Fast algorithms for determining the linear complexities of sequences over GF(pm) with the period 3n

In this paper, for the the primes p such that 3 is a divisor of p − 1, we prove a result which reduces the computation of the linear complexity of a sequence over GF(p ^m) (any positive integer m) with the period 3n (n and p ^m − 1 are coprime) to the computation of the linear complexities of three sequences with the period n. Combined with some known algorithms such as generalized Games-Chan algorithm, Berlekamp-Massey algorithm and Xiao-Wei-Lam-Imamura algorithm, we can determine the linear complexity of any sequence over GF(p ^m) with the period 3n (n and p ^m − 1 are coprime) more efficiently.     

Fast Algorithms for Centro-Symmetric and Centro-Skewsymmetric Toeplitz-Plus-Hankel Matrices

Subject of the paper are centro-symmetric and centro-skewsymmetric Toeplitz-plus-Hankel matrices with the property that all central submatrices are nonsingular. Fast algorithms are presented that solve an n×n system of equations with O(n ²) operations in sequential and O(n) operations in parallel processing and compute the ZW-factorization with the same computational complexity. These algorithms are more efficient than existing algorithms because they fully exploit the symmetry properties of the matrices.     

The power of adaptive algorithms for functions with singularities

This is an overview of recent results on complexity and optimality of adaptive algorithms for integrating and approximating scalar piecewise r-smooth functions with unknown singular points. We provide adaptive algorithms that use at most n function samples and have the worst case errors proportional to n^−r for functions with at most one unknown singularity. This is a tremendous improvement over nonadaptive algorithms whose worst case errors are at best proportional to n⁻¹ for integration and n^−1/p for the L^p approximation problem. For functions with multiple singular points the adaptive algorithms cease to dominate the nonadaptive ones in the worst case setting. Fortunately, they regain their superiority in the asymptotic setting. Indeed, they yield convergence of order n^−r for piecewise r-smooth functions with an arbitrary (unknown but finite) number of singularities. None of these results hold for the L^∞ approximation. However, they hold for the Skorohodmetric, which we argue to be more appropriate than L^∞ for dealing with discontinuous functions. Numerical test results and possible extensions are also discussed.     

An improved algorithm for selecting <Emphasis Type="Italic">p</Emphasis> items with uncertain returns according to the minmax-regret criterion

We consider the problem of selecting a subset of p investments of maximum total return out of a set of n available investments with uncertain returns, where uncertainty is represented by interval estimates for the returns, and the minmax regret objective is used. We develop an algorithm that solves this problem in O(min{p,n–p}n) time. This improves the previously known complexity O(min{p,n–p}²n).This research has been supported by the Spanish Science and Technology Ministry and FEDER Grant No. BFM2002-04525-C02-02.Received: October 2002 / Accepted: September 2003     

A parallel search algorithm for directed acyclic graphs

A parallel algorithm for depth-first searching of a directed acyclic graph (DAG) on a shared memory model of a SIMD computer is proposed. The algorithm uses two parallel tree traversal algorithms, one for the preorder traversal and the other for therpostorder traversal of an ordered tree. Each of these traversal algorithms has a time complexity ofO(logn) whenO(n) processors are used,n being the number of vertices in the tree. The parallel depth-first search algorithm for a directed acyclic graphG withn vertices has a time complexity ofO((logn)²) whenO(n ^2.81/logn) processors are used.     

Asymptotic Behaviour of Best lp-Approximations from Affi ne Subspaces

In this paper we consider the problem of best approximation in ℓ_pⁿ, 1<p∞. If h_p, 1<p<∞, denotes the best ℓ_p-approximation of the element h ⁿ from a proper affine subspace K of ⁿ, hK, then lim_p→∞h_p=h_∞^*, where h_∞^* is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to prove that for all r there are α_j ⁿ, 1jr, such that

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, with γ_p^{(r) n} and γ_p^(r)= (p^−r−1).     

Strongly polynomial dual simplex methods for the maximum flow problem

This paper presents dual network simplex algorithms that require at most 2nm pivots and O(n ² m) time for solving a maximum flow problem on a network ofn nodes andm arcs. Refined implementations of these algorithms and a related simplex variant that is not strictly speaking a dual simplex algorithm are shown to have a complexity of O(n ³). The algorithms are based on the concept of apreflow and depend upon the use of node labels that are underestimates of the distances from the nodes to the sink node in the extended residual graph associated with the current flow. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research was supported by NSF Grants DMS 91-06195, DMS 94-14438 and CDR 84-21402 and DOE Grant DE-FG02-92ER25126.Research was supported by NSF Grant CDR 84-21402 at Columbia University.     

Optimal location with equitable loads

The problem considered in this paper is to find p locations for p facilities such that the weights attracted to each facility will be as close as possible to one another. We model this problem as minimizing the maximum among all the total weights attracted to the various facilities. We propose solution procedures for the problem on a network, and for the special cases of the problem on a tree or on a path. The complexity of the problem is analyzed, O(n) algorithms and an O(pn ³) dynamic programming algorithm are proposed for the problem on a path respectively for p=2 and p>2 facilities. Heuristic algorithms (two types of a steepest descent approach and tabu search) are proposed for its solution. Extensive computational results are presented.     

Chebyshev approximation by discrete superposition. Application to neural networks

In this paper, we develop two algorithms for Chebyshev approximation of continuous functions on [0, 1] ⁿ using the modulus of continuity and the maximum norm estimated by a given finite data system. The algorithms are based on constructive versions of Kolmogorov's superposition theorem. One of the algorithms we apply to neural networks.     

On the Implementation of Lie Group Methods on the Stiefel Manifold

There are several applications in which one needs to integrate a system of ODEs whose solution is an n×p matrix with orthonormal columns. In recent papers algorithms of arithmetic complexity order np ² have been proposed. The class of Lie group integrators may seem like a worth while alternative for this class of problems, but it has not been clear how to implement such methods with O(np ²) complexity. In this paper we show how Lie group methods can be implemented in a computationally competitive way, by exploiting that analytic functions of n×n matrices of rank 2p can be computed with O(np ²) complexity.     

Optimal parallel quicksort on EREW PRAM

In this paper, we present parallel quicksort algorithms running inO((n/p+logp) logn) expected time andO((n/p+logp+log logn) logn) deterministic time respectively, and both withO(n) space by usingp processors on EREW PRAM. Whenp=O(n/logn), the cost is optimal, in terms of the product of time and number of processors. These algorithms can be used to obtain parallel algorithms for constructing balanced binary search trees without using sorting algorithms. One of our quicksort algorithms leads to a parallel quickhull algorithm on EREW PRAM.The work of this author was partially supported by a fellowship from the College of Science, Old Dominion University, Norfolk, VA 23529, USA.     

Improved complexity results for several multifacility location problems on trees

In this paper we consider multifacility location problems on tree networks. On general networks, these problems are usually NP-hard. On tree networks, however, often polynomial time algorithms exist; e.g., for the median, center, centdian, or special cases of the ordered median problem. We present an enhanced dynamic programming approach for the ordered median problem that has a time complexity of just O(ps ² n ⁶) for the absolute and O(ps ² n ²) for the node restricted problem, improving on the previous results by a factor of O(n ³). (n and p being the number of nodes and new facilities, respectively, and s (≤n) a value specific for the ordered median problem.) The same reduction in complexity is achieved for the multifacility k-centrum problem leading to O(pk ² n ⁴) (absolute) and O(pk ² n ²) (node restricted) algorithms.     

Short Polynomial Representations for Square Roots Modulo p

Let p be an odd prime number and a a square modulo p. It is well known that the simple formula a mod p gives a square root of a when p 3 mod 4. Let us write p – 1 = 2 ⁿ s with s odd. A fast algorithm due to Shanks, with n steps, allows us to compute a square root of a modulo p. It will be shown that there exists a polynomial of at most 2 ^n–1 terms giving a square root of a. Moreover, if there exists a polynomial in a representing a square root of a modulo p, it will be proved that this polynomial would have at least 2 ^n–1 terms, except for a finite set _n of primes p depending on n.     

On arithmetical algorithms over finite fields

Standard methods for calculating over GF(pⁿ), the finite field of pⁿ elements, require an irreducible polynomial of degree n with coefficients in GF(p). Such a polynomial is usually obtained by choosing it randomly and then verifying that it is irreducible, using a probabilistic algorithm. If it is not, the procedure is repeated. Here we given an explicit basis, with multiplication table, for the fields GF(p^pk), for k = 0, 1, 2,…, and their union. This leads to efficient computational methods, not requiring the preliminary calculation of irreducible polynomials over finite fields and, at the same time, yields a simple recursive formula for irreducible polynomials which generate the fields. The fast Fourier transform (FFT) is a method for efficiently evaluating (or interpolating) a polynomial of degree < n at all of the nth roots of unity, i.e., on the finite multiplicate subgroups of F, in O(nlogn) operations in the underlying field. We give an analogue of the fast Fourier transform which efficiently evaluates a polynomial on some of the additive subgroups ofF. This yields new “fast” algorithms for polynomial computation.     

Exploiting partial knowledge of satisfying assignments

Recently Schöning has shown that a simple local-search algorithm for 3SAT achieves the currently best upper bound, i.e., an expected time of 1.334ⁿ. In this paper, we show that this algorithm can be modified to run much faster if there is some kind of imbalance in satisfying assignments and we have a (partial) knowledge about that. Especially if a satisfying assignment has imbalanced 0's and 1's, i.e., p₁n 1's and (1-p₁)n 0's, then we can find a solution in time 1.260ⁿ when and 1.072ⁿ when p₁=0.1. Such an imbalance often exists in SAT instances reduced from other problems. As a concrete example, we investigate a reduction from 3DM and show our new approach is nontrivially faster than its direct algorithms. Preliminary experimental results are also given.     

How Many Bits have to be Changed to Decrease the Linear Complexity?

The k-error linear complexity of periodic binary sequences is defined to be the smallest linear complexity that can be obtained by changing k or fewer bits of the sequence per period. For the period length p ⁿ, where p is an odd prime and 2 is a primitive root modulo p ², we show a relationship between the linear complexity and the minimum value k for which the k-error linear complexity is strictly less than the linear complexity. Moreover, we describe an algorithm to determine the k-error linear complexity of a given p ⁿ-periodic binary sequence.     

Computing homotopic shortest paths in the plane

We address the problem of computing homotopic shortest paths in the presence of obstacles in the plane. Problems on homotopy of paths received attention very recently [Cabello et al., in: Proc. 18th Annu. ACM Sympos. Comput. Geom., 2002, pp. 160–169; Efrat et al., in: Proc. 10th Annu. European Sympos. Algorithms, 2002, pp. 411–423]. We present two output-sensitive algorithms, for simple paths and non-simple paths. The algorithm for simple paths improves the previous algorithm [Efrat et al., in: Proc. 10th Annu. European Sympos. Algorithms, 2002, pp. 411–423]. The algorithm for non-simple paths achieves O(log²n) time per output vertex which is an improvement by a factor of O(n/log²n) of the previous algorithm [Hershberger, Snoeyink, Comput. Geom. Theory Appl. 4 (1994) 63–98], where n is the number of obstacles. The running time has an overhead O(n²⁺) for any positive constant . In the case k<n²⁺, where k is the total size of the input and output, we improve the running to O((n+k+(nk)^2/3)log^O(1)n).     

Algorithms for coloring semi-random graphs

The graph coloring problem is to color a given graph with the minimum number of colors. This problem is known to be NP-hard even if we are only aiming at approximate solutions. On the other hand, the best known approximation algorithms require n^δ (δ>0) colors even for bounded chromatic (k-colorable for fixed k) n-vertex graphs. The situation changes dramatically if we look at the average performance of an algorithm rather than its worst case performance. A k-colorable graph drawn from certain classes of distributions can be k-colored almost surely in polynomial time. It is also possible to k-color such random graphs in polynomial average time. In this paper, we present polynomial time algorithms for k-coloring graphs drawn from the semirandom model. In this model, the graph is supplied by an adversary each of whose decisions regarding inclusion of edges is reversed with some probability p. In terms of randomness, this model lies between the worst case model and the usual random model where each edge is chosen with equal probability. We present polynomial time algorithms of two different types. The first type of algorithms always run in polynomial time and succeed almost surely. Blum and Spencer [J. Algorithms, 19 , 204–234 (1995)] have also obtained independently such algorithms, but our results are based on different proof techniques which are interesting in their own right. The second type of algorithms always succeed and have polynomial running time on the average. Such algorithms are more useful and more difficult to obtain than the first type of algorithms. Our algorithms work for semirandom graphs drawn from a wide range of distributions and work as long as p≥n^−α(k)+ϵ where α(k)=(2k)/((k−1)(k+2)) and ϵ is a positive constant. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 125–158 (1998)     

The Compact Law of the Iterated Logarithm for Multivariate Stochastic Approximation Algorithms

Abstract

We consider a sequence (Z _n ) _n≥1 defined by a general multivariate stochastic approximation algorithm and assume that (Z _n ) converges to a solution z* almost surely. We establish the compact law of the iterated logarithm for Z _n by proving that, with probability one, the limit set of the sequence (Z _n  ? z*) suitably normalized is an ellipsoid. We also give the law of the iterated logarithm for the l ^p norms, p ∈ [1, ∞], of (Z _n  ? z*).     

The single facility location problem with average-distances

This paper considers a location problem in ℝ ⁿ , where the demand is not necessarily concentrated at points but it is distributed in hypercubes following a Uniform probability distribution. The goal is to locate a service facility minimizing the weighted sum of average distances (measured with ℓ _p norms) to these demand hypercubes. In order to do that, we present an iterative scheme that provides a sequence converging to an optimal solution of the problem for p∈[1,2]. For the planar case, analytical expressions of this iterative procedure are obtained for p=2 and p=1, where two different approaches are proposed. The paper ends with a computational analysis of the proposed methodology, comparing its efficiency with a standard minimizer.