Rates of uniform convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions

Assume that {Xn} is a strictly stationary β-mixing random sequence with the β-mixing coefficient βk = O(k-r), 0 < r ≤1. Yu (1994) obtained convergence rates of empirical processes of strictly stationary β-mixing random sequence indexed by bounded classes of functions. Here, a new truncation method is proposed and used to study the convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions. The research results show that if the envelope of the index class of functions is in Lp, p > 2 or p > 4, uniform convergence rates of empirical processes of strictly stationary β-mixing random sequence over the index classes can reach O((nr/(l+r)/logn)-1/2) or O((nr/(1+r)/ log n)-3/4) and that the Central Limit Theorem does not always hold for the empirical processes.``     

On strategy improvement algorithms for simple stochastic games

The study of simple stochastic games (SSGs) was initiated by Condon for analyzing the computational power of randomized space-bounded alternating Turing machines. The game is played by two players, MAX and MIN, on a directed multigraph, and when the play terminates at a sink vertex s, MAX wins from MIN a payoff p(s)∈[0,1]. Condon proved that the problem SSG-VALUE—given a SSG, determine whether the expected payoff won by MAX is greater than 1/2 when both players use their optimal strategies—is in NP∩coNP. However, the exact complexity of this problem remains open, as it is not known whether the problem is in P or is hard for some natural complexity class. In this paper, we study the computational complexity of a strategy improvement algorithm by Hoffman and Karp for this problem. The Hoffman–Karp algorithm converges to optimal strategies of a given SSG, but no non-trivial bounds were previously known on its running time. We prove a bound of O(n²/n) on the convergence time of the Hoffman–Karp algorithm, and a bound of O(²0.78n) on a randomized variant. These are the first non-trivial upper bounds on the convergence time of these strategy improvement algorithms.     

Approximation Algorithms for Min–Max Tree Partition

We consider the problem of partitioning the node set of a graph intopequal sized subsets. The objective is to minimize the maximum length, over these subsets, of a minimum spanning tree. We show that no polynomial algorithm with bounded error ratio can be given for the problem unless P = NP. We present anO(n²) time algorithm for the problem, wherenis the number of nodes in the graph. Assuming that the edge lengths satisfy the triangle inequality, its error ratio is at most 2p − 1. We also present an improved algorithm that obtains as an input a positive integerx. It runs inO(2^{(p + x)p}n²) time, and its error ratio is at most (2 − x/(x + p − 1))p.     

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.     

Routing through a generalized switchbox

We present an algorithm for the routing problem for two-terminal nets in generalized switchboxes. A generalized switchbox is any subset R of the planar rectangular grid with no nontrivial holes, i.e., every finite face has exactly four incident vertices. A net is a pair of nodes of nonmaximal degree on the boundary of R. A solution is a set of edge-disjoint paths, one for each net. Our algorithm solves standard generalized switchbox routing problems in time O(n(log n)²) where n is the number of vertices of R, i.e., it either finds a solution or indicates that there is none. A problem is standard if deg(ν) + ter(ν) is even for all vertices ν where deg(ν) is the degree of ν and ter(ν) is the number of nets which have ν as a terminal. For nonstandard problems we can find a solution in time O(n(log n)² + |U|²) where U is the set of vertices ν with deg(ν) + ter(ν) is odd.     

Block-Coordinate Gradient Descent Method for?Linearly Constrained Nonsmooth Separable Optimization

We consider the problem of minimizing the weighted sum of a smooth function f and a convex function P of n real variables subject to m linear equality constraints. We propose a block-coordinate gradient descent method for solving this problem, with the coordinate block chosen by a Gauss-Southwell-q rule based on sufficient predicted descent. We establish global convergence to first-order stationarity for this method and, under a local error bound assumption, linear rate of convergence. If f is convex with Lipschitz continuous gradient, then the method terminates in O(n ²/ε) iterations with an ε-optimal solution. If P is separable, then the Gauss-Southwell-q rule is implementable in O(n) operations when m=1 and in O(n ²) operations when m>1. In the special case of support vector machines training, for which f is convex quadratic, P is separable, and m=1, this complexity bound is comparable to the best known bound for decomposition methods. If f is convex, then, by gradually reducing the weight on P to zero, the method can be adapted to solve the bilevel problem of minimizing P over the set of minima of f+δ _X , where X denotes the closure of the feasible set. This has application in the least 1-norm solution of maximum-likelihood estimation. This research was supported by the National Science Foundation, Grant No. DMS-0511283.     

An Algorithm for the Modular Decomposition of Hypergraphs

We propose an O(n⁴) algorithm to build the modular decomposition tree of hypergraphs of dimension three and show how this algorithm can be generalized to compute in O(n^{3k − 5}) time the decomposition of hypergraphs of any fixed dimension k.     

A Randomized Parallel Algorithm for Planar Graph Isomorphism

We present a parallel randomized algorithm running on aCRCW PRAM, to determine whether two planar graphs are isomorphic, and if so to find the isomorphism. We assume that we have a tree of separators for each planar graph (which can be computed by known algorithms inO(log² n) time withn^{1 + ε}processors, for any ε > 0). Ifnis the number of vertices, our algorithm takesO(log(n)) time with processors and with a probability of failure of 1/nat most. The algorithm needs 2 · log(m) − log(n) + O(log(n)) random bits. The number of random bits can be decreased toO(log(n)) by increasing the number of processors ton^{3/2 + ε}, for any ε > 0. Our parallel algorithm has significantly improved processor efficiency, compared to the previous logarithmic time parallel algorithm of Miller and Reif (Siam J. Comput.20(1991), 1128–1147), which requiresn⁴randomized processors orn⁵deterministic processors.     

Note on multidimensional empirical processes for φ-mixing random vectors

In 1974, Sen proved weak convergence of the empirical processes (in the J₁-topology on D^p[0, 1]) for a stationary φ-mixing sequence of stochastic p( 1)-vectors. In this note, we show that Sen's theorem on weak convergence of the multidimensional empirical process for a stationary φ-mixing sequence of stochastic vectors remains true under a less restrictive condition on the mixing constants {φ_n}, i.e., φ_n = O(n^−1−δ) for some δ > 0.     

An algorithm of MCMC method for solving F(X) =0

An algorithm of continuous stage-space MCMC method for solving algebra equation f(x)=0 is given. It is available for the case that the sign of f(x) changes frequently or the derivative f′(x) does not exist in the neighborhood of the root, while the Newton method is hard to work. Let n be the number of random variables created by computer in our algorithm. Then after m=O(n) transactions from the initial value x ₀,x* can be got such that |f(x*)|<e ^−cm |f(x ₀)| by choosing suitable positive constant c. An illustration is also given with the discussion of convergence by adjusting the parameters in the algorithm. Supported by the National Natural Science Foundation of China (70171008).     

A note on optimal multiway split trees

Split trees are suitable data structures for storing records with different access frequencies. Under assumption that the access frequencies are all distinct, Huang has proposed anO(n ⁴ logm) time algorithm to construct an (m+1)-way split tree for a set ofn keys. In this paper, we generalize Huang's algorithm to deal with the case of non-distinct access frequencies. The technique used in the generalized algorithm is a generalization of Hesteret al.'s, where the binary case was considered. The generalized algorithm runs inO(n ⁵ logm) time.     

A parallel wilf algorithm for complex zeros of a polynomial

A global recursive bisection algorithm is described for computing the complex zeros of a polynomial. It has complexityO(n ³ p) wheren is the degree of the polynomial andp the bit precision requirement. Ifn processors are available, it can be realized in parallel with complexityO(n ² p); also it can be implemented using exact arithmetic. A combined Wilf-Hansen algorithm is suggested for reduction in complexity.     

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).     

Explicit Construction of RIP Matrices Is Ramsey-Hard

Matrices Φ ∈ ℝ^{n × p} satisfying the restricted isometry property (RIP) are an important ingredient of the compressive sensing methods. While it is known that random matrices satisfy the RIP with high probability even for n = log^O(1)p , the explicit deteministic construction of such matrices defied the repeated efforts, and most of the known approaches hit the so-called sparsity bottleneck. The notable exception is the work by Bourgain et al. constructing an n × p RIP matrix with sparsity s = Θ(n^{1/2 + ϵ}) , but in the regime n = Ω(p^{1 − δ}) . In this short note we resolve this open question by showing that an explicit construction of a matrix satisfying the RIP in the regime n = O(log²p) and s = Θ(n^1/2) implies an explicit construction of a three-colored Ramsey graph on p nodes with clique sizes bounded by O(log²p) — a question in the field of extremal combinatorics that has been open for decades. © 2019 Wiley Periodicals, Inc.     

MAX k‐CUT and approximating the chromatic number of random graphs

We consider the MAX k‐CUT problem on random graphs G_n,p. First, we bound the probable weight of a MAX k‐CUT using probabilistic counting arguments and by analyzing a simple greedy heuristic. Then, we give an algorithm that approximates MAX k‐CUT in expected polynomial time, with approximation ratio 1 + O((np)^‐1/2). Our main technical tool is a new bound on the probable value of Frieze and Jerrum's semidefinite programming (SDP)‐relaxation of MAX k‐CUT on random graphs. To obtain this bound, we show that the value of the SDP is tightly concentrated. As a further application of our bound on the probable value of the SDP, we obtain an algorithm for approximating the chromatic number of G_n,p, 1/n ≤ p ≤ 0.99, within a factor of O((np)^1/2) in polynomial expected time, thereby answering a question of Krivelevich and Vu. We give similar algorithms for random regular graphs. The techniques for studying the SDP apply to a variety of SDP relaxations of further NP‐hard problems on random structures and may therefore be of independent interest. For instance, to bound the SDP we estimate the eigenvalues of random graphs with given degree sequences. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Fast Construction of Optimal Circulant Preconditioners for Matrices from the Fast Dense Matrix Method

In this paper, we consider solving non-convolution type integral equations by the preconditioned conjugate gradient method. The fast dense matrix method is a fast multiplication scheme that provides a dense discretization matrix A approximating a given integral equation. The dense matrix A can be constructed in O(n) operations and requires only O(n) storage where n is the size of the matrix. Moreover, the matrix-vector multiplication A xcan be done in O(n log n) operations. Thus if the conjugate gradient method is used to solve the discretized system, the cost per iteration is O(n log n) operations. However, for some integral equations, such as the Fredholm integral equations of the first kind, the system will be ill-conditioned and therefore the convergence rate of the method will be slow. In these cases, preconditioning is required to speed up the convergence rate of the method. A good choice of preconditioner is the optimal circulant preconditioner which is the minimizer of C – A _F in Frobenius norm over all circulant matrices C. It can be obtained by taking arithmetic averages of all the entries of A and therefore the cost of constructing the preconditioner is of O(n ²) operations for general dense matrices. In this paper, we develop an O(n log n) method of constructing the preconditioner for dense matrices A obtained from the fast dense matrix method. Application of these ideas to boundary integral equations from potential theory will be given. These equations are ill-conditioned whereas their optimal circulant preconditioned equations will be well-conditioned. The accuracy of the approximation A, the fast construction of the preconditioner and the fast convergence of the preconditioned systems will be illustrated by numerical examples.     

Greedily Finding a Dense Subgraph

Given an n-vertex graph with nonnegative edge weights and a positive integer k ≤ n, our goal is to find a k-vertex subgraph with the maximum weight. We study the following greedy algorithm for this problem: repeatedly remove a vertex with the minimum weighted-degree in the currently remaining graph, until exactly k vertices are left. We derive tight bounds on the worst case approximation ratio R of this greedy algorithm: (1/2 + n/2k)² − O(n ^− 1/3) ≤ R ≤ (1/2 + n/2k)² + O(1/n) for k in the range n/3 ≤ k ≤ n and 2(n/k − 1) − O(1/k) ≤ R ≤ 2(n/k − 1) + O(n/k²) for k < n/3. For k = n/2, for example, these bounds are 9/4 ± O(1/n), improving on naive lower and upper bounds of 2 and 4, respectively. The upper bound for general k compares well with currently the best (and much more complicated) approximation algorithm based on semidefinite programming.     

Optimal Partition Trees

We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Matoušek (Discrete Comput. Geom. 10:157–182, 1993) gave a partition tree method for d-dimensional simplex range searching achieving O(n) space and O(n ^1−1/d ) query time. Although this method is generally believed to be optimal, it is complicated and requires O(n ^1+ε ) preprocessing time for any fixed ε>0. An earlier method by Matoušek (Discrete Comput. Geom. 8:315–334, 1992) requires O(nlogn) preprocessing time but O(n ^1−1/d log ^O(1) n) query time. We give a new method that achieves simultaneously O(nlogn) preprocessing time, O(n) space, and O(n ^1−1/d ) query time with high probability. Our method has several advantages:

On rank and null space computation of the generalized Sylvester matrix

In this paper, we present new approaches computing the rank and the null space of the (m n + p)×(n + p) generalized Sylvester matrix of (m + 1) polynomials of maximal degrees n,p. We introduce an algorithm which handles directly a modification of the generalized Sylvester matrix, computing efficiently its rank and null space and replacing n by log ₂ n in the required complexity of the classical methods. We propose also a modification of the Gauss-Jordan factorization method applied to the appropriately modified Sylvester matrix of two polynomials for computing simultaneously its rank and null space. The methods can work numerically and symbolically as well and are compared in respect of their error analysis, complexity and efficiency. Applications where the computation of the null space of the generalized Sylvester matrix is required, are also given.     

L p -Version of the Dubins–Savage Inequality and Some Exponential Inequalities

The Dubins–Savage inequality is generalized by using the pth (1<p≤2) conditional moment of the martingale differences. This inequality is further extended under suitable conditions when p>2. Another martingale inequality due to Freedman is also generalized when 0<p≤2. Implications of these inequalities for strong convergence are discussed. Some general exponential inequalities are also given for martingales (supermartingales) under suitable conditions.