., fewer than 104points in dimension 30) then nearly the lowest possibleL2-discrepancy is attained by a pathological point set, and hence theL2-discrepancy may not be very relevant for relatively small sets. Recently, Hickernell obtained a formula for the expectedL2-discrepancy of certain randomized low-discrepancy set constructions introduced by Owen. We note that his formula remains valid also for several modifications of these constructions which admit a very simple and efficient implementation. We also report results of computational experiments with various constructions of low-discrepancy sets. Finally, we present a fairly precise formula for the performance of a recent algorithm due to Heinrich for computing theL2-discrepancy
Approximating Band- and Energy-Limited Signals in the Presence of Jitter

AbstractThis paper deals with recovering band- and energy-limited signals from a finite set of their perturbed samples taken at slightly wrong points. The perturbations in sample points reading (called jitter) and in the measurements of the corresponding samples are assumed to be bounded byγandδ, respectively. The goal is to analyze how the minimal worst-case recovery error depends onγandδ. This is accomplished by proving tight upper and lower bounds on the diameter of information. The main conclusion is that the jitter causes the error of orderγΔ3/2+δ, whereΔis the bandwidth
Lower Bounds on the Depth of Monotone Arithmetic Computations

AbstractConsider an arithmetic expression of lengthninvolving only the operations {+,×} and non-negative constants. We prove lower bounds on the depth of any binary computation tree over the same sets of operations and constants that computes such an expression. We exhibit a family of arithmetic expressions that requires computation trees of depth at least 1.5log2n−O(1), thus proving a conjecture of S. R. Kosaraju (1986,in“Proc. of the 18th ACM Symp. on Theory Computing,” pp. 231–239). In contrast, Kosaraju showed how to compute such expressions with computation trees of depth 2log2n+O(1)
On-line Construction of Two-Dimensional Suffix Trees

AbstractWe say that a data structure is builton-lineif, at any instant, we have the data structure corresponding to the input we have seen up to that instant. For instance, consider the suffix tree of a stringx[1,n]. An algorithm building iton-lineis such that, when we have read the firstisymbols ofx[1,n], we have the suffix tree forx[1,i]. We present a new technique, which we refer to asimplicit updates, based on which we obtain: (a) an algorithm for theon-lineconstruction of the Lsuffix tree of ann×nmatrixA—this data structure is the two-dimensional analog of the suffix tree of a string; (b) simple algorithms implementing primitive operations forLZ1-typeon-line losslessimage compression methods. Those methods, recently introduced by Storer, are generalizations ofLZ1-typecompression methods for strings. For the problem in (a), we get nearly an order of magnitude improvement over algorithms that can be derived from known techniques. For the problem in (b), we do not get an asymptotic speed-up with respect to what can be done with known techniques; rather we show that our algorithms are a natural support for the primitive operations. This may lead to faster implementations of those primitive operations. To the best of our knowledge, our technique is the first one that effectively addresses problems related to theon-lineconstruction of two-dimensional suffix trees
Parallel Construction and Query of Index Data Structures for Pattern Matching on Square Matrices

AbstractWe describe fast parallel algorithms for building index data structures that can be used to gather various statistics on square matrices. The main data structure is the Lsuffix tree, which is a generalization of the classical suffix tree for strings. Given ann×ntext matrixA, we build our data structures inO(logn) time withn2processors on a CRCW PRAM, so that we can quickly processAin parallel as follows: (i) report some statistical information aboutA, e.g., find the largest repeated square submatrices that appear at least twice inAor determine, for each position inA, the smallest submatrix that occurs only there; (ii) given, on-line, anm×mpattern matrixPAT, check whether it occurs inA. We refer to the above two kinds of operations as queries and point out that they have applications to visual databases and two-dimensional data compression. Query (i) takesO(logn) time withn2/lognprocessors and query (ii) takesO(logm) time withm2/logmprocessors. The query algorithms are work optimal while the construction algorithm is work optimal only for arbitrary and large alphabets
Efficient Algorithms for Approximate String Matching with Swaps

AbstractMost research on the edit distance problem and thek-differences problem considered the set of edit operations consisting of changes, insertions, and deletions. In this paper we include theswapoperation that interchanges two adjacent characters into the set of allowable edit operations, and we present anO(tmin(m,n))-time algorithm for the extended edit distance problem, wheretis the edit distance between the given strings, and anO(kn)-time algorithm for the extendedk-differences problem. That is, we add swaps into the set of edit operations without increasing the time complexities of previous algorithms that consider only changes, insertions, and deletions for the edit distance andk-differences problems
Trade-offs between Communication Throughput and Parallel Time

AbstractWe study the effect of limited communication throughput on parallel computation in a setting where the number of processors is much smaller than the length of the input. Our model haspprocessors that communicate through a shared memory of sizem. The input has sizenand can be read directly by all the processors. We will be primarily interested in studying cases wheren⪢p⪢m. As a test case we study the list reversal problem. For this problem we prove a time lower bound ofΩ(n/mp). (A similar lower bound holds also for the problems of sorting, finding all unique elements, convolution, and universal hashing.) This result demonstrates that limiting the communication (i.e., smallm) could have significant effect on parallel computation. We show an almost matching upper bound ofO((n/mp)logO(1)n). The upper bound requires the development of a few interesting techniques which can alleviate the limited communication in some general settings. Specifically, we show how to emulate a large shared memory on a limited shared memory efficiently. The lower bound applies even to randomized machines, and the upper bound is a randomized algorithm. We also argue that some standard methodology for designing parallel algorithms appears to require a relatively high level of communication throughput. Our results suggest that new alternative methodologies that need a lower such level must be invented for parallel machines that enable a low level of communication throughput, since otherwise those machines will be severly handicapped as general purpose parallel machines. Although we do not rule that out, we cannot offer any encouraging evidence to suggest that such new methodologies are likely to be found
A Note on the Complexity of Solving Poisson's Equation for Spaces of Bounded Mixed Derivatives

AbstractWe study the complexity of solving the d-dimensional Poisson equation on ]0, 1[d. We restrict ourselves to cases where the solution u lies in some space of functions of bounded mixed derivatives (with respect to the L∞- or the L2-norm) up to ∂2d/∏dj=1∂x2j. An upper bound for the complexity of computing a solution of some prescribed accuracy ε with respect to the energy norm is given, which is proportional to ε−1. We show this result in a constructive manner by proposing a finite element method in a special sparse grid space, which is obtained by an a priori grid optimization process based on the energy norm. Thus, the result of this paper is twofold: First, from a theoretical point of view concerning the complexity of solving elliptic PDEs, a strong tractability result of the order O(ε−1) is given, and, second, we provide a practically usable hierarchical basis finite element method of this complexity O(ε−1), i.e., without logarithmic terms growing exponentially in d, at least for our sparse grid setting with its underlying smoothness requirements
Stochastic Analog Networks and Computational Complexity

AbstractThe model of analog recurrent neural networks (ARNN) is typically perceived as based on either the practical powerful tool of automatic learning or on biological metaphors, yet it constitutes an appealing model of computation. This paper provides rigorous foundations for ARNN, when they are allowed to exhibit stochastic and random behavior of discrete nature. Our model is an extension of the von Neumann model of unreliable interconnection of components and incorporates a generalization of Shannon's random-noise philosophy. In the general case the computational class (P/poly) is associated with both deterministic and stochastic networks. However, when the weights are restricted to rational numbers, stochasticity adds power to the computation. As part of the proof, we show that probabilistic Turing machines that use a coin with a real probability rather than an exactly random (12) coin, compute the nonuniform version BPP/log* instead of the recursive class BPP. We also show that in the case of real probabilities only their first logarithmic number of bits are relevant for the computation
Intractability Results for Positive Quadrature Formulas and Extremal Problems for Trigonometric Polynomials

AbstractLower bounds for the error of quadrature formulas with positive weights are proved. We get intractability results for quasi-Monte Carlo methods and, more generally, for positive formulas. We consider general classes of functions but concentrate on lower bounds for relatively small classes of trigonometric polynomials. We also conjecture that similar lower bounds hold for arbitrary quadrature formulas and state different equivalent conjectures concerning positive definiteness of certain matrices and certain extremal problems for trigonometric polynomials. We also study classes of functions with weighted norms where some variables are “more important” than others. Positive quadrature formulas are then tractable iff the sum of the weights is bounded
Monte Carlo Complexity of Parametric Integration

AbstractThe Monte Carlo complexity of computing integrals depending on a parameter is analyzed for smooth integrands. An optimal algorithm is developed on the basis of a multigrid variance reduction technique. The complexity analysis implies that our algorithm attains a higher convergence rate than any deterministic algorithm. Moreover, because of savings due to computation on multiple grids, this rate is also higher than that of previously developed Monte Carlo algorithms for parametric integration
Quasi-Monte Carlo Simulation of Diffusion

AbstractA particle method adapted to the simulation of diffusion problems is presented. Time is discretized into increments of length Δt. During each time step, the particles are allowed to random walk to any point by taking steps sampled from a Gaussian distribution centered at the current particle position with variance related to the time discretization Δt. Quasi-random samples are used and the particles are relabeled according to their position at each time step. Convergence is proved for the pure initial-value problem in s space dimensions. For some simple demonstration problems, the numerical results indicate that an improvement is achieved over standard random walk simulation
Where Does Smoothness Count the Most for Two-Point Boundary-Value Problems?

AbstractWe study the complexity of scalar 2mth order elliptic two-point boundary-value problems Lu=f, error being measured in the energy norm. Previous work on the complexity of these problems has generally assumed that we had partial information about the right-hand side f and complete information about the coefficients of L. In this paper, we study the complexity of such problems when, in addition to partial information about f, we have only partial information about the coefficients of L. More precisely, we suppose that f has r derivatives in the Lp-sense, with r⩾−m and p∈[2, ∞], and that L has the usual divergence form Lv=∑0⩽i, j⩽m(−1)iDi(ai, jDjv), with ai, j being ri, j-times continuously differentiable, where ri, j⩾0. We first suppose that continuous linear information is available. Let r=min{r, min0⩽i, j⩽m{ri, j−i}}. If r=−m, the problem is unsolvable; for r>−m, we find that the ε-complexity is proportional to (1/ε)1/(r+m), and we show that a finite element method (FEM) is optimal. We next suppose that only standard information (consisting of function and/or derivative evaluations) is available. Let rmin=min{r, min0⩽i, j⩽m{ri, j}}. If rmin=0, the problem is unsolvable; for rmin>0, we find that the ε-complexity is proportional to (1/ε)1/rmin, and we show that a modified FEM (which uses only function evaluations, and not derivatives) is optimal
Weighted Tensor Product Algorithms for Linear Multivariate Problems

AbstractWe study the ε-approximation of linear multivariate problems defined over weighted tensor product Hilbert spaces of functions f of d variables. A class of weighted tensor product (WTP) algorithms is defined which depends on a number of parameters. Two classes of permissible information are studied. Λall consists of all linear functionals while Λstd consists of evaluations of f or its derivatives. We show that these multivariate problems are sometimes tractable even with a worst-case assurance. We study problem tractability by investigating when a WTP algorithm is a polynomial-time algorithm, that is, when the minimal number of information evaluations is a polynomial in 1/ε and d. For Λall we construct an optimal WTP algorithm and provide a necessary and sufficient condition for tractability in terms of the sequence of weights and the sequence of singular values for d=1. ForΛstd we obtain a weaker result by constructing a WTP algorithm which is optimal only for some weight sequences
Quadrature Formulas for the Wiener Measure

AbstractWe present a new method for the approximation of Wiener integrals and provide an explicit error bound for a class F of smooth integrands. The purely deterministic algorithm is a sequence of quadrature formulas for the Wiener measure, where the knots are piecewise linear functions. It uses ideas of Smolyak, as well as the multiscale decomposition of the Wiener measure due to Lévy and Ciesielski. For the class F we obtain n(ε)⩽max(1, 2ε−4), where n(ε) is the number of integrand evaluations needed to guarantee an error at most ε for f∈F
Randomized Algorithms over Finite Fields for the Exact Parity Base Problem

AbstractWe present three randomized pseudo-polynomial algorithms for the problem of finding a base of specified value in a weighted represented matroid subject to parity conditions. These algorithms, the first two being an improved version of those presented by P. M. Camerini et al. (1992, J. Algorithms13, 258–273) use fast arithmetic working over a finite field chosen at random among a set of appropriate fields. We show that the choice of a best algorithm among those presented depends on a conjecture related to the best value of the so-called Linnik constant concerning the distribution of prime numbers in arithmetic progressions. This conjecture, which we call the C-conjecture, is a strengthened version of a conjecture formulated in 1934 by S. Chowla. If the C-conjecture is true, the choice of a best algorithm is simple, since the last algorithm exhibits the best performance, either when the performance is measured in arithmetic operations, or when it is measured in bit operations and mild assumptions hold. If the C-conjecture is false we are still able to identify a best algorithm, but in this case the choice is between the first two algorithms and depends on the asymptotic growth of m with respect to those of U and n, where 2n, 2m, U are the rank, the number of elements, and the maximum weight assigned to the elements of the matroid, respectively
Polynomial Time Samplable Distributions

AbstractThis paper studies the complexity of the polynomial-time samplable (P-samplable) distributions, which can be approximated within an exponentially small factor by sampling algorithms in time polynomial in the length of their outputs. The paper shows that common assumptions in complexity theory yield the separation of polynomial-time samplable distributions from the polynomial-time computable distributions with respect to polynomial domination, average-polynomial domination, polynomial equivalence, and average-polynomial equivalence
Time-Space Tradeoffs in Algebraic Complexity Theory

AbstractWe exhibit a new method for showing lower bounds for time-space tradeoffs of polynomial evaluation procedures given by straight-line programs. From the tradeoff results obtained by this method we deduce lower space bounds for polynomial evaluation procedures running in optimal nonscalar time. Time, denoted by L, is measured in terms of nonscalar arithmetic operations and space, denoted by S, is measured by the maximal number of pebbles (registers) used during the given evaluation procedure. The time-space tradeoff function considered in this paper is LS2. We show that for “almost all” univariate polynomials of degree at most d our time-space tradeoff functions satisfy the inequality LS2⩾d8. From this we conclude that for “almost all” degree d univariate polynomials, any nonscalar time optimal evaluation procedure requires space at least S⩾cd, where c>0 is a suitable universal constant. The main part of this paper is devoted to the exhibition of specific families of univariate polynomials which are “hard to compute” in the sense of time-space tradeoff (this means that our tradeoff function increases linearly in the degree)
A Fast Version of the Schur–Cohn Algorithm

AbstractIn this paper we describe a fast algorithm for counting the number of roots of complex polynomials in the open unit disk, classically called the Schur–Cohn problem. For degree d polynomials our bound is of order dlog2d in terms of field operations and comparisons, but our approach nicely allow us to integrate the bit size as a further complexity parameter as well. For bit size σ of input polynomials of Z[X] our running time is of order d2σ·log(dσ)·loglog(dσ)·log(d) in the multi-tape Turing machine model, thus improving all previously proposed approaches