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)).     

Adaptive h-p procedures for high accuracy finite element analysis of two-dimensional linear elastic problems

Two adaptive procedures of the h-p version for a high accuracy finite element analysis of two-dimensional elastic problems are studied. These are based on a strategy of first using an h-version to predict a nearly optimal mesh up to a certain accuracy and then following up with a p-version to achieve a higher accuracy. The h-version, using linear triangular elements, is developed by coupling a code, ADMESH, with an error estimation in energy norm. Following the h-version, two alternative procedures of non-uniform p-refinements are then performed. In procedure I, p-refinements are made in one step by selectively adding hierarchical shape functions of order p = 2 and 3 based on the estimated error in energy norm. In procedure II, p-refinements are made in a step-by-step way by which, in the kth step of the p-refinements, hierarchical shape functions of order p = k + 1 are selectively introduced. In the first step of p-refinements of procedure II, hierarchical variables are selected by means of the estimated errors in energy norm, whereas in the later steps, they are selected with a guidance of an error estimate which evaluates the local average error of stresses. The performances of both procedures and the rate of convergence are studied in numerical examples. Numerical tests for the error estimates being used are also made. Obtained results indicate that both procedures can achieve a high accuracy (say, error below 5% measured in energy norm) in an exponential rate of convergence.     

Two minimum spanning forest algorithms on fixed-size hypercube computers

Two parallel algorithms for finding minimum spanning forest (MSF) of a weighted undirected graph on hypercube computers, consisting of a fixed number of processors, are presented. One algorithm is suited for sparse graphs, the other for dense graphs. Our design strategy is based on successive elimination of non-MSF edges. The input graph is partitioned equally among different processors, which then repeatedly eliminate non-MSF edges and merge results to gradually construct the desired MSF of the entire graph. Low communication overhead is achieved by restricting the message-flow to between the neighboring processors in the hypercube topology. The correctness of our approach is due to a theorem which states that with total-ordered edges, if an edge of an arbitrary subgraph does not belong to its MSF, then it does not belong to the MSF of the entire graph. For a graph of n vertices and m edges, our first algorithm finds an MSF in O(m log m)/p) time using p processors for p ≤ (mlog m)/n(1+log(m/n)). The second algorithm, efficient for dense graphs, requires O(n²/p) time for p≤n/log n.     

Optimal bounds for the approximation of boolean functions and some applications

This paper presents an optimal bound on the Shannon function L(n,m,) that gives the worstcase circuit-size complexity to approximate, within an approximation degree at least , partial boolean functions having n inputs and domain size m. That is

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. Our bound applies to any partial boolean function and any approximation degree, and thus completes the study of boolean function approximation introduced by Pippenger (1977).

Our results give an upper bound for the hardness function h(ƒ), introduced by Nisan and Wigderson (1994), which denotes the minimum value l for which there exists a circuit of size at most l that approximates a boolean function ƒ with degree at least 1/l. Indeed, if H(n) denotes the maximum hardness value achieved by boolean functions with n inputs, we prove that for almost every nH(n)n/3 + n² + O(1). The exponent n/3 in the above inequality implies that no family of boolean functions exists which has ‘full’ hardness. This fact establishes connections with Allender and Strauss' (1994) work that explores the structure of BPP.

Finally, we show that for almost every n and for almost every boolean function ƒ of n inputs we have h(ƒ)2^{n/3−2 log n}. The contribution in the proof of the upper bound for L(n, m, ) can be viewed as a set of technical results that globally show how boolean linear operators are ‘well’ distributed on the class of 4-regular domains. This property is then applied to approximate partial boolean functions on general domains using a suitable composition of boolean linear operators.     

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.     

Parallel algorithms for planar dominance counting

Given n points in the plane the planar dominance counting problem is to determine for each point the number of points dominated by it. Point p is said to dominate point q if x(q)x(p) and y(q)y(p), when x(p) and y(p) are the x− and y-coordinate of p, respectively. We present two CREW PRAM parallel algorithms for the problem, one running in O(log n loglog n) time and and the other in O(lognloglogn/logloglogn) time both using O(n) processors. Some applicationsare also given.     

Blossoms are polar forms

Consider the functions H(t):=t² and h(u,v):=uv. The identity H(t)=h(t,t) shows that h is the restriction of h to the diagonal u=v in the uv-plane. Yet, in many ways, a bilinear function like h is simpler than a homogeneous quadratic function like H. More generally, if F(t) is some n-ic polynomial function, it is often helpful to study the polar form of F, which is the unique symmetric, multiaffine function ƒ(u_1,…u_n) satisfying the identity F(t)=f(t,…,t). The mathematical theory underlying splines is one area where polar forms can be particularly helpful, because two pieces F and G of an n-ic spline meet at a point r with C^k parametric continuity if and only if their polar forms ƒ and g agree on all sequences of n arguments that contain at least n-k copies of r.

The polar approach to the theory of splines emerged in rather different guises in three independent research efforts: Paul de Faget Casteljau called it ‘shapes through poles’; Carl de Boor called it ‘B-splines without divided differences’; and Lyle Ramshaw called it ‘blossoming’. This paper reviews the work of de Casteljau, de Boor, and Ramshaw in an attempt to clarify the basic principles that underly the polar approach. It also proposes a consistent system of nomenclature as a possible standard.     

ANTS: Agents on Networks, Trees, and Subgraphs

Efficient exploration of large networks is a central issue in data mining and network maintenance applications. In most existing work there is a distinction between the active ‘searcher’ which both executes the algorithm and holds the memory and the passive ‘searched graph’ over which the searcher has no control at all. Large dynamic networks like the Internet, where the nodes are powerful computers and the links have narrow bandwidth and are heavily-loaded, call for a different paradigm, in which a noncentralized group of one or more lightweight autonomous agents traverse the network in a completely distributed and parallelizable way. Potential advantages of such a paradigm would be fault tolerance against network and agent failures, and reduced load on the busy nodes due to the small amount of memory and computing resources required by the agent in each node. Algorithms for network covering based on this paradigm could be used in today’s Internet as a support for data mining and network control algorithms. Recently, a vertex ant walk ( ) method has been suggested [I.A. Wagner, M. Lindenbaum, A.M. Bruckstein, Ann. Math. Artificial Intelligence 24 (1998) 211–223] for searching an undirected, connected graph by an a(ge)nt that walks along the edges of the graph, occasionally leaving ‘pheromone’ traces at nodes, and using those traces to guide its exploration. It was shown there that the ant can cover a static graph within time nd, where n is the number of vertices and d the diameter of the graph. In this work we further investigate the performance of the method on dynamic graphs, where edges may appear or disappear during the search process. In particular we prove that (a) if a certain spanning subgraph S is stable during the period of covering, then the method is guaranteed to cover the graph within time nd_s, where d_s is the diameter of S, and (b) if a failure occurs on each edge with probability p, then the expected cover time is bounded from above by nd((logΔ/log(1/p))+((1+p)/(1−p))), where Δ is the maximum vertex degree in the graph. We also show that (c) if G is a static tree then it is covered within time 2n.     

An h-p-n adaptive finite element scheme for shell problems

We propose an adaptive finite element algorithm for shells which, in addition to the usual h-p adaption also shows adaptivity with respect to the order n of the dimension reduction. The idea of the algorithm is to adaptively capture and resolve the various length scales that may occur in shells. The algorithm presented in the paper is limited to axisymmetric problems, which reduces the h-p part of the problem to one dimension only. The performance of the algorithm is tested in some example cases where the shell is cylindrical. For comparison, we test the algorithm also when n is limited so that n k, where K = 1, 2 or 3. Choosing k = 2 essentially corresponds to the classical shell models.     

The p-valued-input, q-valued-output threshold logic and the (p, q)-polypheck-like function

We have previously proposed a concept of p-valued-input, q-valued-output threshold logic, namely (p, q)-threshold logic, where 2 qp, 3p, and suggested that p-valued logical networks with costs as low as those of 2-valued logical networks could be obtained, by using the (p, q)-threshold elements with small values of q. In this paper, we describe (1) the condition under which there is a 2-place (p, q)-adic function such that the output-closed set F, generated only from , is (p, q)-logically complete, and (2) the fact that any n-place(p, q)-adic function can be realized using at most O(n) elements in the above F.     

The symmetric all-furthest- neighbor problem

Given a set of n points on the plane, a symmetric furthest-neighbor (SFN) pair of points p, q is one such that both p and q are furthest from each other among the points in . A pair of points is antipodal if it admits parallel lines of support. In this paper it is shown that a SFN pair of is both a set of extreme points of and an antipodal pair of . It is also shown that an asymmetric furthest-neighbor (ASFN) pair is not necessarily antipodal. Furthermore, if is such that no two distances are equal, it is shown that as many as, and no more than, n/2 pairs of points are SFN pairs. A polygon is unimodal if for each vertex p_k, k = 1,…,n the distance function defined by the euclidean distance between p_k and the remaining vertices (traversed in order) contains only one local maximum. The fastest existing algorithms for computing all the ASFN or SFN pairs of either a set of points, a simple polygon, or a convex polygon, require 0(n log n) running time. It is shown that the above results lead to an 0(n) algorithm for computing all the SFN pairs of vertices of a unimodal polygon.     

An optimal algorithm for Gaussian elimination of band matrices on an MIMD computer

This paper describes a parallel algorithm for the LU decomposition of band matrices using Gaussian elimination. The matrix dimension is n × n with 2r−1 diagonals. In the case when 1 r 2 p an optimal number of the processors, , is determined according to the equation . When 2 p r n a number of processors, p, statged by Veldhorst is adopted (see [7]). For band matrix with 2r-1 diagonals (1 r 2p) the task scheduling procedure with the aim to obtain maximal parallelism in system operation, i.e. good load balancing, is defined. The architecture of the system is of MIMD type. The connection between the processors is realised via a common bus. Communication and synchronization is performed by message passing technique.     

Learning counting functions with queries

We investigate the problem of learning disjunctions of counting functions, which are general cases of parity and modulo functions, with equivalence and membership queries. We prove that, for any prime number p, the class of disjunctions of integer-weighted counting functions with modulus p over the domain Z_qⁿ (or Zⁿ) for any given integer q 2 is polynomial time learnable using at most n + 1 equivalence queries, where the hypotheses issued by the learner are disjunctions of at most n counting functions with weights from Z_p. In general, a counting function may have a composite modulus. We prove that, for any given integer q 2, over the domain Z₂ⁿ, the class of read-once disjunctions of Boolean-weighted counting functions with modulus q is polynomial-time learnable with only one equivalence query and O(n^q) membership queries.     

Recent experiences with error estimation and adaptivity, Part II: Error estimation for h-adaptive approximations on grids of triangles and quadrilaterals

In Part I, residual and flux projection error estimators for finite element approximations of scalar elliptic problems were reviewed; numerical studies on the performance of these estimators were presented for finite element approximations of the solution of Poisson's equation on uniform grids of hierarchic triangles of order p (1 p 7). Here further numerical experiments are given which also include error estimators for the vector-valued problem of plane elastostatics and implementations for h-adaptive grids of triangles and quadrilaterals which are constructed using an algorithm of equidistribution of error coupled with h-refinement or h-remeshing schemes. A detailed numerical study of several flux-projectors for h-adaptive grids of bilinear and biquadratic quadrilaterals is conducted; a flux equilibration iteration, which may be employed in some cases to improve flux projection estimates, is also included. FAor the case of grids of quadrilaterals, several versions of the element residual estimators, which differ by the approximate flux employed for the calculation of the boundary integral term in the definition of the local problems, are compared. The numerical experiments confirm the good overall performance of residual estimates and indicate that flux projection estimates, which are now operational in several commercial codes, may be divergent when they are employed to estimate the error in even order h-adaptive approximations.     

(h,k)-Arbiters for h-out-of-k mutual exclusion problem

h-Out-of-k mutual exclusion is a generalization of the 1-mutual exclusion problem, where there are k units of shared resources and each process requests h (1hk) units at the same time. Though k-arbiter has been shown to be a quorum-based solution to this problem, quorums in k-arbiter are much larger than those in the 1-coterie for 1-mutual exclusion. Thus, the algorithm based on k-arbiter needs many messages. This paper introduces the new notion that each request uses different quorums depending on the number of units of its request. Based on the notion, this paper defines two (h,k)-arbiters for h-out-of-k mutual exclusion: a uniform (h,k)-arbiter and a (k+1)-cube (h,k)-arbiter. The quorums in each (h,k)-arbiter are not larger than the ones in the corresponding k-arbiter; consequently, it is more efficient to use (h,k)-arbiters than the k-arbiters. A uniform (h,k)-arbiter is a generalization of the majority coterie for 1-mutual exclusion. A (k+1)-cube (h,k)-arbiter is a generalization of square grid coterie for 1-mutual exclusion.     

Optimal simulation of tree arrays by linear arrays

We show that an n-node complete binary tree can be embedded into an n-node linear array such that the maximum edge length is n/log n, and the distribution of the tree nodes is regular in the sense that if each node in the tree array is a finite-state machine and each edge is a communication channel, then the message exchanges (i.e., communication) among the tree nodes can be easily simulated by the linear array which has also finite-state machines for its nodes. The embedding is optimal with respect to the maximum edge length. The motivation for this is the proof of the following result: every cellular (or (log n)-depth iterative) tree array running in T(n) time can be simulated by a linear cellular (or iterative) array in nT(n)/log n time.     

New bounds for multi-label interval routing

Interval routing (IR) is a space-efficient routing method for computer networks. For longest routing path analysis, researchers have focused on lower bounds for many years. For any n-node graph G of diameter D, there exists an upper bound of 2D for IR using one or more labels, and an upper bound of for IR using or more labels. We present two upper bounds in the first part of the paper. We show that for every integer i>0, every n-node graph of diameter D has a k-dominating set of size for . This result implies a new upper bound of for IR using or more labels, where i is any positive integer constant. We apply the result by Kutten and Peleg [8] to achieve an upper bound of (1+)D for IR using O(n/D) or more labels, where is any constant in (0,1). The second part of the paper offers some lower bounds for planar graphs. For any M-label interval routing scheme (M-IRS), where , we derive a lower bound of [(2M+1)/(2M)]D−1 on the longest path for , and a lower bound of [(2(1+δ)M+1)/(2(1+δ)M)]D, where δ(0,1], for . The latter result implies a lower bound of on the number of labels needed to achieve optimality.     

The p- and h-p versions of the finite element method in solid mechanics

Applications of p- and h-p extension procedures in solid mechanics are discussed. Error estimation and quality control procedures made possible by p-extensions are described and illustrated by examples.     

Rational functions and pseudo-Newton algorithms

Newton's method for calculating the zeros of a polynomial almost always has periodic orbits or even stable periodic orbits. We identify a rational algorithm that converges globally to a root of the polynomial. We then proceed to examine the topology of the class of all rational functions T that have the property that for a given polynomial p(x), T(x) = x iff p(x) = 0. The topology of this set of rational functions is more complicated than the topology of Rat(n).     

The limitedness problem on distance automata: Hashiguchi''s method revisited

Hashiguchi has studied the limitedness problem of distance automata (DA) in a series of paper [(J. Comput System Sci. 24 (1982) 233; Theoret. Comput. Sci. 72 (1990) 27; Theoret. Comput. Sci. 233 (2000) 19)]. The distance of a DA can be limited or unbounded. Given that the distance of a DA is limited, Hashiguchi has proved in Hashiguchi (2000) that the distance of the automaton is bounded by 2^{4n3+nlg(n+2)+n}, where n is the number of states. In this paper, we study again Hashiguchi's solution to the limitedness problem. We have made a number of simplification and improvement on Hashiguchi's method. We are able to improve the upper bound to 2^{3n3+nlgn+n−1}.