Minimum Number Of Nodes Research Articles

Hardness of identifying the minimum ordered binary decision diagram

An ordered binary decision diagram (OBDD) is a graph representation of a Boolean function. We consider minimum OBDD identification problems: given positive and negative examples of a Boolean function, identify the OBDD with minimum number of nodes (or with minimum width) that is consistent with all the examples. We prove in this paper that the problems are NP-complete. The result implies that f( n)-width OBDD and f( n)-node OBDD are not learnable for some fixed f( n) under the PAC-learning model unless NP = RP. We also show that the problems are still NP-hard even if we restrict the functions to monotone functions.

Heuristic for setting up a stack of WDM rings with wavelength reuse

A heuristic methodology is proposed for the setting up of a stack of wavelength-division-multiplexing (WDM) rings with wavelength reuse when the design traffic exceeds the capacity of a single ring and no wavelength conversion is employed. A ring stack consists of an overlay of rings routed over the same physical route, and it can be setup and dimensioned in a myriad of ways. The design traffic comprises of a set of bidirectional lightpaths or wavelength connections. There exists a tradeoff between the number of nodes and the number of rings required to carry this traffic, and it is demonstrated that both cannot be minimized simultaneously. For certain traffic patterns, we identify stacks requiring the minimum number of nodes or WADM's, which is desirable from a cost point of view, and stacks requiring the minimum number of rings. An algorithm is presented that manipulates the tradeoff phenomenon to produce a spectrum of designs with deterministic composition. We finally conclude by identifying factors that may influence the choice of design.

Minimizing the complexity of an activity network

When an activity-on-arc (AoA) project network is to be constructed, one typically seeks to minimize the number of dummy arcs. Recent investigations have shown, however, that the computational effort of many network-oriented project management techniques depends strongly on the so-called complexity index (CI) of a network. We show other justifications for minimizing the Cl rather than the number of dummy arcs. We also present a polynomial time algorithm for constructing an AoA network with the minimum Cl in the class of all AoA networks having the minimum number of nodes.

Edge connectivity between nodes and node-subsets

Let G = (V, E) be a graph where V and E are a set of nodes and a set of edges, respectively. Let X = {V1, V2, …, Vp}, Vi ⊆ V be a family of node-subsets. Each node-subset Vi is called an area, and a pair of G and X is called an area graph. A node ν ∈ V and an area Vi ∈ X are called k-NA (node-to-area)-connected if the minimum size of a cut separating ν and Vi is at least k. We say that an area graph (G, X) is k-NA-edge-connected when each ν ∈ V and Vi ∈ X are k-NA-edge-connected. This paper gives a necessary and sufficient condition for a given (G, X) to be k-NA-edge-connected: (G, X) is k-NA-edge-connected iff, for all positive integers h ≤ k, every h-edge-connected component of G includes at least one node from each area or has at least k edges between the component and the rest of the nodes. This paper also studied the Minimum Area Augmentation Problem, i.e., the problem of determining whether or not a given area graph (G, X) is k-NA-edge-connected and of choosing the minimum number of nodes to be included in appropriate areas to make the area graph k-NA-edge-connected (if (G, X) is not k-NA-edge-connected). This problem can be regarded as one of the location problems, which arises from allocating service-nodes on multimedia networks. We propose an O(|E| + |V|2 + L′ + min {|E|, k|V|} min {k|V|, k + |V|2}) time algorithm for solving this problem, where L′ is a space required to represent output areas. For a fixed k, this algorithm also runs in linear time when the h-edge-connected components of G are available for all h = 1, 2, …, k. © 1998 John Wiley & Sons, Inc. Networks 31: 157–163, 1998

Delay-period activity is affected by visual cues presented outside the memory field

In the wake of rapid development of multiprocessor systems, fault tolerance of processors plays an even more vital role in measuring the reliability of a multiprocessor system. The g-extra connectivity of a multiprocessor system modeled by graph G, denoted by κo(g)(G), is the minimum number of nodes whose deletion will disconnect the network and every remaining component has more than g vertices. The g-extra conditional diagnosability of multiprocessor system G, denoted by tg˜(G), is the maximum number of faulty vertices that the system can guarantee to identify under the condition that every fault-free component contains at least g+1 vertices. Only a few achievements have been established on κo(g)(G) for some special graphs with small g. In this paper, we first determine that the g-extra connectivity of (n,k)-star network Sn,k is κo(g)(Sn,k)=n+g(k−2)−1 for 2≤k≤n−1 and 0≤g≤n−k and then show that the g-extra conditional diagnosability of Sn,k under the PMC model (n≥4, 2≤k≤n−1 and 1≤g≤n−k) and the MM* model (n≥6, 3≤k≤n−3 and 1≤g≤min⁡{n−k+14,k−2}) is tg˜(Sn,k)=n+g(k−1)−1, respectively. Meanwhile, we also show that Sn,k is (n+m(k−2)−1)/m-diagnosable under the pessimistic diagnostic strategy. As by-products, we derive the g-extra conditional diagnosability of n-dimensional star graph Sn (n≥4, g=1) and n-dimensional alternating group network ANn (n≥4, 1≤g≤2) under the PMC model.

Biorthogonal decompositions of the Radon transform by multivariate interpolation

The concept of biorthogonal and singular value decompositions is a valuable tool in the examination of ill-posed inverse problems such as the inversion of the Radon transform. By application of the theory of multivariate interpolation, e. g. the set of Lagrange polynomials with respect to the space of homogeneous spherical polynomials, we determine new biorthogonal decompositions of the Radon transform. We consider the case of functions with support in the unit ball and the case of functions with support ℝr. In both cases we assume that the functions are square integrable with respect to some weight functions. In the important special case of square integrable functions with respect to the unit ball the structure of the biorthogonal decompositions is easier in comparison with the known singular and biorthogonal decompositions. Especially the calculation of the unknown expansion coefficients can be done by using arbitrary fundamental systems (μ-resolving data set in terms of tomography with a minimum number of nodes) and simplifies essentially. The decompositions are based on a system of zonal (ridge) Gegenbauer (ultraspherical) polynomials which are used in the theory of the Radon transform and in the field of numerical algorithms for the inversion of the transform.

A class of non‐canonic, driving‐point immittance realizations of passive, common‐ground, transformerless, two‐element‐kind RLC networks

AbstractA new class of non‐canonic, driving‐point immittance realizations of passive, common‐ground, transformerless, two‐element‐kind RLC networks with minimum number of nodes is presented. Six new algorithms of realization are developed by application of canonic, orthogonal matrix transformations with reduced number of degrees of freedom. the number of applied RLC elements of one kind can be minimized and of the other kind reduced, while their parameter values are calculated directly in a linear way.

The geodetic number of a graph

We introduce a new graph theoretic parameter, the geodetic number of a connected graph G = ( V, E) as follows. The geodetic closure of a node set S⊂ V is the set of all nodes uϵ V which lie in some geodesic in G joining two nodes x and y of S. The geodetic number of a connected graph G, denoted by g( G), is the minimum number of nodes on a set S ∗ whose geodetic closure is all of V. We show that the determination of g( G) is an NP-hard problem and its decision problem is NP-complete, and present an algorithm for finding g( G).

On the minimum dummy-arc problem

A precedence relation can be represented non-uniquely by an activity on arc (AoA) directed acyclic graph (dag). This paper deals with the NP-hard problem of constructing an AoA dag having the minimum number of arcs among those that have the minimum number of nodes. We show how this problem can be reduced in polynomial time to the set-cover problem so that the known methods of solving the set-cover problem can be applied. Several special cases that lead to easy set-cover problems are discussed. Return+ I'm' ft~luiif,n Jt /Aˆ,,' t'lit'ni'r ps'iit it,, rt~pr~~~~~~t~~~~ .Jut,,' in^tiiii'is- nun-un~qtit 1 par u,t ~idplii'dirfi tm \iln/uf\rJuium l~rfutcwnrur, i iA\.-Ii f'fturiii If tr~ilt'dn prt.hI>ittr NP.JtIfti.ile la ~~,ns~ru~rt~,,~ d'm XJU .ld mwtl le numhrt m,nvn,d d.~r, purnt, wua uw mt It, ,~,mI,r~~

An information theoretic design and training algorithm for neural networks

An algorithmic approach to designing feedforward neural networks for pattern classification is presented. The technique computes a single hidden layer of nodes by adding one node at a time until the desired classification has been achieved. At each iteration, a node that maximizes an information theoretic measure is selected from a collection of candidates. The methodology is heuristic in nature, intending to solve the NP-hard problem of constructing a neural network with a minimum number of nodes. Two strategies for computing a collection of candidate nodes are presented and some experimental results obtained by using the strategies are reported.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Minimum graphs of specified diameter, connectivity and valence. II

For positive integers d , c and v , the minimum number of nodes for a c -connected v -regular graph of diameter d is determined.

Cubature formulae for the approximate integration of functions of three variables

Cubature formulae of high trigonometric accuracy are constructed for functions of three variables with a number of integration nodes close to the minimum. Formulae are constructed with the minimum number of nodes that accurately integrate trigonometric polynomials in n variables of not higher than the third order.

Piecewise Chebyshev approximation of experimental data

The problem of approximating experimental data with an a priori fixed error by means of piecewise polynomial functions (with respect to a Chebyshev system) with free conjunction nodes is considered. The properties of the approximation with the minimum number of nodes are studied. A computational scheme for constructing the optimal approximation is proposed.

Simplified finite element mesh for dynamic analysis of a nuclear reactor building

AbstractThe design of nuclear facilities requires dynamic analysis to evaluate their performance during an earthquake. In modal analysis, a method recommended for dynamic analysis by the Nuclear Regulatory Commission, the natural frequencies and mode shapes of the structure are commonly calculated using a stiffness matrix computed using the finite element method.A primary goal of this study was to design a finite element mesh, capable of modelling a reactor structure, with a minimum number of nodes and elements. This is necessary since analysis using a mesh fine enough to include all of the structural elements is not feasible because of exorbitant computer time and storage requirements. It was found that for lateral motion, a representative reactor structure could be accurately modelled using a simplified mesh with only one isoparametric quadratic quadrilateral element per floor. In this mesh essentially infinite springs must be placed between the masses at each floor level to prevent internal movements of the lumped masses within the quadratic quadrilateral elements.It was also found that a structure with rigid heavy bottom floors and a light top floor can be subjected to large displacements in the top floor during earthquakes. It was concluded that the properties of that light top floor (especially shear area) controls the value of the fundamental frequency of the building.

Classification and enumeration of minimum ( d, 3, 3)-graphs for odd d

Abstract : A (d,c,v)-graph is a c-connected graph of diameter d in which each node is of valence v. A minimum (d,c,v)-graph is one with the minimum number of nodes. Each minimum (d,c,v)-graph corresponds to an efficient way of arranging the stations of a communication network so that if any c-1 stations are incapacitated, the rest of the network is still connected, and so that in case of breakdown or other difficulty each station can rely for assistance on precisely v others. Here the minimum (d,3,3)-graphs are classified and counted for odd d. (Author)

The node-deletion problem for hereditary properties is NP-complete

We consider the family of graph problems called node-deletion problems, defined as follows; For a fixed graph property Π, what is the minimum number of nodes which must be deleted from a given graph so that the resulting subgraph satisfies Π? We show that if Π is nontrivial and hereditary on induced subgraphs, then the node-deletion problem for Π is NP-complete for both undirected and directed graphs.

Regular and canonical colorings

A general framework for coloring problems is described; the concept of regular coloring is introduced; it simply means that one specifies in each edge the maximum and the minimum number of nodes which may have the same color. For several types of regular colorings, one defines canonical colorings where colors form an ordered set and where one always tries to use first the “smallest” colors. It is shown that for some classes of multigraphs including bipartite multigraphs, regular edge colorings corresponding to maximal color feasible sequences are canonical.

Construction of Algebraic Cubature Rules Using Polynomial Ideal Theory

Cubature formulae of fixed degree using the minimum number of nodes, the common zeros of a set of polynomials, are constructed by a number of techniques based on the theory of polynomial ideals. Examples demonstrate that known lower bounds to the number of nodes can be attained though usually these bounds are too severe. One example is shown to give rise to an interlacing family of rules, a two dimensional analogue of the Clenshaw–Curtis quadrature. An effective numerical procedure is also given for finding all the common zeros of a set of polynomials.

An Algorithm for Constructing Optimal Binary Decision Trees

We consider the problem of optimally partitioning an n-dimensional lattice, L = L, X ... X LN, where Lj is a one-dimensional lattice with kj elements, by means of a binary tree into specified (labeled) subsets of L. Such lattices arise from problems in pattern classification, in nonlinear regression, in defining logical equations, and a number of related areas. When viewed as the partitioning of a vector space, each point in the lattice corresponds to a subregion of the space which is relatively homogeneous with respect to classification or range of a dependent variable. Optimality is defined in terms of a general cost function which includes the following: 1) min-max path length (i. e., minimize the maximum number of nodes traversed in making a decision); 2) minimum number of nodes in the tree; and 3) expected path length. It is shown that an optimal tree can be recursively constructed through the application of invariant imbedding (dynamic programming). An algorithm is detailed which embodies this recursive approach. The algorithm allows the assignment of a don't care label to elements of L.

Kubaturformeln mit minimaler Knotenzahl

For two-fold integrals, a lower bound is derived for the number of nodes in a cubature formula of degree 2s-1. There is a formula of degree 2s-1 for which the number of nodes attains this lower bound, iff a certain condition is fullfilled. By this condition, all formulas of degree 2s-1 with that minimal number of nodes can be constructed. Examples and a generalization to then-dimensional case are given.