Resolvability and the upper dimension of graphs

For an ordered set W = {w₁, w₂,…, w_k} of vertices and a vertex v in a connected graph G, the (metric) representation of v with respect to W is the k-vector r(v | W) = (d(v, w₁), d(v, w₂),…, d(v, w_k)), where d(x, y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct representations. A new sharp lower bound for the dimension of a graph G in terms of its maximum degree is presented.

A resolving set of minimum cardinality is a basis for G and the number of vertices in a basis is its (metric) dimension dim(G). A resolving set S of G is a minimal resolving set if no proper subset of S is a resolving set. The maximum cardinality of a minimal resolving set is the upper dimension dim⁺(G). The resolving number res(G) of a connected graph G is the minimum k such that every k-set W of vertices of G is also a resolving set of G. Then 1 ≤ dim(G) ≤ dim⁺(G) ≤ res(G) ≤ n − 1 for every nontrivial connected graph G of order n. It is shown that dim⁺(G) = res(G) = n − 1 if and only if G = K_n, while dim⁺(G) = res(G) = 2 if and only if G is a path of order at least 4 or an odd cycle.

The resolving numbers and upper dimensions of some well-known graphs are determined. It is shown that for every pair a, b of integers with 2 ≤ a ≤ b, there exists a connected graph G with dim(G) = dim⁺(G) = a and res(G) = b. Also, for every positive integer N, there exists a connected graph G with res(G) − dim⁺(G) ≥ N and dim⁺(G) − dim(G) ≥ N.     

Integration of polynomials over n-dimensional linear polyhedra

This paper is concerned with explicit integration formulae for computing integrals of n-variate polynomials over linear polyhedra in n-dimensional space ⁿ. Two different approaches are discussed; the first set of formulae is obtained by mapping the polyhedron in n-dimensional space ⁿ into a standard n-simplex in ⁿ, while the second set of formulae is obtained by reducing the n-dimensional integral to a sum of n − 1 dimensional integrals which are n + 1 in number. These formulae are followed by an application example for which we have explained the detailed computational scheme. The symbolic integration formulae presented in this paper may lead to an easy and systematic incorporation of global properties of solid objects, such as, for example, volume, centre of mass, moments of inertia etc., required in engineering design problems.     

Boundary matching algorithm for connected component labelling using linear quadtrees

A computationally fast top-down recursive algorithm for connected component labelling using linear quadtrees is presented. The input data structure used is a linear quadtree representing only black leaf nodes. The boundary matching approach used ensures that at most two adjacencies of any black leaf node are considered. Neighbour searching is carried out within restricted subsets of the input quadtree. The time and space complexities of the algorithm are O(Bn) and O(B) respectively for a linear quadtree with B black leaves constructed from a binary array of size 2ⁿ × 2ⁿ. Simulations show the algorithm to be twice as fast as an existing algorithm that uses an identical input data structure. The top-down algorithm presented can also be used to efficiently generate a linear quadtree representing all nodes — ‘grey’, ‘black’ and ‘white’ — in preorder when given an input linear quadtree representing only ‘black’ leaf nodes. The boundary matching algorithm is computationally fast and has low static and dynamic storage costs, making it useful for applications where linear quadtrees are held in main memory.     

On space-efficient algorithms for certain NP-complete problems

Some recent results claimed the existence of a class of algorithms for certain NP-complete problems, with running time O(n^{1g k} 2^n/2) and storage requirements O(k 2^n/k), for 2 kn. In this note we show that those results do not hold, implying that an algorithm with time O(n 2^n/2) and space O(2^n/4) is still the best-known solution for such class of NP-complete problems.     

Four-moduli set (2, 2−1, 2+2−1, 2+2−1) simplies the residue to binary converters based on CRT II

A multiplier-free residue to binary converter architecture based on the Chinese remainder theorem II (CRT II) [1] is presented. The paper also includes a binary to residue converter. This is achieved by introducing a new moduli set (2, 2ⁿ − 1, 2ⁿ + 2ⁿ⁻¹ − 1, 2ⁿ⁺¹ + 2ⁿ − 1) for RNS application. The complexity of conversion has been greatly reduced using CRT II with the new moduli set. The proposed hardware architecture replaces the necessary multiplication by shift-left operations. A similar hardware architecture is presented for the binary to residue conversion.     

Experimental numerical studies on a supercomputer of natural convection in an enclosure with localized heating

We present particle simulations of natural convection of a symmetrical, nonlinear, three-dimensional cavity flow problem. Qualitative studies are made in an enclosure with localized heating. The assumption is that particles interact locally by means of a compensating Lennard-Jones type force F, whose magnitude is given by −G/r^p + H/r^q.

In this formula, the parameters G, H, p, q depend upon the nature of the interacting particles and r is the distance between two particles. We also consider the system to be under the influence of gravity. Assuming that there are n particles, the equations relating position, velocity and acceleration at time t_k = kΔt, K = 0, 1, 2, …, are solved simultaneously using the “leap-frog” formulas. The basic formulas relating force and acceleration are Newton's dynamical equations F_i,k = m_ia_i,k, I = 1, 2, 3, …, n, where m_i is the mass of the ith particle.

Extensive and varied computations on a CRAY X - MP/24 are described and discussed, and comparisons are made with the results of others.     

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.     

Subspace selection algorithms to be used with the nonlinear projection methods in solving systems of nonlinear equations

The nonlinear projection methods are minimization procedures for solving systems of nonlinear equations. They permit reevaluation of n_k, 1 ≤ n_k ≤ n, components of the approximate solution vector at each iteration step where n is the dimension of the system. At iteration step k, the reduction in the norm of the residue vector depends upon the n_k components which are reevaluated. These n_k components are obtained by solving a linear system.

We present two algorithms for determining the components to be modified at each iteration of the nonlinear projection method and compare the use of these algorithms to Newton's method. The computational examples demonstrate that Newton's method, which reevaluates all components of the approximate solution vector at each iteration, can be accelerated by using the projection techniques.     

A new routing algorithm for multirate rearrangeable Clos networks

In this paper, we study the problem of finding routing algorithms on the multirate rearrangeable Clos networks which use as few number of middle-stage switches as possible. We propose a new routing algorithm called the “grouping algorithm”. This is a simple algorithm which uses fewer middle-stage switches than all known strategies, given that the number of input-stage switches and output-stage switches are relatively small compared to the size of input and output switches. In particular, the grouping algorithm implies that m = 2n+(n−1)/2^k is a sufficient number of middle-stage switches for the symmetric three-stage Clos network C(n,m,r) to be multirate rearrangeable, where k is any positive integer and rn/(2^k−1).     

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.     

Contrast-optimal k out of n secret sharing schemes in visual cryptography

Visual cryptography and (k,n)-visual secret sharing schemes were introduced by Naor and Shamir (Advances in Cryptology — Eurocrypt 94, Springer, Berlin, 1995, pp. 1–12). A sender wishing to transmit a secret message distributes n transparencies amongst n recipients, where the transparencies contain seemingly random pictures. A (k,n)-scheme achieves the following situation: If any k recipients stack their transparencies together, then a secret message is revealed visually. On the other hand, if only k−1 recipients stack their transparencies, or analyze them by any other means, they are not able to obtain any information about the secret message. The important parameters of a scheme are its contrast, i.e., the clarity with which the message becomes visible, and the number of subpixels needed to encode one pixel of the original picture. Naor and Shamir constructed (k,k)-schemes with contrast 2^−(k−1). By an intricate result from Linial (Combinatorica 10 (1990) 349–365), they were also able to prove the optimality of these schemes. They also proved that for all fixed kn, there are (k,n)-schemes with contrast . For k=2,3,4 the contrast is approximately and . In this paper, we show that by solving a simple linear program, one is able to compute exactly the best contrast achievable in any (k,n)-scheme. The solution of the linear program also provides a representation of a corresponding scheme. For small k as well as for k=n, we are able to analytically solve the linear program. For k=2,3,4, we obtain that the optimal contrast is at least and . For k=n, we obtain a very simple proof of the optimality of Naor's and Shamir's (k,k)-schemes. In the case k=2, we are able to use a different approach via coding theory which allows us to prove an optimal tradeoff between the contrast and the number of subpixels.     

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.     

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.     

Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues

In many calculations, spectral discretization in space is coupled with a standard ordinary differential equation formula in time. To analyze the stability of such a combination, one would like simply to test whether the eigenvalues of the spatial discretization operator (appropriately scaled by the time step k) lie in the stability region for the o.d.e. formula, but it is well known that this kind of analysis is in general invalid. In the present paper we rehabilitate the use of stability regions by proving that a discrete linear multistep ‘method of lines’ approximation to a partial differential equation is Lax-stable, within a small algebraic factor, if and only if all of the -pseudo-eigenvalues of the spatial discretization operator lie within O() of the stability region as → 0. An -pseudo-eigenvalue of a matrix A is any number that is an eigenvalue of some matrix A + E with E ; our arguments make use of resolvents and are closely related to the Kreiss matrix theorem. As an application of our general result, we show that an explicit N-point Chebyshev collocation approximation of u_t = −xu_x on [−1, 1] is Lax-stable if and only if the time step satisfies k = O(N⁻²), although eigenvalue analysis would suggest a much weaker restriction of the form k CN⁻¹.     

Simultaneous H control of a finite collection of linear plants with a single nonlinear digital controller

This paper considers the problem of simultaneous H^∞ control of a finite collection of linear time-invariant systems via a nonlinear digital output feedback controller. The main result is given in terms of the existence of suitable solutions to Riccati algebraic equations and a dynamic programming equation. Our main result shows that if the simultaneous H^∞ control problem for k linear time-invariant plants of orders n₁,n₂,…,n_k can be solved, then this problem can be solved via a nonlinear time-invariant controller of order nn₁+n₂++n_k.     

Embedding a complete binary tree into the supercube

Consider a complete binary tree with 2ⁿ − 1 nodes and a supercube with the same number of nodes. We present a new embedding method to map the complete binary tree into the supercube with dilation 1. Our simple mapping method is quite competitive with the previous result.     

Algorithms for four variants of the exact satisfiability problem

We present four polynomial space and exponential time algorithms for variants of the E S problem. First, an O(1.1120ⁿ) (where n is the number of variables) time algorithm for the NP-complete decision problem of E 3-S , and then an O(1.1907ⁿ) time algorithm for the general decision problem of E S . The best previous algorithms run in O(1.1193ⁿ) and O(1.2299ⁿ) time, respectively. For the #P-complete problem of counting the number of models for E 3-S we present an O(1.1487ⁿ) time algorithm. We also present an O(1.2190ⁿ) time algorithm for the general problem of counting the number of models for E S ; presenting a simple reduction, we show how this algorithm can be used for computing the permanent of a 0/1 matrix.     

On the error in Padé approximations for functions defined by Stieltjes integrals

Consdier I(z) = ∫^b_a w(t)f(t, z) dt, f(t, z) = (1 + t/z)⁻¹. It is known that generalized Gaussian quadrature of I(z) leads to approximations which occupy the (n, n + r − 1) positions of the Padé matrix table for I(z). Here r is a positive integer or zero. In a previous paper the author developed a series representation for the error in Gaussian quadrature. This approach is now used to study the error in the Padé approximations noted. Three important examples are treated. Two of the examples are generalized to the case where f(t, z) = (1 + t/z)^−v.     

Adaptive wormhole routing in k-ary n-cubes

Distributed memory multiprocessor (DMMP) systems have gained much attention because their performance can be easily scaled up by increasing the number of processor-memory modules. The k-ary n-cube is the most popular interconnection network topology currently used in DMMPs. Wormhole routing is one of the most promising switching technology and has been used in many new generation multicomputers. Wormhole routing makes the communication latency insensitive to the network diameter and reduces the size of the channel buffer of each router. The concept of virtual channels and virtual networks are widely invented for deadlock-free design. A fully adaptive wormhole routing method for k-ary n-cubes has been proposed by Linder in 1991 [10]. Unfortunately, the need of 2^{n − 1} virtual networks makes it unreasonable. In this paper, we propose a virtual network system to support an adaptive, minimal and deadlock free routing in k-ary n-cubes. It uses only four virtual networks but can get a higher degree of adaptability and higher traffic capacity. Simulation results are presented to verify the performance.     

Optimal and nearly optimal algorithms for approximating polynomial zeros

We substantially improve the known algorithms for approximating all the complex zeros of an n^th degree polynomial p(x). Our new algorithms save both Boolean and arithmetic sequential time, versus the previous best algorithms of Schönhage [1], Pan [2], and Neff and Reif [3]. In parallel (NC) implementation, we dramatically decrease the number of processors, versus the parallel algorithm of Neff [4], which was the only NC algorithm known for this problem so far. Specifically, under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p(x) to the unit disc x : |x| ≤ 1, our algorithms (which promise to be practically effective) approximate all the zeros of p(x) within the absolute error bound 2^−b, by using order of n arithmetic operations and order of (b + n)n² Boolean (bitwise) operations (in both cases up to within polylogarithmic factors). The algorithms allow their optimal (work preserving) NC parallelization, so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or (b + n)n² Boolean processors. All the cited bounds on the computational complexity are within polylogarithmic factors from the optimum (in terms of n and b) under both arithmetic and Boolean models of computation (in the Boolean case, under the additional (realistic) assumption that n = O(b)).