Two-dimensional convolution on a pyramid computer

An algorithm for convolving a k×k window of weighting coefficients with an n×n image matrix on a pyramid computer of O(n²) processors in time O(logn+k²), excluding the time to load the image matrix, is presented. If k=Ω (√log n), which is typical in practice, the algorithm has a processor-time product O(n ² k²) which is optimal with respect to the usual sequential algorithm. A feature of the algorithm is that the mechanism for controlling the transmission and distribution of data in each processor is finite state, independent of the values of n and k. Thus, for convolving two {0, 1}-valued matrices using Boolean operations rather than the typical sum and product operations, the processors of the pyramid computer are finite-state     

Some explicit formulas for the matrix exponential

Formulas are derived for the exponential of an arbitrary 2×2 matrix in terms of either its eigenvalues or entries. These results are then applied to the second-order mechanical vibration equation with weak or strong damping. Some formulas for the exponential of n×n matrices are given for matrices that satisfy an arbitrary quadratic polynomial. Besides the above-mentioned 2×2 matrices, these results encompass involutory, rank 1, and idempotent matrices. Consideration is then given to n×n matrices that satisfy a special cubic polynomial. These results are applied to the case of a 3×3 skew symmetric matrix whose exponential represents the constant rotation of a rigid body about a fixed axis     

Constant time algorithms for the transitive closure and somerelated graph problems on processor arrays with reconfigurable bussystems

The transitive closure problem in O(1) time is solved by a new method that is far different from the conventional solution method. On processor arrays with reconfigurable bus systems, two O (1) time algorithms are proposed for computing the transitive closure of an undirected graph. One is designed on a three-dimensional n×n×n processor array with a reconfigurable bus system, and the other is designed on a two-dimensional n²×n² processor array with a reconfigurable bus system, where n is the number of vertices in the graph. Using the O(1) time transitive closure algorithms, many other graph problems are solved in O(1) time. These problems include recognizing bipartite graphs and finding connected components, articulation points, biconnected components, bridges, and minimum spanning trees in undirected graphs     

An algorithm computing the general entry of the nthKronecker power of a matrix

The number of distinct entries among the m²ⁿ entries of the nth Kronecker power of an m×m matrix is derived. An algorithm to find the value of each entry of the Kronecker power is presented     

Serial and parallel algorithms for the medial axis transform

An O(n²) time serial algorithm is developed for obtaining the medial axis transform (MAT) of an n×n image. An O(log n) time CREW PRAM algorithm and an O(log² n) time SIMD hypercube parallel algorithm for the MAT are also developed. Both of these use O(n²) processors. Two problems associated with the MAT, the area and perimeter reporting problem, are studied. An O(log n) time hypercube algorithm is developed for both of them, where n is the number of squares in the MAT, and the algorithms use O(n²) processors     

Mixed H2/H∞ control:a convex optimization approach

The problem of finding an internally stabilizing controller that minimizes a mixed H₂/H_∞ performance measure subject to an inequality constraint on the H_∞ norm of another closed-loop transfer function is considered. This problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint. Both the state-feedback and output-feedback problems are considered. It is shown that in the state-feedback case one can come arbitrarily close to the optimal (even over full information controllers) mixed H₂/H_∞ performance measure using constant gain state feedback. Moreover, the state-feedback problem can be converted into a convex optimization problem over a bounded subset of (n×n and n ×q, where n and q are, respectively, the state and input dimensions) real matrices. Using the central H_∞ estimator, it is shown that the output feedback problem can be reduced to a state-feedback problem. In this case, the dimension of the resulting controller does not exceed the dimension of the generalized plant     

A processor-time-minimal systolic array for cubical mesh algorithms

Using a directed acyclic graph (DAG) model of algorithms, the paper focuses on time-minimal multiprocessor schedules that use as few processors as possible. Such a processor-time-minimal scheduling of an algorithm's DAG first is illustrated using a triangular shaped 2-D directed mesh (representing, for example, an algorithm for solving a triangular system of linear equations). Then, algorithms represented by an n×n×n directed mesh are investigated. This cubical directed mesh is fundamental; it represents the standard algorithm for computing matrix product as well as many other algorithms. Completion of the cubical mesh required 3n-2 steps. It is shown that the number of processing elements needed to achieve this time bound is at least [3n^2/4]. A systolic array for the cubical directed mesh is then presented. It completes the mesh using the minimum number of steps and exactly [3n ^2/4] processing elements it is processor-time-minimal. The systolic array's topology is that of a hexagonally shaped, cylindrically connected, 2-D directed mesh     

The minimal dimension of stable faces required to guaranteestability of a matrix polytope

Considers the problem of determining whether each point in a polytope n×n matrices is stable. The approach is to check stability of certain faces of the polytope. For n⩾3, the authors show that stability of each point in every (2n-4)-dimensional face guarantees stability of the entire polytope. Furthermore, they prove that, for any k⩽n², there exists a k-dimensional polytope containing a strictly unstable point and such that all its subpolytopes of dimension min {k-1,2n-5} are stable     

The minimal dimension of stable faces required to guaranteestability of a matrix polytope: D-stability

The problem of determining whether a polytope P of n ×n matrices is D-stable-i.e. whether each point in P has all its eigenvalues in a given nonempty, open, convex, conjugate-symmetric subset D of the complex plane-is discussed. An approach which checks the D-stability of certain faces of P is used. In particular, for each D and n the smallest integer m such that D-stability of every m-dimensional face guarantees D-stability of P is determined. It is shown that, without further information describing the particular structure of a polytope, either (2n-4)-dimensional or (2n-2)-dimensional faces need to be checked for D-stability, depending on the structure of D. Thus more work needs to be done before a computationally tractable algorithm for checking D-stability can be devised     

Fast image labeling using local operators on mesh-connectedcomputers

A new parallel algorithm is proposed for fat image labeling using local operators on image pixels. The algorithm can be implemented on an n×n mesh-connected computer such that, for any integer k in the range [1, log (2n)], the algorithm requires Θ(kn¹k/) bits of local memory per processor and takes Θ(kn) time. Bit-serial processors and communication links can be used without affecting the asymptotic time complexity of the algorithm. The time complexity of the algorithm has very small leading constant factors, which makes it superior to previous mesh computer labeling algorithms for most practical image sizes (e.g. up to 4096×4096 images). Furthermore, the algorithm is based on using stacks that can be realized using very fast shift registers within each processing element     

Convolution on mesh connected multicomputers

An efficient parallel algorithm is presented for convolution on a mesh-connected computer with wraparound. The algorithm does not require a broadcast feature for data values, as assumed by previously proposed algorithms. As a result, the algorithm is applicable to both SIMD and MIMD meshes. For an N×N image and a M×M template, the previous algorithms take O (M²q) time on an N×N mesh-connected multicomputer (q is the number of bits in each entry of the convolution matrix). The algorithms have complexity O(M²r), where r=max {number of bits in an image entry, number of bits in a template entry}. In addition to not requiring a broadcast capability, these algorithms are faster for binary images     

An efficient heuristic for permutation packet routing on mesheswith low buffer requirements

Even though exact algorithms exist for permutation routine of n² messages on a n×n mesh of processors which require constant size queues, the constants are very large and the algorithms very complicated to implement. A novel, simple heuristic for the above problem is presented. It uses constant and very small size queues (size=2). For all the simulations run on randomly generated data, the number of routing steps that is required by the algorithm is almost equal to the maximum distance a packet has to travel. A pathological case is demonstrated where the routing takes more than the optimal, and it is proved that the upper bound on the number of required steps is O(n²). Furthermore, it is shown that the heuristic routes in optimal time inversion, transposition, and rotations, three special routing problems that appear very often in the design of parallel algorithms     

A linear algorithm for generating random numbers with a givendistribution

Let ξ be a random variable over a finite set with an arbitrary probability distribution. Improvements to a fast method of generating sample values for ξ in constant time are suggested. The proposed modification reduces the time required for initialization to O( n). For a simple genetic algorithm, this improvement changes an O(g n 1n n) algorithm into an O(g n) algorithm (where g is the number of generations, and n is the population size)     

Parallel binary search

Two arrays of numbers sorted in nondecreasing order are given: an array A of size n and an array B of size m, where n<m. It is required to determine, for every element of A, the smallest element of B (if one exists) that is larger than or equal to it. It is shown how to solve this problem on the EREW PRAM (exclusive-read exclusive-write parallel random-access machine) in O(logm logn/log log m) time using n processors. The solution is then extended to the case in which fewer than n processors are available. This yields an EREW PRAM algorithm for the problem whose cost is O(n log m, which is O(m)) for n⩽m/log m. It is shown how the solution obtained leads to an improved parallel merging algorithm     

Balanced realization of stable transfer function matrices

Simple formulas are presented to compute the internally balanced minimal realization and the singular decomposition of the Hankel operator of a given continuous-time p×m stable transfer function matrix E(s)/d(s). The proposed formulas involve the Schwarz numbers of d(s) and the singular eigenvalues-eigenmatrices of a suitable finite matrix. Similar results are also obtained for a given discrete-time transfer function matrix     

On time mapping of uniform dependence algorithms into lowerdimensional processor arrays

Most existing methods of mapping algorithms into processor arrays are restricted to the case where n-dimensional algorithms, or algorithms with n nested loops, are mapped into (n-1)-dimensional arrays. However, in practice, it is interesting to map n-dimensional algorithms into (k-1)-dimensional arrays where k<n. A computational conflict occurs if two or more computations of an algorithm are mapped into the same execution time. Based on the Hermite normal form of the mapping matrix, necessary and sufficient conditions are derived to identify mapping without computational conflicts. These conditions are used to find time mappings of n-dimensional algorithms into (k-1)-dimensional arrays, k<n , without computational conflicts. For some applications, the mapping is time-optimal     

Parallel implementation of the extended square-root covariancefilter for tracking applications

Parallel implementations of the extended square-root covariance filter (ESRCF) for tracking applications are developed. The decoupling technique and special properties used in the tracking Kalman filter (KF) are employed to reduce computational requirements and to increase parallelism. The application of the decoupling technique to the ESRCF results in the time and measurement updates of m decoupled (n/m)-dimensional matrices instead of one coupled n-dimensional matrix, where m denotes the tracking dimension and n denotes the number of state elements. The updates of m decoupled matrices are found to require approximately m fewer processing elements and clock cycles than the updates of one coupled matrix. The transformation of the Kalman gain which accounts for the decoupling is found to be straightforward to implement. The sparse nature of the measurement matrix and the sparse, band nature of the transition matrix are explored to simplify matrix multiplications     

Odd even shifts in SIMD hypercubes

A linear-time algorithm is developed to perform all odd (even) length circular shifts of data in an SIMD (single-instruction-stream, multiple-data-stream) hypercube. As an application, the algorithm is used to obtain an O(M²+log N) time and O(1) memory per processor algorithm to compute the two-dimensional convolution of an N×N image and an M×M template on an N² processor SIMD hypercube. This improves the previous best complexity of O(M² log M+log N)     

Computing the width of a set

For a set of points P in three-dimensional space, the width of P, W (P), is defined as the minimum distance between parallel planes of support of P. It is shown that W(P) can be computed in O(n log n +I) time and O(n) space, where I is the number of antipodal pairs of edges of the convex hull of P, and n is the number of vertices; in the worst case, I=O( n²). For a convex polyhedra the time complexity becomes O(n+I). If P is a set of points in the plane, the complexity can be reduced to O(nlog n). For simple polygons, linear time suffices     

A smoothly parameterized family of stabilizable, observable linearsystems containing realizations of all transfer functions of McMillandegree not exceeding n

It is shown that there is a continuously parameterized family F of n-dimensional single-input single-output (SISO) stabilizable detectable linear system Σ(p) which contains at least one realization of each reduced, strictly proper transfer function of McMillan degree not exceeding n. The parameterization map p→Σ(p) is a polynomial function in 2n indeterminates from an open convex polyhedron in R²ⁿ to the linear space of all SISO n-dimensional linear systems