Lyapunov exponents of the logistic map with periodic forcing

The iterative map x_n+1 = r_nx_n (1 - x_n) is investigated with r_n changing periodically between two values A and B. Different periodicities are assumed, e.g., {r_n} = {BABA …} or {r_n} = {BBABA BBABA …}. The Lyapunov exponent (a measure of average stability) is displayed with high resolution on the A-B-plane. The resulting images have aesthetically appealing self-similar structures. Furthermore, these images allow with one glimpse the identification of a number of system properties: coexistence of attractors, superstable curves, order by alternation of chaotic processes, and chaos by periodic resetting from a stable into an unstable fixed point.     

Efficient algorithms for computing two nearest-neighbor problems on a rap

This paper makes an improvement of computing two nearest-neighbor problems of images on a reconfigurable array of processors (RAP) by increasing the bus width between processors. Based on a base-n system, a constant time algorithm is first presented for computing the maximum/minimum of N log N-bit unsigned integers on a RAP using N processors each with N^1/c-bit bus width, where c is a constant and c ≥ 1. Then, two basic operations such as image component labeling and border following are also derived from it. Finally, these algorithms are used to design two constant time algorithms for the nearest neighbor black pixel and the nearest neighbor component problems on an N^1/2 × N^1/2 image using N^1/2 × N^1/2 processors each with N^1/c-bit bus width, where c is a constant and c ≥ 1. Another contribution of this paper is that the execution time of the proposed algorithms is tunable by the bus width.     

Scheduling parallel iterative methods on multiprocessor systems

The paper describes the implementation of the Successive Overrelaxation (SOR) method on an asynchronous multiprocessor computer for solving large, linear systems. The parallel algorithm is derived by dividing the serial SOR method into noninterfering tasks which are then combined with an optimal schedule of a feasible number of processors. The important features of the algorithm are: (i) achieves a speedup S_p O(N/3) and an efficiency E_p 2/3 using P = [N/2] processors, where N is the number of the equations, (ii) contains a high level of inherent parallelism, whereas on the other hand, the convergence theory of the parallel SOR method is the same as its sequential counterpart and (iii) may be modified to use block methods in order to minimise the overhead due to communication and synchronisation of the processors.     

A robustness measure for hypercube networks

Hypercube networks offer a feasible cost-effective solution to parallel computing. Here, a large number of low-cost processors with their own local memories are connected to form an n-cube (B_n) or one of its variants; and the inter-processor communication takes place by message passing instead of shared variables. This paper addresses a constrained two-terminal reliability measure referred to as distance reliability (DR) as it considers the probability that a message can be delivered in optimal time from a given node s to a node t. The problem is equivalent to that of having an operational optimal path (not just any path) between the two nodes. In B_n, the Hamming distance between labels of s and t or H(s, t) determines the length of the optimal path between the two nodes. The shortest distance restriction guarantees optimal communication delay between processors and high link/node utilization across the network. Moreover, it provides a measure for the robustness of symmetric networks. In particular, when H(s, t) = n in B_n, DR will yield the probability of degradation in the diameter, a concept which directly relates to fault-diameter. The paper proposes two schemes to evaluate DR in B_n. The first scheme uses a combinatorial approach by limiting the number of faulty components to (2H(s, t) − 2), while the second outlines paths of length H(s, t) and, then generates a recursive closed-form solution to compute DR. The theoretical results have been verified by simulation. The discrepancy between the theoretical and simulation results is in most cases below 1% and in the worst case 4.6%.     

A sum-of-product neural network (SOPNN)

This paper presents a sum-of-product neural network (SOPNN) structure. The SOPNN can learn to implement static mapping that multilayer neural networks and radial basis function networks normally perform. The output of the neural network has the sum-of-product form ∑Npi=1∏Nvj=1 fij (xj), where xj's are inputs, Nv is the number of inputs, fij( ) is a function generated through network training, and Np is the number of product terms. The function fij(xj) can be expressed as ∑kwijkBjk(xj), where Bjk( ) is a single-variable basis function and Wijk's are weight values. Linear memory arrays can be used to store the weights. If Bjk( ) is a Gaussian function, the new neural network degenerates to a Gaussian function network. This paper focuses on the use of overlapped rectangular pulses as the basis functions. With such basis functions, WijkBjk(xj) will equal either zero or Wijk, and the computation of fij(xj) becomes a simple addition of some retrieved Wijk's. The structure can be viewed as a basis function network with a flexible form for the basis functions. Learning can start with a small set of submodules and have new submodules added when it becomes necessary. The new neural network structure demonstrates excellent learning convergence characteristics and requires small memory space. It has merits over multilayer neural networks, radial basis function networks and CMAC in function approximation and mapping in high-dimensional input space. The technique has been tested for function approximation, prediction of a time series, learning control, and classification.     

Nonlinear robust stabilization of multi-parameter families of systems

We show that given any family of asymptotically stabilizable LTI systems depending continuously on a parameter that lies in some subset [a₁,b₁]××[a_p,b_p] of , there exists a C⁰ time-varying state feedback law v(t,x) (resp. a C⁰ time-invariant feedback law v(x)) which robustly globally exponentially stabilizes (resp. which robustly stabilizes, not asymptotically) the family. Further, if these systems are obtained by linearizing some nonlinear systems, then v(t,x) locally exponentially stabilizes these nonlinear systems. Finally, v(t,x) globally exponentially stabilizes any time-varying system which switches “slowly enough” between the given LTI systems.     

A limit characterization for the number of spanning trees of graphs

In this paper we propose a limit characterization of the behaviour of classes of graphs with respect to their number of spanning trees. Let {G_n} be a sequence of graphs G₀,G₁,G₂,… that belong to a particular class. We consider graphs of the form K_n−G_n that result from the complete graph K_n after removing a set of edges that span G_n. We study the spanning tree behaviour of the sequence {K_n−G_n} when n→∞ and the number of edges of G_n scales according to n. More specifically, we define the spanning tree indicator ({G_n}), a quantity that characterizes the spanning tree behaviour of {K_n−G_n}. We derive closed formulas for the spanning tree indicators for certain well-known classes of graphs. Finally, we demonstrate that the indicator can be used to compare the spanning tree behaviour of different classes of graphs (even when their members never happen to have the same number of edges).     

A novel algorithm for fast computation of Zernike moments

Zernike moments (ZMs) have been successfully used in pattern recognition and image analysis due to their good properties of orthogonality and rotation invariance. However, their computation by a direct method is too expensive, which limits the application of ZMs. In this paper, we present a novel algorithm for fast computation of Zernike moments. By using the recursive property of Zernike polynomials, the inter-relationship of the Zernike moments can be established. As a result, the Zernike moment of order n with repetition m, Z_nm, can be expressed as a combination of Z_n−2,m and Z_n−4,m. Based on this relationship, the Zernike moment Z_nm, for n>m, can be deduced from Z_mm. To reduce the computational complexity, we adopt an algorithm known as systolic array for computing these latter moments. Using such a strategy, the multiplication number required in the moment calculation of Z_mm can be decreased significantly. Comparison with known methods shows that our algorithm is as accurate as the existing methods, but is more efficient.     

Efficient enumeration of all minimal separators in a graph

This paper presents an efficient algorithm for enumerating all minimal a-b separators separating given non-adjacent vertices a and b in an undirected connected simple graph G = (V, E), Our algorithm requires O(n³R_ab) time, which improves the known result of O(n⁴R_ab) time for solving this problem, where ¦V¦= n and R_ab is the number of minimal a-b separators. The algorithm can be generalized for enumerating all minimal A-B separators that separate non-adjacent vertex sets A, B < V, and it requires O(n²(n − n_A − n_b)R_AB) time in this case, where n_a = ¦A¦, n_B = ¦B¦ and r_AB is the number of all minimal A−B separators. Using the algorithm above as a routine, an efficient algorithm for enumerating all minimal separators of G separating G into at least two connected components is constructed. The algorithm runs in time O(n³R⁺_Σ + n⁴R_Σ), which improves the known result of O(n⁶R_Σ) time, where R_σ is the number of all minimal separators of G and R_ΣR⁺_Σ = ∑_1i, v_j) ER_vivj n − 1)/2 − m)R_Σ. Efficient parallelization of these algorithms is also discussed. It is shown that the first algorithm requires at most O((n/log n)R_ab) time and the second one runs in time O((n/log n)R⁺_Σ+n log nR_Σ) on a CREW PRAM with O(n³) processors.     

Periodic solutions of a class of nonlinear difference equations via critical point method

In this paper, we obtain some new sufficient conditions for the existence of nontrivial m-periodic solutions of the following nonlinear difference equation

by using the critical point method, where f: Z × R → R is continuous in the second variable, m ≥ 2 is a given positive integer, p_n+m = p_n for any n  Z and f(t + m, z) = f(t, z) for any (t, z)  Z × R, (−1)^δ = −1 and δ > 0.     

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.     

Computational complexity of distributed detection problems with information constraints

For a system consisting of a set of sensors S = {S₁, S₂, …, S_m} and a set of objects O = {O₁, O₂, …, O_n}, there are information constraints given by a relation R S × O such that (S_i, O_j) R if and only if S_i is capable of detecting O_j. Each (S_i, O_j) R is assigned a confidence factor (a positive real number) which is either explicitly given or can be efficiently computed. Given that a subset of sensors have detected obstacles, the detection problem is to identify a subset H O with the maximum confidence value. The computational complexity of the detection problem, which depends on the nature of the confidence factor and the information constraints, is the main focus of this paper. This problem exhibits a myriad of complexity levels ranging from a worst-case exponential (in n) lower bound in a general case to an O(m + n) time solvability. We show that the following simple versions of a detection problem are computationally intractable: (a) deterministic formulation, where confidence factors are either 0 or 1; (b) uniform formulation where (S_i, O_j) R, for all S_i S, O_j O; (c) decomposable systems under multiplication operation. We then show that the following versions are solvable in polynomial (in n) time: (a) single object detection; (b) probabilistically independent detection; (c) decomposable systems under additive and nonfractional multiplicative measures; and (d) matroid systems.     

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.     

Mutilated chessboard problem is exponentially hard for resolution

Mutilated chessboard principle CB_n says that it is impossible to cover by domino tiles the chessboard 2n×2n with two diagonally opposite corners removed. We prove lower bound on the size of minimal resolution refutation of CB_n.     

Efficient and scalable quicksort on a linear array with a reconfigurable pipelined bus system

Based on the current fiber optic technology, a new computational model, called a linear array with a reconfigurable pipelined abus system (LARPBS), is proposed in this paper. A parallel quicksort algorithm is implemented on the model, and its time complexity is analyzed. For a set of N numbers, the quicksort algorithm reported in this paper runs in O(log₂ N) average time on a linear array with a reconfigurable pipelined bus system of size N. If the number of processors available is reduced to P, where P < N, the algorithm runs in O((N/P) log₂ N) average time and is still scalable. Besides proposing a new algorithm on the model, some basic data movement operations involved in the algorithm are discussed. We believe that these operations can be used to design other parallel algorithms on the same model. Future research in this area is also identified in this paper.     

The Variance of the height of binary search trees

By using analytic tools it is shown that the variance E(H_n−EH_n)² of the height H_n of binary search trees of size n is bounded.     

On some dynamical subsets of the rauzy fractal

Let ( ,(+1)_n) be the adic system associated to the substitution: 1 → 12,…,(n − 1) → 1n, n → 1. In Sirvent (1996) it was shown that there exist a subset Cⁿ of and a map h_n: C → Cⁿ such that the dynamical system (C, h_n) is semiconjugate to ( ). In this paper we compute the Hausdorff and Billingsley dimensions of the geometrical realizations of the set Cⁿ on the (n− l)-dimensional torus. We also show that the dynamical system (Cⁿ,h_n) cannot be realized on the (n − 1)-torus.     

A systolic algorithm for solving dense linear systems

For an arbitrary n × n matrix A and an n × 1 column vector b, we present a systolic algorithm to solve the dense linear equations Ax = b. An important consideration is that the pivot row can be changed during the execution of our systolic algorithm. The computational model consists of n linear systolic arrays. For 1 ≤ i ≤ n, the i^th linear array is responsible to eliminate the i^th unknown variable x_i of x. This algorithm requires 4n time steps to solve the linear system. The elapsed time unit within a time step is independent of the problem size n. Since the structure of a PE is simple and the same type PE executes the identical instructions, it is very suitable for VLSI implementation. The design process and correctness proof are considered in detail. Moreover, this algorithm can detect whether A is singular or not.     

Automatic rotational symmetry determination for shape analysis

The determination of the rotational symmetry of a shape is useful for object recognition and shape analysis in computer vision applications. In this paper, a simple, but effective, algorithm to analyse the rotational symmetry of a given closed-curve shape S is proposed. A circle C with the centroid of S as the circle center and the average radius of S as the circle radius is superimposed on S, resulting in the intersection of C and S at a set of points. By theoretical analysis, the relationship between the order N_s of the rotational symmetry of S and the number of intersection points between C and S is established. All the possible values for N_s are determined. And finally N_s is determined by evaluating the similarity between S and its rotated versions. In the proposed algorithm, only simple pixel operations and second-order moment function computation are involved. Several problems caused by the use of discrete image coordinates are analysed and solved for correct decision-making in the algorithm. Good experimental results are also included.     

Distribution of black nodes at various levels in a linear quadtree

The distribution of black leaf nodes at each level of a linear quadtree is of significant interest in the context of estimation of time and space complexities of linear quadtree based algorithms. The maximum number of black nodes of a given level that can be fitted in a square grid of size 2ⁿ × 2ⁿ can readily be estimated from the ratio of areas. We show that the actual value of the maximum number of nodes of a level is much less than the maximum obtained from the ratio of the areas. This is due to the fact that the number of nodes possible at a level k, 0≤k≤n − 1, should consider the sum of areas occupied by the actual number of nodes present at levels k + 1, k + 2, …, n − 1.