Given two strings X and Y of lengths m and n, respectively, the all-substrings longest common subsequence (ALCS) problem obtains the lengths of the subsequences common to X and any substring of Y . The sequential algorithm takes O(mn) time and O(n) space. We present a parallel algorithm for ALCS on a coarse-grained multicomputer (BSP/CGM) model with p < p m processors that takes O(mn=p) time an...

Simple Monte Carlo is a versatile computational method with a convergence rate of O(n−1/2). It can be used to estimate the means of random variables whose distributions are unknown. Bernoulli random variables, Y , are widely used to model success (failure) of complex systems. Here Y = 1 denotes a success (failure), and p = E(Y ) denotes the probability of that success (failure). Another applica...

In this work we solve numerically a boundary value problem for second order fuzzy differential equations under generalized differentiability in the form y′′(t) = p(t)y′(t) + q(t)y(t) + F(t) y(0) = γ, y(`) = λ where t ∈ T = [0, `], p(t) ≥ 0, q(t) ≥ 0 are continuous functions on [0, `] and [γ]α = [γα ,γα ], [λ ] α = [λ α ,λ α ] are fuzzy numbers. There are four different solutions of the problem ...

At a high level, the AKS algorithm for finding the shortest vector in a lattice works by taking a large sphere or box and choosing many (e.g. ∼ 2) random lattice points in the box by taking random combinations of the basis. The points are then refined through an iterative routine called sieving. The sieving procedure Sieve(Y ) will have the following properties. Sieve(Y ) = C ⊆ Y such that: • F...

We provide a new algorithm for generating the Baker–Campbell–Hausdorff (BCH) series Z = log(ee ) in an arbitrary generalized Hall basis of the free Lie algebra L(X,Y ) generated by X and Y . It is based on the close relationship of L(X,Y ) with a Lie algebraic structure of labeled rooted trees. With this algorithm, the computation of the BCH series up to degree 20 (111013 independent elements i...

We prove that a polynomial f ∈ R[x, y] with t non-zero terms, restricted to a real line y = ax + b, either has at most 6t − 4 zeros or vanishes over the whole line. As a consequence, we derive an alternative algorithm for deciding whether a linear polynomial y − ax − b ∈ K [x, y] divides a lacunary polynomial f ∈ K [x, y], where K is a real number field. The number of bit operations performed b...

Abstract An affine de Casteljau type algorithm to compute q−Bernstein Bézier curves is introduced and its intermediate points are obtained explicitly in two ways. Furthermore we define a tensor product patch, based on this algorithm, depending on two parameters. Degree elevation procedure is studied. The matrix representation of tensor product patch is given and we find the transformation matri...

As a consequence of Algorithm 1 below, there always exist integers x, y ∈ Z such that ax+by = gcd(a, b). We say that a and b are co-prime (or relatively prime) if gcd(a, b) = 1, i.e., ax = 1 mod b. From this, x is the multiplicative inverse of a modulo b, and likewise y is the multiplicative inverse of b modulo a. The following deterministic algorithm shows that gcd(a, b) (and additionally, the...

Given topological spaces X,Y , a fundamental problem of algebraic topology is understanding the structure of all continuous maps X → Y . We consider a computational version, where X,Y are given as finite simplicial complexes, and the goal is to compute [X,Y ], i.e., all homotopy classes of such maps. We solve this problem in the stable range, where for some d ≥ 2, we have dimX ≤ 2d−2 and Y is (...

The expectation-maximization (EM) algorithm aims to nd the maximum of a log-likelihood function, by alternating between conditional expectation (E) step and maximization (M) step. This survey rst introduces the general structure of the EM algorithm and the convergence guarantee. Then Gaussian Mixture Model (GMM) are employed to demonstrate how EM algorithm could be applied under Maximum-Likelih...