394,446 research outputs found
The complexity of class polynomial computation via floating point approximations
We analyse the complexity of computing class polynomials, that are an
important ingredient for CM constructions of elliptic curves, via complex
floating point approximations of their roots. The heart of the algorithm is the
evaluation of modular functions in several arguments. The fastest one of the
presented approaches uses a technique devised by Dupont to evaluate modular
functions by Newton iterations on an expression involving the
arithmetic-geometric mean. It runs in time for any , where
is the CM discriminant and is the degree of the class polynomial.
Another fast algorithm uses multipoint evaluation techniques known from
symbolic computation; its asymptotic complexity is worse by a factor of . Up to logarithmic factors, this running time matches the size of the
constructed polynomials. The estimate also relies on a new result concerning
the complexity of enumerating the class group of an imaginary-quadratic order
and on a rigorously proven upper bound for the height of class polynomials
PTHash: Revisiting FCH Minimal Perfect Hashing
Given a set S of n distinct keys, a function f that bijectively maps the keys of S into the range (0,...,n-1) is called a minimal perfect hash function for S. Algorithms that find such functions when n is large and retain constant evaluation time are of practical interest; for instance, search engines and databases typically use minimal perfect hash functions to quickly assign identifiers to static sets of variable-length keys such as strings. The challenge is to design an algorithm which is efficient in three different aspects: time to find f (construction time), time to evaluate f on a key of S (lookup time), and space of representation for f. Several algorithms have been proposed to trade-off between these aspects. In 1992, Fox, Chen, and Heath (FCH) presented an algorithm at SIGIR providing very fast lookup evaluation. However, the approach received little attention because of its large construction time and higher space consumption compared to other subsequent techniques. Almost thirty years later we revisit their framework and present an improved algorithm that scales well to large sets and reduces space consumption altogether, without compromising the lookup time. We conduct an extensive experimental assessment and show that the algorithm finds functions that are competitive in space with state-of-the art techniques and provide 2-4x better lookup time
A (Slightly Less Brutal) Method for Numerically Evaluating Structure Functions
A fast numerical algorithm for the evolution of parton distributions in x
space is described. The method is close in spirit to `brute' force techniques.
The necessary integrals are performed by summing the approximate contributions
from small steps of the integration region. Because it is a numerical
evaluation it shares the advantage with brute force numerical integration that
there are no restrictions placed on the functional form of the distributions to
be evolved. However, an improvement in the approximation technique results in a
significant reduction in the number of integration steps and a savings in time
on the order of three hundred fifty. The method has been implemented for the
structure functions F_2 and g_1 at next-to-leading order.Comment: 21 pages, LaTeX, 11 epsf figures include
Moments of spectral functions: Monte Carlo evaluation and verification
The subject of the present study is the Monte Carlo path-integral evaluation
of the moments of spectral functions. Such moments can be computed by formal
differentiation of certain estimating functionals that are
infinitely-differentiable against time whenever the potential function is
arbitrarily smooth. Here, I demonstrate that the numerical differentiation of
the estimating functionals can be more successfully implemented by means of
pseudospectral methods (e.g., exact differentiation of a Chebyshev polynomial
interpolant), which utilize information from the entire interval . The algorithmic detail that leads to robust numerical
approximations is the fact that the path integral action and not the actual
estimating functional are interpolated. Although the resulting approximation to
the estimating functional is non-linear, the derivatives can be computed from
it in a fast and stable way by contour integration in the complex plane, with
the help of the Cauchy integral formula (e.g., by Lyness' method). An
interesting aspect of the present development is that Hamburger's conditions
for a finite sequence of numbers to be a moment sequence provide the necessary
and sufficient criteria for the computed data to be compatible with the
existence of an inversion algorithm. Finally, the issue of appearance of the
sign problem in the computation of moments, albeit in a milder form than for
other quantities, is addressed.Comment: 13 pages, 2 figure
Energy-Optimal Routes for Electric Vehicles
Abstract. We study the problem of electric vehicle route planning, where an important aspect is computing paths that minimize energy consumption. Thereby, any method must cope with specific properties, such as recuperation, battery constraints (over- and under-charging), and frequently changing cost functions (e. g., due to weather conditions). This work presents a practical algorithm that quickly computes energy-optimal routes for networks of continental scale. Exploiting multi-level overlay graphs [26, 31], we extend the Customizable Route Planning approach [8] to our scenario in a sound manner. This includes the efficient computation of profile queries and the adaption of bidirectional search to battery constraints. Our experimental study uses detailed consumption data measured from a production vehicle (Peugeot iOn). It reveals for the network of Europe that a new cost function can be incorporated in about five seconds, after which we answer random queries within 0.3ms on average. Additional evaluation on an artificial but realistic [22, 36] vehicle model with unlimited range demonstrates the excellent scalability of our algorithm: Even for long-range queries across Europe it achieves query times below 5ms on average—fast enough for interactive applications. Altogether, our algorithm exhibits faster query times than previous approaches, while improving (metric-dependent) preprocessing time by three orders of magnitude.
Lattice Green's Functions for High Order Finite Difference Stencils
Lattice Green's Functions (LGFs) are fundamental solutions to discretized
linear operators, and as such they are a useful tool for solving discretized
elliptic PDEs on domains that are unbounded in one or more directions. The
majority of existing numerical solvers that make use of LGFs rely on a
second-order discretization and operate on domains with free-space boundary
conditions in all directions. Under these conditions, fast expansion methods
are available that enable precomputation of 2D or 3D LGFs in linear time,
avoiding the need for brute-force multi-dimensional quadrature of numerically
unstable integrals. Here we focus on higher-order discretizations of the
Laplace operator on domains with more general boundary conditions, by (1)
providing an algorithm for fast and accurate evaluation of the LGFs associated
with high-order dimension-split centered finite differences on unbounded
domains, and (2) deriving closed-form expressions for the LGFs associated with
both dimension-split and Mehrstellen discretizations on domains with one
unbounded dimension. Through numerical experiments we demonstrate that these
techniques provide LGF evaluations with near machine-precision accuracy, and
that the resulting LGFs allow for numerically consistent solutions to
high-order discretizations of the Poisson's equation on fully or partially
unbounded 3D domains
Efficient solution of two-dimensional wave propagation problems by Cq-Wavelet BEM: Algorithm and applications
In this paper we consider wave propagation problems in two-dimensional unbounded domains, including dissipative effects, reformulated in terms of space-time boundary integral equa- tions. For their solution, we employ a convolution quadrature (CQ) for the temporal and a Galerkin boundary element method (BEM) for the spatial discretization. It is known that one of the main advantages of the CQ-BEMs is the use of the FFT algorithm to retrieve the discrete time integral operators with an optimal linear complexity in time, up to a logarithmic term. It is also known that a key ingredient for the success of such methods is the efficient and accurate evaluation of all the integrals that define the matrix entries associated to the full space-time discretization. This topic has been successfully addressed when standard Lagrangian basis functions are considered for the space discretization. However, it results that, for such a choice of the basis, the BEM matrices are in general fully populated, a drawback that prevents the application of CQ-BEMs to large-scale problems. In this paper, as a possible remedy to reduce the global complexity of the method, we consider approximant functions of wavelet type. In particular, we propose a numerical procedure that, by taking advantage of the fast wavelet transform, allows us on the one hand to compute the matrix entries associated to the choice of wavelet basis functions by maintaining the accuracy of those associated to the Lagrangian basis ones and, on the other hand, to generate sparse matrices without the need of storing a priori the fully populated ones. Such an approach allows in principle the use of wavelet basis of any type and order, combined with CQ based on any stable ordinary differential equations solver. Several numerical results, showing the accuracy of the solution and the gain in terms of computer memory saving, are presented and discussed
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