Confluent general-factorial difference and derivative operators and polynomial-exponential interpolation

A previous application of the Newton divided difference series of the displacement function E^z = (1 + Δ)^z = e ^Dz, where the operators Δ and D are the variables, to purely exponential interpolation employing general-factorial differences and derivatives, {Pi;^m_i=0 (Δ - S_i)}f(0) and {Pi;^m_i=0 (D - t_i)}f(0), in which the s_i's and t_i's are distinct[1], is here extended to mixed polynomial-exponential interpolation where the s_i's and t_i's are no longer distinct.     

The unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan

In this paper we consider the unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan. We show that this problem is strongly NP-hard, and give an O(n(n/m+1)^m) time dynamic programming algorithm and an O(mk^k+1P^2k−1) time dynamic programming algorithm, where n is the number of jobs, m is the number of families, k is the number of distinct release dates and P is the sum of the processing times of all families. We further give a heuristic with a performance ratio 2. We also give a polynomial-time approximation scheme for the problem.     

Parallel clustering algorithms

Clustering techniques play an important role in exploratory pattern analysis, unsupervised learning and image segmentation applications. Many clustering algorithms, both partitional clustering and hierarchical clustering, require intensive computation, even for a modest number of patterns. This paper presents two parallel clustering algorithms. For a clustering problem with N = 2ⁿ patterns and M = 2^m features, the time complexity of the traditional partitional clustering algorithm on a single processor computer is O(MNK), where K is the number of clusters. The proposed algorithm on anSIMD computer with MN processors has a time complexity O(K(n + m)). The time complexity of the proposed single-link hierarchical clustering algorithm is reduced from O(MN²) of the uniprocessor algorithm to O(nN) with MN processors.     

Parallel routing algorithms for incomplete hypercube interconnection networks

Hypercube interconnection networks have been receiving considerable attention in the supercomputing environment. However, the number of processors must be exactly 2^r for an r-cube complete hypercube. This restriction severely limits its applicability. In this paper, we address three variant hypercube topologies with more flexibility in system sizes, the labelled hypercubes I_m^r, I_M^r, and I_A^r. Incomplete hypercube I_m^r consists of an r-cube and an m-cube complete hypercubes; I_m^r is composed of 2^r and Σ_{m ε M} 2^m nodes; I_Ar comes from an r-cube complete hypercube which operates in a degraded manner and allows that the missing nodes to be arbitrarily distributed. Specifically, we focus on the parallel paths routing algorithms for these three classes of incomplete hypercubes. Parallel paths between any given two nodes mean that these paths have the same source and destination nodes but with different intermediate nodes. Parallel communication is important as it will allow us to use the full bandwidth of the multiprocessors for the data transfer operation between any two nodes, and3these redundant paths can increase system fault-tolerance and communication reliability. With these parallel routing algorithms, one can use them as a criterion to design multiprocessor systems.     

Heaps and heapsort on secondary storage

A heap structure designed for secondary storage is suggested that tries to make the best use of the available buffer space in primary memory. The heap is a complete multi-way tree, with multi-page blocks of records as nodes, satisfying a generalized heap property. A special feature of the tree is that the nodes may be partially filled, as in B-trees. The structure is complemented with priority-queue operations insert and delete-max. When handling a sequence of S operations, the number of page transfers performed is shown to be O(∑_{i = 1}^S(1/P) log_(M/P)(N_i/P)), where P denotes the number of records fitting into a page, M the capacity of the buffer space in records, and N_i, the number of records in the heap prior to the ith operation (assuming P 1 and S> M c · P, where c is a small positive constant). The number of comparisons required when handling the sequence is O(∑_{i = 1}^S log₂ N_i). Using the suggested data structure we obtain an optimal external heapsort that performs O((N/P) log_(M/P)(N/P)) page transfers and O(N log₂ N) comparisons in the worst case when sorting N records.     

Least adaptive optimal search with unreliable tests

We consider the basic problem of searching for an unknown m-bit number by asking the minimum possible number of yes–no questions, when up to a finite number e of the answers may be erroneous. In case the (i+1)th question is adaptively asked after receiving the answer to the ith question, the problem was posed by Ulam and Rényi and is strictly related to Berlekamp's theory of error correcting communication with noiseless feedback. Conversely, in the fully non-adaptive model when all questions are asked before knowing any answer, the problem amounts to finding a shortest e-error correcting code. Let q_e(m) be the smallest integer q satisfying Berlekamp’s bound . Then at least q_e(m) questions are necessary, in the adaptive, as well as in the non-adaptive model. In the fully adaptive case, optimal searching strategies using exactly q_e(m) questions always exist up to finitely many exceptional m's. At the opposite non-adaptive case, searching strategies with exactly q_e(m) questions—or equivalently, e-error correcting codes with 2^m codewords of length q_e(m)—are rather the exception, already for e=2, and are generally not known to exist for e>2. In this paper, for each e>1 and all sufficiently large m, we exhibit searching strategies that use a first batch of m non-adaptive questions and then, only depending on the answers to these m questions, a second batch of q_e(m)−m non-adaptive questions. These strategies are automatically optimal. Since even in the fully adaptive case, q_e(m)−1 questions do not suffice to find the unknown number, and q_e(m) questions generally do not suffice in the non-adaptive case, the results of our paper provide e fault tolerant searching strategies with minimum adaptiveness and minimum number of tests.     

Comparing parallel Newton''s method with parallel Laguerre''s method

Newton's and Laguerre's methods can be used to concurrently refine all separated zeros of a polynomial P(z). This paper analyses the rate convergence of both procedures, and its implication on the attainable number n of correct figures. In two special cases the number m of iterations required to reach an accuracy η = 10⁻ⁿ is shown to grow as log_λ n, where λ = 3 for Newton's and λ = 4 for Laguerre's. In the general case m is shown to grow linearly with n for both procedures. An assessment of the efficiency of the two methods is also given, by evaluating the computational complexity of operations in circular arithmetic, and the efficiency indices of the two iterative schemes.     

Learning counting functions with queries

We investigate the problem of learning disjunctions of counting functions, which are general cases of parity and modulo functions, with equivalence and membership queries. We prove that, for any prime number p, the class of disjunctions of integer-weighted counting functions with modulus p over the domain Z_qⁿ (or Zⁿ) for any given integer q 2 is polynomial time learnable using at most n + 1 equivalence queries, where the hypotheses issued by the learner are disjunctions of at most n counting functions with weights from Z_p. In general, a counting function may have a composite modulus. We prove that, for any given integer q 2, over the domain Z₂ⁿ, the class of read-once disjunctions of Boolean-weighted counting functions with modulus q is polynomial-time learnable with only one equivalence query and O(n^q) membership queries.     

Algorithms of discretization of algebraic spatial curves on homogeneous cubical grids

In this paper new methods of discretization (integer approximation) of algebraic spatial curves in the form of intersecting surfaces P(x, y, z) = 0 and Q(x, y, z) = 0 are analyzed.

The use of homogeneous cubical grids G(h³) to discretize a curve is the essence of the method. Two new algorithms of discretization (on 6-connected grid G_6c(h³) and 26-connected grid G₂₆(h³)) are presented based on the method above. Implementation of the algorithms for algebraic spatial curves is suggested. The elaborated algorithms are adjusted for application in computer graphics and numerical control of machine tools.     

On-line parallel heuristics, processor scheduling and robot searching under the competitive framework

In this paper we investigate parallel searches on m concurrent rays for a point target t located at some unknown distance along one of the rays. A group of p agents or robots moving at unit speed searches for t. The search succeeds when an agent reaches the point t. Given a strategy S the competitive ratio is the ratio of the time needed by the agents to find t using S and the time needed if the location of t had been known in advance. We provide a strategy with competitive ratio of 1+2(m/p−1)(m/(m−p))^m/p and prove that this is optimal. This problem has applications in multiple heuristic searches in AI as well as robot motion planning. The case p = 1 is known in the literature as the cow path problem.     

C- and C-continuous spline-interpolation of a regular triangular net of points

The investigations focus on the construction of a C^k-continuous (k=0,1,2) interpolating spline-surface for a given data set consisting of points P_ijk arranged in a regular triangular net and corresponding barycentric parameter triples (u_i,v_j,w_k). We try to generalize an algorithm by A.W. Overhauser who solved the analogous problem for the case of a univariate data set. As a straightforward generalization does not work out we adapt the Overhauser-construction. We use some blending of basic surfaces with uniquely determined basic functions. This yields a spline-surface with a polynomial parametric representation which display C¹- or C²-continuity along the common curve of two adjacent sub-patches. Local control of the emerging spline surface is provided which means moving one data point P changes only some of the sub-patches around P and does not affect regions lying far away.     

Discontinuous quasivariational-like inequalities

In this note, we deal with the following problem: given X Rⁿ, a multification gG : X → 2^X, two (single-valued) maps f : X → Rⁿ, η : X × X → Rⁿ, find a point x* X such that x* Γ (x*) and f(x*), η(x,x*) ≥ 0 for all x Γ(x*). We prove an existence theorem in which, in particular, the function f is not supposed to be continuous.     

Bivariate hermite subdivision

A subdivision scheme for constructing smooth surfaces interpolating scattered data in R³ is proposed. It is also possible to impose derivative constraints in these points. In the case of functional data, i.e., data are given in a properly triangulated set of points {(x_i, y_i)}_i=1^N from which none of the pairs (x_i,y_i) and (x_j,y_j) with i≠j coincide, it is proved that the resulting surface (function) is C¹. The method is based on the construction of a sequence of continuous splines of degree 3. Another subdivision method, based on constructing a sequence of splines of degree 5 which are once differentiable, yields a function which is C² if the data are not ‘too irregular’. Finally the approximation properties of the methods are investigated.     

Minimizing the distance to one evader while chasing another

To approach a simple game Δ² of P and E = {E₁, E₂} with no a priori evaders' role assignment and the payoff equal to the distance to one evader at an instant of catching another, we introduce a concept of casting and study the games Δ_1,2 and Δ_2,1 for preassigned and Δ_p² for open-loop casting procedures. Since Δ_p² is reduced to Δ_1,2 or Δ_2,1 which, in turn, are distinguished only by their notations, we focus attention mainly on Δ_1,2. According to the tenet of transition, Δ_1,2 is divided into a concatenation of Δ_1,2^b (basic) and Δ_1,2^a (auxiliary) games that model the problem before and after the first instant of E₁ capture. The games Δ_1,2^a, Δ_1,2^b, Δ_1,2 are studied one after another with use of the Isaacs' approach extended by Berkowitz, Breakwell, Bernhard et al.     

Parallel nested dissection

Nested dissection is a very popular direct method for solving sparse linear systems that arise from finite difference and finite element methods. Worley and Schreiber [16] give a fine grain algorithm for a square array of processors. Their algorithm uses O(N²) processors, each with O(N) memory, to factor an N² by N² sparse matrix whose graphs is an N × N mesh. The efficiency of their method is between 1/46 and 1/12. George et al. [6] [8] give a medium grain algorithm for hypercube architecture, while George et al. [7] give an algorithm for shared memory machines. These papers present a column oriented approach which can exploit O(N) parallelism and yield efficiencies up to 50%. Lucas [11] also gives a column oriented scheme which achieves up to 75% efficiency and O(N) parallelism. In this paper, we present a medium to fine grain algorithm for a P × P array of processors with local memory. This algorithm can exploit up to O(N²) parallelism. The efficiency of the fine grain version is comparable to [16] while as a medium grain algorithm achieves about 49% efficiency. The strength of the method is due to three factors: its ability to pipeline much of the computation, overlapping computation and communication, and the use of level 3 BLAS like primitives. In addition to its high efficiency its memory requirement is optimal, only O(N² log N/P²) words memory is needed per processor.     

A family of network topologies with multiple loops and logarithmic diameter

A new family of network topologies containing multiple loops is discussed in this paper. In the proposed structure, N processors are interconnected to form a graph G(m, N), m 3, where m is a parameter of the graph such that N is an even multiple of m and (m − 1) × 2[(^{m− l)/2}]+ < N m × 2[^m/2]+1. The graph G(m, N) is hamiltonian with an average node degree (3 + l/m), when m is even and exactly 3 when m is odd. Whereas, the maximum node degree is 4. The diameter of G(m, N) is upper bounded by [11m/8]+ 1. A point to point routing algorithm has been presented. Implementation of ASCEND/DESCEND algorithms in O(m) time has been discussed. It has been shown that in case of a single node failure, the diameter increases by at most 6.     

The role of coherence in eliciting and handling imprecise probabilities and its application to medical diagnosis

We refer to an arbitrary family of events (hypotheses), i.e., has neither any particular algebraic structure nor is a partition of the certain event Ω. We detect logical relations among the given events (the latter could represent some possible diseases), and some further information is carried by probability assessments, relative to an event E (e.g., a symptom) conditionally to some of the H_i's (“partial likelihood”). If we assess (prior) probabilities for the events H_i's, then the ensuing problems are: (i) is this assessment coherent? (ii) is the partial likelihood coherent “per se”? (iii) is the global assignment (the initial one together with the likelihood) coherent? If the relevant answers are all YES, then we may try to “update” (coherently) the priors P(H_i) into the posteriors P(H_i|E). This is an instance of a more general issue, the problem of coherent extensions: a very particular case is Bayes' updating for exhaustive and mutually exclusive hypotheses, in which this extension is unique. In the general case the lack of uniqueness gives rise to upper and lower updated probabilities, and we could now update again the latter, given a new event F and a corresponding (possibly partial) likelihood. In this paper, many relevant features of this problem are discussed, keeping an eye on the distinction between semantic and syntactic aspects.     

A parallel algorithm for generating combinations

A parallel algorithm for generating all combinations of m items out of n given items in lexicographic order is presented. The computational model is a linear systolic array consisting of m identical processing elements. It takes (_mⁿ) time-steps to generate all the (_mⁿ) combinations. Since any processing element is identical and executes the same procedure, it is suitable for VLSI implementation. Based on mathematical induction, such algorithm is proved to be correct.     

Fast evaluation of radial basis functions: I

This paper describes some new techniques for the rapid evaluation and fitting of radial basic functions. The techniques are based on the hierarchical and multipole expansions recently introduced by several authors for the calculation of many-body potentials. Consider in particular the N term thin-plate spline, s(x) = Σ_j=1^N d_jφ(x−x_j), where φ(u) = |u|²log|u|, in 2-dimensions. The direct evaluation of s at a single extra point requires an extra O(N) operations. This paper shows that, with judicious use of series expansions, the incremental cost of evaluating s(x) to within precision ε, can be cut to O(1+|log ε|) operations. In particular, if A is the interpolation matrix, a_i,j = φ(x_i−x_j, the technique allows computation of the matrix-vector product Ad in O(N), rather than the previously required O(N²) operations, and using only O(N) storage. Fast, storage-efficient, computation of this matrix-vector product makes pre-conditioned conjugate-gradient methods very attractive as solvers of the interpolation equations, Ad = y, when N is large.     

Integration of beam functions

The beam functions satisfying d⁴φ/dξ⁴ = β⁴φ with the associated boundary conditions are convenient admissible comparison functions for the approximated solutions of complex structural problems by the well known Rayleigh-Ritz method. Reliable integration formulae for products of various beam functions φ and ψ with an arbitrary function θ are essential. Felgar's recurrence formula for ∫ θφψ dξ is found to be in error and is corrected here. The condition when the characteristic values of θ and ψ are identical is also considered. A simple subroutine is given to evaluate ∫ θ(ξ)(dⁿφ/dξⁿ)(d^mψ/dξ^mdξ for all possible sets of boundary conditions. Applications to the free vibration of nonuniform beams and plate systems are demonstrated.