We present a fast algorithm for the subset convolution problem:given functions f and g defined on the lattice of subsets of ann-element set n, compute their subset convolution f*g, defined for S⊆ N by [ (f * g)(S) = [T ⊆ S] f(T) g(S/T),,]where addition and multiplication is carried out in an arbitrary ring. Via Mobius transform and inversion, our algorithm evaluates the subset convolution in O(n2 2n) additions and multiplications, substanti y improving upon the straightforward O(3n) algorithm. Specifically, if the input functions have aninteger range [-M,-M+1,...,M], their subset convolution over the ordinary sum--product ring can be computed in O(2n log M) time; the notation O suppresses polylogarithmic factors.Furthermore, using a standard embedding technique we can compute the subset convolution over the max--sum or min--sum semiring in O(2n M) time.To demonstrate the applicability of fast subset convolution, wepresent the first O(2k n2 + n m) algorithm for the Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights, improving upon the O(3kn + 2k n2 + n m) time bound of the classical Dreyfus-Wagner algorithm. We also discuss extensions to recent O(2n)-time algorithms for covering and partitioning problems (Bjorklund and Husfeldt, FOCS 2006; Koivisto, FOCS 2006).

Fourier meets Möbius: fast subset convolution

Modern Computer Arithmetic focuses on arbitrary-precision algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics such as modular arithmetic, greatest common divisors, the Fast Fourier Transform (FFT), and the computation of elementary and special functions. Brent and Zimmermann present algorithms that are ready to implement in your favorite language, while keeping a high-level description and avoiding too low-level or machine-dependent details. The book is intended for anyone interested in the design and implementation of efficient high-precision algorithms for computer arithmetic, and more generally efficient multiple-precision numerical algorithms. It may also be used in a graduate course in mathematics or computer science, for which exercises are included. These vary considerably in difficulty, from easy to small research projects, and expand on topics discussed in the text. Solutions are available from the authors.

/pdf/modern-computer-arithmetic-15pgrhiwb0.pdf

Modern Computer Arithmetic

We present an analysis of proton number fluctuations in $\sqrt{s_{NN}}$ = 2.4 GeV Au+Au collisions measured with the High-Acceptance DiElectron Spectrometer (HADES) at GSI. With the help of extensive detector simulations done with IQMD transport model events including nuclear clusters, various nuisance effects influencing the observed proton cumulants have been investigated. Acceptance and efficiency corrections have been applied as a function of fine grained rapidity and transverse momentum bins, as well as considering local track density dependencies. Next, the effects of volume changes within particular centrality selections have been considered and beyond-leading-order corrections have been applied to the data. The efficiency and volume corrected proton number moments and cumulants Kn of orders n = 1, . . . , 4 have been obtained as a function of centrality and phase-space bin, as well as the corresponding correlators C_n . We find that the observed correlators show a power-law scaling with the mean number of protons, i.e. $C_n \propto ^n$, indicative of mostly long-range multi-particle correlations in momentum space. We also present a comparison of our results with Au+Au collision data obtained at RHIC at similar centralities, but higher $\sqrt{s_{NN}}$.

Proton number fluctuations in $\sqrt{s_{NN}}$ = 2.4 GeV Au+Au collisions studied with HADES

A system and method for interconnecting nodes and routing data packets in high radix networks includes constructing or redefining a network structure to provide improved performance. Computation and data storage nodes are connected to a network of switching nodes that provide near optimum bandwidth and latency for networks of any size. Specialized interconnection patterns and addressing methods ensure reliable data delivery in very large networks with high data traffic volume.

Flexible Radix Switch

A combinatorial Gray code for a class of objects is a listing that contains each object from the class exactly once such that any two consecutive objects in the list differ only by a `small change'. Such listings are known for many different combinatorial objects, including bitstrings, combinations, permutations, partitions, triangulations, but also for objects defined with respect to a fixed graph, such as spanning trees, perfect matchings or vertex colorings. This survey provides a comprehensive picture of the state-of-the-art of the research on combinatorial Gray codes. In particular, it gives an update on Savage's influential survey [C. D. Savage. A survey of combinatorial Gray codes. SIAM Rev., 39(4):605-629, 1997.], incorporating many more recent developments. We also elaborate on the connections to closely related problems in graph theory, algebra, order theory, geometry and algorithms, which embeds this research area into a broader context. Lastly, we collect and propose a number of challenging research problems, thus stimulating new research endeavors.

Combinatorial Gray codes-an updated survey

This book provides algorithms and ideas for computationalists. Subjects treated include low-level algorithms, bit wizardry, combinatorial generation, fast transforms like the Fourier transform, and fast arithmetic for both real numbers and finite fields. Various optimization techniques are described and the actual performance of many given implementations is examined. The focus is on material that does not usually appear in textbooks on algorithms. The implementations are done in C++ and the GP language, written for POSIX-compliant platforms such as the Linux and BSD operating systems.

Matters Computational: Ideas, Algorithms, Source Code

Low level algorithms.- Bit wizardry.- Permutations and their operations.- Sorting and searching.- Data structures.- Combinatorial generation.- Conventions and considerations.- Combinations.- Compositions.- Subsets.- Mixed radix numbers.- Permutations.- Multisets.- Gray codes for string with restrictions.- Parenthesis strings.- Integer partitions.- Set partitions.- Necklaces and Lyndon words.- Hadamard and conference matrices.- Searching paths in directed graphs.- Fast transforms.- The Fourier transform.- Convolution, correlation, and more FFT algorithms.- The Walsh transform and its relatives.- The Haar transform.- The Hartley transform.- Number theoretic transforms (NTTs).- Fast wavelet transforms.- Fast arithmetic.- Fast multiplication and exponentiation.- Root extraction.- Iterations for the inversion of a function.- The AGM, elliptic integrals, and algorithms for computing.- Logarithm and exponential function.- Computing the elementary functions with limited resources.- Numerical evaluation of power series.- Cyclotomic polynomials, product forms, and continued fractions.- Synthetic Iterations.-. Algorithms for finite fields.- Modular arithmetic and some number theory.- Binary polynomials.- Shift registers.- Binary finite fields.- The electronic version of the book.- Machine used for benchmarking.- The GP language.- Bibliography.- Index.

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Matters Computational: Ideas, Algorithms, Source Code

Matters Computational: Ideas, Algorithms, Source Code