On the gap between the structural controllability of time-varyingand time-invariant systems

Structural controllability of time-invariant and time-varying systems when the input control sequences have a restricted length k is compared. The dimensions of controllable space coincide in the following three special cases: the input sequences have length k=2; the input sequences have k=n, where n is the size of the system (i.e., the ultimate controllability is the same in both cases); and for every length of input sequences provided that the system has a single input only. It is proved that there may appear a gap for every input length k such that 2< k⩽n/2. The case when n/2<k<n is left open     

Optimal resilient distributed algorithms for ring election

The problem of electing a leader in a dynamic ring in which processors are permitted to fail and recover during election is discussed. It is shown that &thetas;(n log n+k_r) messages, counting only messages sent by functional processors, are necessary and sufficient for dynamic ring election, where k_r is the number of processor recoveries experienced     

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     

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     

Network communication in edge-colored graphs: gossiping

A mechanism for scheduling communications in a network in which individuals exchange information periodically according to a fixed schedule is presented. A proper k edge-coloring of the network is considered to be a schedule of allowed communications such that an edge of color i can be used only at times i modulo k. Within this communication scheduling mechanism, the information exchange problem known as gossiping is considered. It is proved that there is a proper k edge-coloring such that gossip can be completed in a path of n edges in a certain time for n⩾k⩾1. Gossip can not be completed in such a path any earlier under any proper k edge-coloring. In any tree of bounded degree Δ and diameter d, gossip can be completed under a proper Δ edge-coloring in time (Δ-1)d +1. In a k edge-colored cycle of n vertices, other time requirements of gossip are determined     

Fault-tolerant embedding of complete binary trees in hypercubes

The focus is on the following graph-theoretic question associated with the simulation of complete binary trees by faulty hypercubes: if a certain number of nodes or links are removed from an n-cube, will an (n-1)-tree still exists as a subgraph? While the general problem of determining whether a k-tree, k< n, still exists when an arbitrary number of nodes/links are removed from the n-cube is found to be NP-complete, an upper bound is found on how many nodes/links can be removed and an (n-1)-tree still be guaranteed to exist. In fact, as a corollary of this, it is found that if no more than n-3 nodes/links are removed from an (n-1)-subcube of the n-cube, an (n-1)-tree is also guaranteed to exist     

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     

Explicit expressions for cascade factorization of generalnonminimum phase systems

Explicit expressions for two different cascade factorizations of any detectable left invertible nonminimum phase systems are given. The first one is a well known minimum phase/all-pass factorization by which all nonminimum phase zeros of a transfer function G(s) are collected into an all-pass factor V(s), and G (s) is written G_m(s)V$ where G_ms is considered as a minimum phase image of G(s). The second one is a new cascade factorization by which G(s) is rewritten as G_M( s)U(s) where U(s) collects all `awkward' zeros including all nonminimum phase zeros of G( s). Both G_m(s) and G_M(s) retain the given infinite zero structure of G(s). Further properties of G _m(s), G_M(s), and U (s) are discussed. These factorizations are useful in several applications including loop transfer recovery     

Vibrational feedback control in the problem of acoustic stability

The problem of absolute stability in a vibrational feedback controller is introduced and discussed. It is shown that for any rational G(s)=n(s)/d(s ) with d(s) Hurwitz and deg d(s) -deg n(s)=1 there exists a linear dynamic periodic controller that ensures, in a certain sense, the infinite sector of absolute stability. This implies that an additional dynamical element, inserted in the feedback loop, may lead to improvements in the robustness of nonlinear systems     

An efficient digital search algorithm by using a double-arraystructure

An efficient digital search algorithm that is based on an internal array structure called a double array, which combines the fast access of a matrix form with the compactness of a list form, is presented. Each arc of a digital search tree, called a DS-tree, can be computed from the double array in 0(1) time; that is to say, the worst-case time complexity for retrieving a key becomes 0(k) for the length k of that key. The double array is modified to make the size compact while maintaining fast access, and algorithms for retrieval, insertion, and deletion are presented. If the size of the double array is n+cm, where n is the number of nodes of the DS-tree, m is the number of input symbols, and c is a constant particular to each double array, then it is theoretically proved that the worst-case times of deletion and insertion are proportional to cm and cm², respectively, and are independent of n. Experimental results of building the double array incrementally for various sets of keys show that c has an extremely small value, ranging from 0.17 to 1.13     

Systolic computation of multivariable frequency response

An algorithm intended for software implementation on a programmable systolic/wavefront computer is presented for the computation of a complex-valued frequency-response matrix G. Typically, real-valued state-space model matrices are given and the calculation of G must be performed for a very large number of values of the scalar frequency parameter. The algorithm is an orthogonal version of an algorithm described previously by A.J. Laub (ibid., vol.26, no.4, p.407-8, 1981). The system matrix A is reduced initially to an upper Hessenberg form which is preserved as the frequency varies subsequently. A systolic QR factorization of a certain complex-valued matrix is then implemented for effecting the necessary linear system solution (inversion). The critical computational component is the back solve. This computational component's process dependency graph is embedded optimally in space and time through the use of a nonlinear spacetime transformation. The computational period of the algorithm is O(n) where n is the order of the matrix A     

Balanced parallel sort on hypercube multiprocessors

A parallel sorting algorithm for sorting n elements evenly distributed over 2^d p nodes of a d-dimensional hypercube is presented. The average running time of the algorithm is O((n log n)/p+p log 2n). The algorithm maintains a perfect load balance in the nodes by determining the (kn/p)th elements (k1,. . ., (p-1)) of the final sorted list in advance. These p-1 keys are used to partition the sorted sublists in each node to redistribute data to the nodes to be merged in parallel. The nodes finish the sort with an equal number of elements (n/ p) regardless of the data distribution. A parallel selection algorithm for determining the balanced partition keys in O(p log2n) time is presented. The speed of the sorting algorithm is further enhanced by the distance-d communication capability of the iPSC/2 hypercube computer and a novel conflict-free routing algorithm. Experimental results on a 16-node hypercube computer show that the sorting algorithm is competitive with the previous algorithms and faster for skewed data distributions     

The stability of a family of polynomials can be deduced from afinite number 0(k3) of frequency checks

Let φ(s,a)=φ₀(s,a)+ a₁φ₁(s)+a₂ φ₂(s)+ . . .+a_kφ _k(s)=φ₀(s)-q(s, a) be a family of real polynomials in s, with coefficients that depend linearly on parameters a_i which are confined in a k-dimensional hypercube Ω_a . Let φ₀(s) be stable of degree n and the φ_i(s) polynomials (i⩾1) of degree less than n. A Nyquist argument shows that the family φ(s) is stable if and only if the complex number φ₀(jω) lies outside the set of complex points -q(jω,Ω_a) for every real ω. In a previous paper (Automat. Contr. Conf., Atlanta, GA, 1988) the authors have shown that -q(jω,Ω_a ), the so-called `-q locus', is a 2k convex parpolygon. The regularity of this figure simplifies the stability test. In the present paper they again exploit this shape and show that to test for stability only a finite number of frequency checks need to be done; this number is polynomial in k, 0(k³), and these critical frequencies correspond to the real nonnegative roots of some polynomials     

Optimal distributed t-resilient election in completenetworks

The problem of distributed leader election in an asynchronous complete network, in the presence of faults that occurred prior to the execution of the election algorithm, is discussed. Failures of this type are encountered, for example, during a recovery from a crash in the network. For a network with n processors, k of which start the algorithm that uses at most O(n log k +n+kt) messages is presented and shown to be optimal. An optimal algorithm for the case where the identities of the neighbors are known is also presented. It is noted that the order of the message complexity of a t-resilient algorithm is not always higher than that of a nonresilient one. The t-resilient algorithm is a systematic modification of an existing algorithm for a fault-free network     

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     

Stabilization by reduced-order controllers

A recent result by A. Linnemann (Syst. Contr. Lett., vol.11, p.27-32, 1988) gives conditions under which a continuous-time single-loop plant of order n can be stabilized by a reduced-order controller. Specifically, if the Euclidean algorithm is applied to the numerator and denominator polynomials of the transfer function and one of the remainders is a kth-order Hurwitz polynomial, then a stabilizing controller of order n-k-1 exists. The author provides an alternative proof of this result     

Optimal parallel algorithms for problems modeled by a family ofintervals

A family of intervals on the real line provides a natural model for a vast number of scheduling and VLSI problems. Recently, a number of parallel algorithms to solve a variety of practical problems on such a family of intervals have been proposed in the literature. The authors develop computational tools and show how they can be used for the purpose of devising cost-optimal parallel algorithms for a number of interval-related problems, including finding a largest subset of pairwise nonoverlapping intervals, a minimum dominating subset of intervals, along with algorithms to compute the shortest path between a pair of intervals and, based on the shortest path, a parallel algorithm to find the center of the family of intervals. More precisely, with an arbitrary family of n intervals as input, all the algorithms run in O(log n) time using O(n) processors in the EREW-PRAM model of computation     

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     

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     

A new method of image compression using irreducible covers ofmaximal rectangles

The binary-image-compression problem is analyzed using irreducible cover of maximal rectangles. A bound on the minimum-rectangular-cover problem for image compression is given under certain conditions that previously have not been analyzed. It is demonstrated that for a simply connected image, the irreducible cover proposed uses less than four times the number of the rectangles in a minimum cover. With n pixels in a square, the parallel algorithm for obtaining the irreducible cover uses (n/log n) concurrent-read-exclusive write (CREW) processors in O(log n) time