Log N Research Articles

Application of Double-Fourier-series Spectral Method to a Large Size Problem: Two-dimensional Simulations of the Shear Instability on the Sphere

Double-Fourier-Series (DFS) spectral method is applied to a large-size problem of barotropic instability of double-shear flow on the sphere. The computing source is the NEC SX-5 parallel vector processors, with the maximum vector length of 512. It is demonstrated that the DFS spectral model is robust and stable even for such a large-sized intensively nonlinear problem, and can simulate well the multiple scale phenomenon without losing accuracy. In addition to the efficiency on serial computing, represented with O(N² log2 N) operations as opposed to O(N³) for the spherical harmonics spectral method, with N the truncation, the DFS spectral model also preserves the efficiency on parallel computing on vector architecture, due to its nature of two dimensional transform. The parallel performance increased slightly with the resolution, and nearly 33.5 percent (26.8 GFLOPS) of the theoretical peak performance (80 GFLOPS) was achieved in the highest-resolution experiment. The zonal-mean absolute vorticity, of which initial condition is characterized as two peaks in both hemispheres, evolves with time into a nearly constant value over the hemisphere. On the other hand, the meridional gradient of the absolute vorticity increases around the equator. The kinetic energy per unit mass is calculated for each total wavenumber, where a disturbance field of a single total wavenumber is separated by an 8th-order spherical harmonics filter. Kinetic energy spectrum shows two distinct subranges, each with a constant slope. The subrange, other than the viscous subrange, shows a slightly increasing slope with time and approaches l-3 (l is the total wavenumber) in the matured stage, when a single large vortex is formed. As the resolution increases, the subrange other than the viscous subrange extends to the higher wavenumber domain, due to low viscosity. Numerical convergence of the solution with a fixed viscosity is discussed in terms of time averaged zonal-mean statistics of the zonal-flow.

Efficient algorithms for checking the equivalence of multistage interconnection networks

In this paper we study the topological equivalence problem of multistage interconnection networks (MINs). We prove a new characterization of topologically equivalent MINs by means of a novel approach. Applying this characterization to log N stage MINs we completely describe the equivalence class which the Reverse Baseline belongs to. Most important, we apply the characterization to (2 log N - 1) stage MINs obtained as concatenation of two log N stage Reverse Baseline equivalent MINs: in this way, we deduce an O(N log N) time algorithm testing the equivalence of two such MINs. This result substantially improves the time complexity of the previously known algorithms (O(N4 log N)). Finally, we determine the number of different equivalence classes of (2 log N - 1) stage MINs and we characterize each of them.

Precise rate in the law of iterated logarithm for ρ-mixing sequence

Let {X, X n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set $$S_n = \sum\nolimits_{k = 1}^n X _k ,M_n = max_{k \leqslant n} \left| {S_k } \right|,n \geqslant 1$$ . Suppose lim n→∞ $$ES_n^2 /n = :\sigma ^2 > 0$$ and $$\sum\limits_{n = 1}^\infty {\rho ^{2/d} \left( {2^n } \right)} < \infty $$ , where d=2, if −1 2(b+1), if b≥0. It is proved that, for any b>−1, $$\begin{gathered} \mathop {lim}\limits_{\varepsilon \searrow 0} \varepsilon ^{2\left( {b + 1} \right)} \sum\limits_{n = 1}^\infty {\frac{{\left( {log logn} \right)^b }}{{nlogn}}} P\left\{ {M_n \geqslant \varepsilon \sigma \sqrt {2nlog logn} } \right\} = \hfill \\ \frac{2}{{\left( {b + 1} \right)\sqrt \pi }}\Gamma \left( {b + 3/2} \right)\sum\limits_{k = 0}^\infty {\frac{{\left( { - 1} \right)^k }}{{\left( {2k + 1} \right)^{2b + 2} }}} \hfill \\ \end{gathered} $$ , where Γ(•) is a Gamma function.

On the persistence and pervasiveness of a new mutation.

It has frequently been assumed that the persistence of a deleterious mutation (the average number of generations before its loss) and its pervasiveness (the average number of individuals carrying the gene before its loss) are equal. This is true for a particular simple, widely used infinite model, but this agreement is not general. If hs >> 1/(4N(e)), where hs is the selective disadvantage of mutant heterozygotes and N(e) is the effective population number, the contribution of homozygous mutants can be neglected and the simple approximate formula 1/hs gives the mean pervasiveness. But the expected persistence is usually much smaller, 2(log(e)(1/2hs) + 1 - gamma) where gamma = 0.5772. For neutral mutations, the total number of heterozygotes until fixation or loss is often the quantity of interest, and its expected value is 2N(e), with remarkable generality for various population structures. In contrast, the number of generations until fixation or loss, 2(N(e)/N)(1 + log(e)2N), is much smaller than the total number of heterozygotes. In general the number of generations is less than the number of individuals.

A Reflective Symmetry Descriptor for 3D Models

Computing reflective symmetries of 2D and 3D shapes is a classical problem in computer vision and computational geometry. Most prior work has focused on finding the main axes of symmetry, or determining that none exists. In this paper we introduce a new reflective symmetry descriptor that represents a measure of reflective symmetry for an arbitrary 3D model for all planes through the model’s center of mass (even if they are not planes of symmetry). The main benefits of this new shape descriptor are that it is defined over a canonical parameterization (the sphere) and describes global properties of a 3D shape. We show how to obtain a voxel grid from arbitrary 3D shapes and, using Fourier methods, we present an algorithm computes the symmetry descriptor in O(N4 log N) time for an N × N × N voxel grid and computes a multiresolution approximation in O(N3 log N) time. In our initial experiments, we have found that the symmetry descriptor is insensitive to noise and stable under point sampling. We have also found that it performs well in shape matching tasks, providing a measure of shape similarity that is orthogonal to existing methods.

Delay Optimization in Quorum Consensus

The management of replicated data in distributed database systems is a classic problem with great practical importance. Quorum consensus is one of the popular methods, combined with eager replication, for managing replicated data. In this paper we investigate the problems of delay-optimal quorum consensus. Firstly, we show that the problem of minimizing the total delay (or mean delay) restricted to a ring can be solved in a constant time in contrast to the existing approximation results. Secondly, we show that the problem of minimizing the total delay (or mean delay) is NP-hard. Thirdly, we present an approximate algorithm with an approximate ratio 2; and the approximate algorithm can guarantee the exact solutions for some specific network topology, such as trees and meshes. Finally, we present an improvement on the existing algorithm to solve the problem of minimizing the maximal delay; this reduces the time complexity from O(n 3 log n) to O(n 3) where n is the number of nodes.

Efficient Algorithms for k-Terminal Cuts on Planar Graphs

The minimum k-terminal cut problem is of considerable theoretical interest and arises in several applied areas such as parallel and distributed computing, VLSI circuit design, and networking. In this paper we present two new approximation and exact algorithms for this problem on an n-vertex undirected weighted planar graph G. For the case when the k terminals are covered by the boundaries of m > 1 faces of G, we give a min{O(n 2 log n logm), O(m 2 n 1.5 log2 n + k n)} time algorithm with a (2–2/k)-approximation ratio (clearly, m \le k). For the case when all k terminals are covered by the boundary of one face of G, we give an O(n k3 + (n log n)k 2) time exact algorithm, or a linear time exact algorithm if k = 3, for computing an optimal k-terminal cut. Our algorithms are based on interesting observations and improve the previous algorithms when they are applied to planar graphs. To our best knowledge, no previous approximation algorithms specifically for solving the k-terminal cut problem on planar graphs were known before. The (2–2/k)-approximation algorithm of Dahlhaus et al. (for general graphs) takes O(k n 2 log n) time when applied to planar graphs. Our approximation algorithm for planar graphs runs faster than that of Dahlhaus et al. by at least an O(k/logm) factor (m \le k).

A multipole Galerkin boundary element method for acoustics

A fast multilevel multipole (FMM) algorithm is derived for the Helmholtz equation and adopted to the symmetric Galerkin boundary element method (BEM) for acoustics. The FMM allows to evaluate a matrix–vector product of the BEM with a computational cost of O(N log 2N), thus leading to a significant reduction of computation time and memory requirements compared to standard BEM formulations. This allows the simulation of large scale acoustic models. The performance of the algorithm is demonstrated on the example of sound radiation from an L-shaped domain with BE discretizations of up to 45,000 elements.

THE LARGEST EMPTY ANNULUS PROBLEM

Given a set of n points S in the Euclidean plane, we address the problem of computing an annulus A, (open region between two concentric circles) of largest width, that partitions S into a subset of points inside and a subset of points outside the circles, such that no point p∈S lies in the interior of A. This problem can be considered as a maximin facility location problem for n points such that the facility is a circumference. We give a characterization of the centres of annuli which are locally optimal and we show that the problem can be solved in O(n3 log n) time and O (n) space. We also consider the case in which the number of points in the inner circle is a fixed value k. When k∈O(n) our algorithm runs in O(n3 log n) time and O(n) space, furthermore, we can simultaneously optimize for all values of k within the same time bound. When k is small, that is a fixed constant, we can solve the problem in O(n log n) time and O(n) space.

A joint functional law for the Wiener process and principal value

We present a joint functional iterated logarithm law for the Wiener process and the principal value of its local times.

Δ-stepping: a parallelizable shortest path algorithm

The single source shortest path problem for arbitrary directed graphs with n nodes, m edges and nonnegative edge weights can sequentially be solved using O(n· logn+m) operations. However, no work-efficient parallel algorithm is known that runs in sublinear time for arbitrary graphs. In this paper we present a rather simple algorithm for the single source shortest path problem. Our new algorithm, which we call Delta-stepping, can be implemented very efficiently in sequential and parallel setting for a large class of graphs. For random edge weights and arbitrary graphs with maximum node degree d, sequential Δ-stepping needs O(n+m+d·L) total average-case time, where L denotes the maximum shortest path weight from the source node s to any node reachable from s. For example, this means linear time on directed graphs with constant maximum degree. Our best parallel version for a PRAM takes O(d·L· logn+ log 2n) time and O(n+m+d·L· logn) work on average. For random graphs, even O( log 2n) time and O(n+m) work on average can be achieved. We also discuss how the algorithm can be adapted to work with nonrandom edge weights and how it can be implemented on distributed memory machines. Experiments indicate that already a simple implementation of the algorithm achieves significant speedup on real machines.

Recognition of some perfectly orderable graph classes

This paper presents new algorithms for recognizing several classes of perfectly orderable graphs. Bipolarizable and P 4-simplicial graphs are recognized in O( n 3.376) time, improving the previous bounds of O( n 4) and O( n 5), respectively. Brittle and semi-simplicial graphs are recognized in O( n 3) time using a randomized algorithm, and O(n 3 log 2n) time if a deterministic algorithm is required. The best previous time bound for recognizing these classes of graphs is O( m 2). Welsh–Powell opposition graphs are recognized in O( n 3) time, improving the previous bound of O( n 4). HHP-free graphs and maxibrittle graphs are recognized in O( mn) and O( n 3.376) time, respectively.

Batch scheduling with controllable setup and processing times to minimize total completion time

We study a problem of scheduling n jobs on a single machine in batches. A batch is a set of jobs processed contiguously and completed together when the processing of all jobs in the batch is finished. Processing of a batch requires a machine setup time dependent on the position of this batch in the batch sequence. Setup times and job processing times are continuously controllable, that is, they are real-valued variables within their lower and upper bounds. A deviation of a setup time or job processing time from its upper bound is called a compression. The problem is to find a job sequence, its partition into batches, and the values for setup times and job processing times such that (a) total job completion time is minimized, subject to an upper bound on total weighted setup time and job processing time compression, or (b) a linear combination of total job completion time, total setup time compression, and total job processing time compression is minimized. Properties of optimal solutions are established. If the lower and upper bounds on job processing times can be similarly ordered or the job sequence is fixed, then O(n3 log n) and O(n5) time algorithms are developed to solve cases (a) and (b), respectively. If all job processing times are fixed or all setup times are fixed, then more efficient algorithms can be devised to solve the problems.

The Reverse Problem of Range Query

We design an efficient algorithm for the reverse problem of the two-dimensional range query. In the range query problem, we specify a range of a rectangular shape in a given (n,n) array, and count the number of points in the range. If the points have weights, we compute the sum of the weights in the range. In the reverse problem, we give a value v and find a range whose sum equals the value. The time for our algorithms is O(n3 log n). We also give an algorithm with O(vn2log2n) time. This is fast for a small v. We briefly compare our problem with the maximum subarray problem where we obtain a subarray that maximizes the sum.

Criteria for irrationality of Euler’s constant

By modifying Beukers’ proof of Apéry’s theorem that ζ ( 3 ) \zeta (3) is irrational, we derive criteria for irrationality of Euler’s constant, γ \gamma . For n &gt; 0 n&gt;0 , we define a double integral I n I_n and a positive integer S n S_n , and prove that with d n = LCM ⁡ ( 1 , … , n ) d_n=\operatorname {LCM}(1,\dotsc ,n) the following are equivalent: 1. The fractional part of log ⁡ S n \log S_n is given by { log ⁡ S n } = d 2 n I n \{\log S_n\}=d_{2n}I_n for some n n . 2. The formula holds for all sufficiently large n n . 3. Euler’s constant is a rational number. A corollary is that if { log ⁡ S n } ≥ 2 − n \{\log S_n\}\ge 2^{-n} infinitely often, then γ \gamma is irrational. Indeed, if the inequality holds for a given n n (we present numerical evidence for 1 ≤ n ≤ 2500 ) 1\le n\le 2500) and γ \gamma is rational, then its denominator does not divide d 2 n ( 2 n n ) d_{2n}\binom {2n}{n} . We prove a new combinatorial identity in order to show that a certain linear form in logarithms is in fact log ⁡ S n \log S_n . A by-product is a rapidly converging asymptotic formula for γ \gamma , used by P. Sebah to compute γ \gamma correct to 18063 decimals.

Optimal Parallel Searching an Array for Certain Repetitions

Abstract Repetitive substructures of two-dimensional arrays have been recently defined and studied in an attempt to parallel some of the analogous developments already known for strings. In the present paper we propose an O (log log n) time O(n3 log n) work optimally fast, optimal parallel CRCW-PRAM algorithm for detecting all repetitions of certain type in an n × n array.

Fast algorithms for discrete polynomial transforms on arbitrary grids

Consider the Vandermonde-like matrix P:=(P k(x M,l)) l,k=0 M,N, where the polynomials P k satisfy a three-term recurrence relation and x M, l ∈[−1,1] are arbitrary nodes. If P k are the Chebyshev polynomials T k , then P coincides with A:=(T k(x M,l)) l=0,k=0 M,N. This paper presents a fast algorithm for the computation of the matrix–vector product Pa in O(N log 2N) arithmetical operations. The algorithm divides into a fast transform which replaces Pa with A a ̃ and a fast cosine transform on arbitrary nodes (NDCT). Since the first part of the algorithm was considered in [Math. Comp. 67 (1998) 1577], we focus on approximative algorithms for the NDCT. Our considerations are completed by numerical tests.

An Edge-Splitting Algorithm in Planar Graphs

For a multigraph G = (V, E), let s ∈ V be a designated vertex which has an even degree, and let λ G (V − s) denote min{c G(X) | O ≠ X ⊂ V − s}, where c G(X) denotes the size of cut X. Splitting two adjacent edges (s, u) and (s, v) means deleting these edges and adding a new edge (u, v). For an integer k, splitting two edges e 1 and e 2 incident to s is called (k, s)-feasible if λG′(V − s) ≥ k holds in the resulting graph G′. In this paper, we prove that, for a planar graph G and an even k or k = 3 with k ≤ λ G (V − s), there exists a complete (k, s)-feasible splitting at s such that the resulting graph G′ is still planar, and present an O(n 3 log n) time algorithm for finding such a splitting, where n = |V|. However, for every odd k ≥ 5, there is a planar graph G with a vertex s which has no complete (k, s)-feasible and planarity-preserving splitting. As an application of this result, we show that for an outerplanar graph G and an even integer k the problem of optimally augmenting G to a k-edge-connected planar graph can be solved in O(n 3 log n) time.

Matrix Decomposition Algorithms for Modified Spline Collocation for Helmholtz Problems

We consider the solution of various boundary value problems for the Helmholtz equation in the unit square using a nodal cubic spline collocation method and modifications of it which produce optimal (fourth-) order approximations. For the solution of the collocation equations, we formulate matrix decomposition algorithms, fast direct methods which employ fast Fourier transforms and require O(N2 log N) operations on an N × N uniform partition of the unit square. A computational study confirms the published analysis for the Dirichlet problem and indicates that similar results hold for Neumann, mixed, and periodic boundary conditions. The numerical results also exhibit superconvergence phenomena not reported in earlier studies.

An efficient deterministic parallel algorithm for two processors precedence constraint scheduling

We present here a new deterministic parallel algorithm for the two-processor scheduling problem. The algorithm uses only O(n 3) processors and takes O( log 2n) time on a CREW PRAM. In order to prove the above bounds we show how to compute in NC the lexicographically first matching for a special kind of convex bipartite graphs.