Deterministic Complexity Research Articles

A Time Hierarchy Theorem for the LOCAL Model

The celebrated time hierarchy theorem for Turing machines states, informally, that more problems can be solved given more time. The extent to which a time hierarchy--type theorem holds in the classic distributed $\mathsf{LOCAL}$ model has been open for many years. In particular, it is consistent with previous results that all natural problems in the $\mathsf{LOCAL}$ model can be classified according to a small constant number of complexities, such as $O(1),O(\log^* n), O(\log n), 2^{O(\sqrt{\log n})}$, etc. In this paper we establish the first time hierarchy theorem for the $\mathsf{LOCAL}$ model and prove that several gaps exist in the $\mathsf{LOCAL}$ time hierarchy. Our main results are as follows: (a) We define an infinite set of simple coloring problems called hierarchical $2\frac{1}{2}$-coloring. A correctly colored graph can be confirmed by simply checking the neighborhood of each vertex, so this problem fits into the class of locally checkable labeling (LCL) problems. However, the complexity of the $k$-level hierarchical $2\frac{1}{2}$-coloring problem is $\Theta(n^{1/k})$ for $k\in\mathbb{Z}^+$. The upper and lower bounds hold for both general graphs and trees and for both randomized and deterministic algorithms. (b) Consider any LCL problem on bounded degree trees. We prove an automatic speedup theorem that states that any randomized $n^{o(1)}$-time algorithm solving the LCL can be transformed into a deterministic $O(\log n)$-time algorithm. Together with a previous result [Y.-J. Chang, T. Kopelowitz, and S. Pettie, Proceedings of FOCS, 2016, pp. 615--624], this establishes that on trees, there are no natural deterministic complexities in the ranges $\omega(\log^* n)$---$o(\log n)$ or $\omega(\log n)$---$n^{o(1)}$. (c) We expose a new gap in the randomized time hierarchy on general graphs. Roughly speaking, any randomized algorithm that solves an LCL problem in sublogarithmic time can be sped up to run in $O(T_{LLL})$ time: the complexity of the distributed Lovász local lemma (LLL) problem. In other words, the LLL is complete for sublogarithmic time. Finally, we revisit Naor and Stockmeyer's characterization of $O(1)$-time $\mathsf{LOCAL}$ algorithms for LCL problems (as order-invariant w.r.t. vertex IDs) and calculate the complexity gaps that are directly implied by their proof. For $n$-rings we see an $\omega(1)$---$o(\log^* n)$ complexity gap, for $(\sqrt{n}\times \sqrt{n})$-tori an $\omega(1)$---$o(\sqrt{\log^* n})$ gap, and for bounded degree trees and general graphs, an $\omega(1)$---$o(\log(\log^* n))$ complexity gap.

Adaptive Regularization Algorithms with Inexact Evaluations for Nonconvex Optimization

A regularization algorithm using inexact function values and inexact derivatives is proposed and its evaluation complexity analyzed. This algorithm is applicable to unconstrained problems and to problems with inexpensive constraints (that is constraints whose evaluation and enforcement has negligible cost) under the assumption that the derivative of highest degree is $\beta$-H\"{o}lder continuous. It features a very flexible adaptive mechanism for determining the inexactness which is allowed, at each iteration, when computing objective function values and derivatives. The complexity analysis covers arbitrary optimality order and arbitrary degree of available approximate derivatives. It extends results of Cartis, Gould and Toint (2018) on the evaluation complexity to the inexact case: if a $q$th order minimizer is sought using approximations to the first $p$ derivatives, it is proved that a suitable approximate minimizer within $\epsilon$ is computed by the proposed algorithm in at most $O(\epsilon^{-\frac{p+\beta}{p-q+\beta}})$ iterations and at most $O(|\log(\epsilon)|\epsilon^{-\frac{p+\beta}{p-q+\beta}})$ approximate evaluations. An algorithmic variant, although more rigid in practice, can be proved to find such an approximate minimizer in $O(|\log(\epsilon)|+\epsilon^{-\frac{p+\beta}{p-q+\beta}})$ evaluations.While the proposed framework remains so far conceptual for high degrees and orders, it is shown to yield simple and computationally realistic inexact methods when specialized to the unconstrained and bound-constrained first- and second-order cases. The deterministic complexity results are finally extended to the stochastic context, yielding adaptive sample-size rules for subsampling methods typical of machine learning.

Non-Iterative DOA Estimation Using Discrete Fourier Transform Interpolation

A fast and accurate non-iterative direction-of-arrival (DOA) estimation algorithm for multiple targets in additive white Gaussian noise is devised in this paper. The proposed estimator makes use of the two highest magnitudes discrete Fourier transform (DFT) coefficients of the input data and two of their associated neighboring bins, resulting in a deterministic complexity of ${\mathcal {O}}(N(1+\log (N)))$ with $N$ being the number of sensors. The bias and mean squares error of the DOA estimates are analyzed. The simulation results are presented to validate the correctness of theoretical derivation and demonstrate the superiority of the devised estimator over several conventional DOA estimators.

Public vs. private randomness in simultaneous multi-party communication complexity

In simultaneous number-in-hand multi-party communication protocols, we have k players, who each receive a private input, and wish to compute a joint function of their inputs. The players simultaneously each send a single message to a referee, who then outputs the value of the function. The cost of the protocol is the total number of bits sent to the referee.For two players, it is known that giving the players a public (shared) random string is much more useful than private randomness: public-coin protocols can be unboundedly better than deterministic protocols, while private-coin protocols can only give a quadratic improvement on deterministic protocols.We extend the two-player gap to multiple players, and show that the private-coin communication complexity of a k-player function f is at least Ω(D(f)) for any k≥2. Perhaps surprisingly, this bound is tight: although one might expect the gap between private-coin and deterministic protocols to grow with the number of players, we show that the All-Equality function, where each player receives n bits of input and the players must determine if their inputs are all the same, can be solved by a private-coin protocol with O˜(nk+k) bits. Since All-Equality has deterministic complexity Θ(nk), this shows that sometimes the gap scales only as the square root of the number of players, and consequently the number of bits each player needs to send actually decreases as the number of players increases. We also consider the Exists-Equality function, where we ask whether there is a pair of players that received the same input, and prove a nearly-tight bound of Θ˜(kn) for it.

Model Development for Risk-Based Safety Assessment of a Geological Disposal System of Radioactive Wastes Generated by Pyroprocessing of Pressurized Water Reactor Spent Fuel in Korea

New methodology for a risk-based safety assessment of a geological disposal system of nuclear waste was implemented using the numerical Korea Atomic Energy Research Institute (KAERI) Performance Assessment Model (K-PAM). K-PAM was applied to a conceptual geological disposal system for pyroprocessed radioactive wastes based on the KAERI Underground Research Tunnel (KURT) site. The methodology was systematically organized for model development considering two types of external events: earthquakes and well intrusion. Following description of its conceptual models and submodules, K-PAM was partially verified by comparing the consequences of two major modules of K-PAM—engineered barrier system and natural barrier system—with those by a well-known, comparable process model using COMSOL. In addition, K-PAM was demonstrated using three scenarios: (1) the reference scenario, in which the normal consequences of the disposal system without external events could be predicted; (2) the deterministic complex scenario, in which the impacts of individual external events on the disposal system could be estimated separately; and (3) the probabilistic complex scenario, in which the efficiency of the new methodology for a risk-based safety assessment could be confirmed numerically by showing the probable maximum dose rate according to any single scenario, the convergence of risk, the dominant impacts contributing to the maximum dose rate, and the probability of occurrence of the scenario groups.

Complex random matrices have no real eigenvalues

Let [Formula: see text] where [Formula: see text] are iid copies of a mean zero, variance one, subgaussian random variable. Let [Formula: see text] be an [Formula: see text] random matrix with entries that are iid copies of [Formula: see text]. We prove that there exists a [Formula: see text] such that the probability that [Formula: see text] has any real eigenvalues is less than [Formula: see text] where [Formula: see text] only depends on the subgaussian moment of [Formula: see text]. The bound is optimal up to the value of the constant [Formula: see text]. The principal component of the proof is an optimal tail bound on the least singular value of matrices of the form [Formula: see text] where [Formula: see text] is a deterministic complex matrix with the condition that [Formula: see text] for some constant [Formula: see text] depending on the subgaussian moment of [Formula: see text]. For this class of random variables, this result improves on the results of Pan–Zhou [Circular law, extreme singular values and potential theory, J. Multivariate Anal. 101(3) (2010) 645–656] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633]. In the proof of the tail bound, we develop an optimal small-ball probability bound for complex random variables that generalizes the Littlewood–Offord theory developed by Tao–Vu [From the Littlewood–Offord problem to the circular law: Universality of the spectral distribution of random matrices, Bull. Amer. Math. Soc.[Formula: see text]N.S.[Formula: see text] 46(3) (2009) 377–396; Inverse Littlewood–Offord theorems and the condition number of random discrete matrices, Ann. of Math.[Formula: see text] 169(2) (2009) 595–632] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633; Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math. 62(12) (2009) 1707–1739].

Deterministic Communication in Radio Networks

In this paper we improve the deterministic complexity of two fundamental communication primitives in the classical model of ad-hoc radio networks with unknown topology: broadcasting and wake-up. We consider an unknown radio network, in which all nodes have no prior knowledge about network topology, and know only the size of the network $n$, the maximum in-degree of any node $\Delta$, and the eccentricity of the network $D$. For such networks, we first give an algorithm for wake-up, based on the existence of small universal synchronizers. This algorithm runs in $O(\frac{\min\{n, D \Delta\} \log n \log \Delta}{\log\log \Delta})$ time, the fastest known in both directed and undirected networks, improving over the previous best $O(n \log^2n)$-time result across all ranges of parameters, but particularly when maximum in-degree is small. Next, we introduce a new combinatorial framework of block synchronizers and prove the existence of such objects of low size. Using this framework, we design a new deterministic algorithm for the fundamental problem of broadcasting, running in $O(n \log D \log\log\frac{D \Delta}{n})$ time. This is the fastest known algorithm for the problem in directed networks, improving upon the $O(n \log n \log \log n)$-time algorithm of De Marco (2010) and the $O(n \log^2 D)$-time algorithm due to Czumaj and Rytter (2003). It is also the first to come within a log-logarithmic factor of the $\Omega(n \log D)$ lower bound due to Clementi et al.\ (2003). Our results also have direct implications on the fastest \emph{deterministic leader election} and \emph{clock synchronization} algorithms in both directed and undirected radio networks, tasks which are commonly used as building blocks for more complex procedures.

State Complexity of Regular Tree Languages for Tree Matching

We study the state complexity of regular tree languages for tree matching problem. Given a tree t and a set of pattern trees L, we can decide whether or not there exists a subtree occurrence of trees in L from the tree t by considering the new language L′ which accepts all trees containing trees in L as subtrees. We consider the case when we are given a set of pattern trees as a regular tree language and investigate the state complexity. Based on the sequential and parallel tree concatenation, we define three types of tree languages for deciding the existence of different types of subtree occurrences. We also study the deterministic top-down state complexity of path-closed languages for the same problem.

State Complexity of Insertion

It is well known that the resulting language obtained by inserting a regular language to a regular language is regular. We study the nondeterministic and deterministic state complexity of the insertion operation. Given two incomplete DFAs of sizes m and n, we give an upper bound (m+2)·2mn−m−1·3m and find a lower bound for an asymp-totically tight bound. We also present the tight nondeterministic state complexity by a fooling set technique. The deterministic state complexity of insertion is 2Θ(mn) and the nondeterministic state complexity of insertion is precisely mn+2m, where m and n are the size of input finite automata. We also consider the state complexity of insertion in the case where the inserted language is bifix-free or non-returning.

On communication complexity of bent functions from Maiorana–McFarland class

In this article we study two party Communication Complexity of Boolean bent functions from Maiorana–McFarland class. In particular, we describe connections between Maiorana–McFarland construction of bent functions and operations on matrix form of Boolean functions and show that bent functions of 2n variables from Maiorana–McFarland class have deterministic communication complexity equal n + 1. Finally, we show that not all bent functions have high communication complexity lower bound by giving the example of such function.

Super-resolution characteristics based on time-reversed single-frequency electromagnetic wave

ABSTRACTA sub-wavelength three-antenna array that is able to perform super-resolution focusing with single-frequency signal excitations is investigated using time-reversal. The antenna array is loaded with a set of uniformly distributed thin metal wires, which when combined with single-frequency electromagnetic waves from a time-reversal mirror yields a 1/35 wavelength super-resolution focusing at the targeted antenna. The length of the metal wires in particular determines the frequency band enabling super-resolution performance. The results have significance for the super-resolution investigation of multi-band compact antenna arrays involving micro-structures, with potential linkage with recent studies on super-resolution mechanisms in deterministic complex media.

Analytical probability density function for the statistics of the ENSO phenomenon: Asymmetry and power law tail

AbstractThis letter has two main goals. The first one is to give a physically reasonable explanation for the use of stochastic models for mimicking the apparent random features of the El Ninõ–Southern Oscillation (ENSO) phenomenon. The second one is to obtain, from the theory, an analytical expression for the equilibrium density function of the anomaly sea surface temperature, an expression that fits the data from observations well, reproducing the asymmetry and the power law tail of the histograms of the NIÑO3 index. We succeed in these tasks exploiting some recent theoretical results of the author in the field of the dynamical origin of the stochastic processes. More precisely, we apply this approach to the celebrated recharge oscillator model (ROM), weakly interacting by a multiplicative term, with a general deterministic complex forcing (Madden‐Julian Oscillations, westerly wind burst, etc.), and we obtain a Fokker‐Planck equation that describes the statistical behavior of the ROM.

CLT for linear spectral statistics of a rescaled sample precision matrix

Central limit theorems (CLTs) of linear spectral statistics (LSS) of general Fisher matrices [Formula: see text] are widely used in multivariate statistical analysis where [Formula: see text] with a deterministic complex matrix [Formula: see text] and two sample covariance matrices [Formula: see text] and [Formula: see text] from two independent samples with sample sizes [Formula: see text] and [Formula: see text]. As the first step to obtain the CLT, it is necessary to establish the CLT for LSS of the random matrix [Formula: see text], or equivalently that of [Formula: see text], that is a sample precision matrix rescaled by a general non-negative definite Hermitian matrix [Formula: see text]. Because the scaling matrix [Formula: see text] in many large-dimensional problems may not be invertible, the result does not simply follow from the celebrated CLT by Bai and Silverstein (2004). Thus, we have to alternatively derive the CLT of LSS of [Formula: see text] where the inverse of [Formula: see text] may not exist, thus extending Bai and Silverstein’s CLT. As a further innovation of the paper, general populations for the sample covariance matrix [Formula: see text] are covered requiring the existence a fourth-order moment of arbitrary value, that is not necessarily matching the values of the Gaussian case.

On Nonlinear Complexity and Shannon’s Entropy of Finite Length Random Sequences

Pseudorandom binary sequences have important uses in many fields, such as spread spectrum communications, statistical sampling and cryptography. There are two kinds of method in evaluating the properties of sequences, one is based on the probability measure, and the other is based on the deterministic complexity measures. However, the relationship between these two methods still remains an interesting open problem. In this paper, we mainly focus on the widely used nonlinear complexity of random sequences, study on its distribution, expectation and variance of memoryless sources. Furthermore, the relationship between nonlinear complexity and Shannon’s entropy is also established here. The results show that the Shannon’s entropy is strictly monotonically decreased with nonlinear complexity.

Controllability of deterministic complex networks

Recently, the controllability of complex networks has attracted wide attention of researchers. The main contribution comes from Liu et al. who proposed the structural controllability as an analytical framework for making predictions regarding the control of directed networks in Nature. Since then, the controllability of many model and real networks has been deeply investigated except deterministic complex networks. In this paper, we focus on studying the controllability of deterministic complex networks. We examine six typical deterministic networks, the simulation results show that the minimum number of driver nodes grows linearly with network size. When the network size is large enough, the controllability approximates to a constant not more than 0.4, indicating that the deterministic networks are relatively easy to control. Furthermore, we investigate the characteristics of driver nodes in deterministic complex networks, finding that the driver nodes tend to avoid high degree nodes but to have high clustering coefficients.

A Mathematical Theory of Super-Resolution by Using a System of Sub-Wavelength Helmholtz Resonators

A rigorous mathematical theory is developed to explain the super-resolution phenomenon observed in the experiment (Lemoult et al., Phys Rev Lett 107:064301, 2011). A key ingredient is the calculation of the resonances and the Green function in the half space with the presence of a system of Helmholtz resonators in the quasi-stationary regime. By using boundary integral equations and generalized Rouché’s theorem, the existence and the leading asymptotic of the resonances are rigorously derived. The integral equation formulation also yields the leading order terms in the asymptotics of the Green function. The methodology developed in the paper provides an elegant and systematic way for calculating resonant frequencies for Helmholtz resonators in assorted space settings, as well as in various frequency regimes. By using the asymptotics of the Green function, the analysis of the imaging functional of the time-reversal wave fields becomes possible, which clearly demonstrates the super-resolution property. The result provides the first mathematical theory of super-resolution in the context of a deterministic medium and sheds light on the mechanism of super-resolution and super-focusing for waves in deterministic complex media.

Static-dynamic stability of the body gymnasts qualifications

Purpose: evaluation of individual ways of postural control gymnasts skilled in solving problems on the stability of the body in the motor tests. Material : The study involved engaged in gymnastics (n = 9). The measurements were carried out on the platform stabilographic Kistler. Tests used: handstand, Biryuk sample, sample Romberg. Results: set individual ways microvibrations parts of the body and makrokolebany in the sagittal and frontal planes. Joined symmetry and asymmetry postural control of the body, the various indicators of energy expenditure. The quality of postural control in the performance of motor tests were deterministic complex conditions of body position on the support, limited visual orientation. Also corresponds to the test specificity of the sport. Conclusions : The method microvibrations in solving problems on the stability of the body in the motor tests gymnasts is the most strategically important for the effective development and management of a system of regulation poses athlete. Confirmation rates are economical expenditure of energy during motor tests.

Modularity and predictability in cell signaling and decision making.

Cells make decisions to differentiate, divide, or apoptose based on multiple signals of internal and external origin. These decisions are discrete outputs from dynamic networks comprised of signaling pathways. Yet the validity of this decomposition of regulatory proteins into distinct pathways is unclear because many regulatory proteins are pleiotropic and interact through cross-talk with components of other pathways. In addition to the deterministic complexity of interconnected networks, there is stochastic complexity arising from the fluctuations in concentrations of regulatory molecules. Even within a genetically identical population of cells grown in the same environment, cell-to-cell variations in mRNA and protein concentrations can be as high as 50% in yeast and even higher in mammalian cells. Thus, if everything is connected and stochastic, what hope could we have for a quantitative understanding of cellular decisions? Here we discuss the implications of recent advances in genomics, single-cell, and single-cell genomics technology for network modularity and cellular decisions. On the basis of these recent advances, we argue that most gene expression stochasticity and pathway interconnectivity is nonfunctional and that cellular decisions are likely much more predictable than previously expected.

Asymptotic distribution of complex zeros of random analytic functions

Let $\xi_0,\xi_1,\ldots$ be independent identically distributed complex- valued random variables such that $\mathbb{E}\log(1+|\xi _0|)<\infty$. We consider random analytic functions of the form \[\mathbf{G}_n(z)=\sum_{k=0}^{\infty}\xi_kf_{k,n}z^k,\] where $f_{k,n}$ are deterministic complex coefficients. Let $\mu_n$ be the random measure counting the complex zeros of $\mathbf{G}_n$ according to their multiplicities. Assuming essentially that $-\frac{1}{n}\log f_{[tn],n}\to u(t)$ as $n\to\infty$, where $u(t)$ is some function, we show that the measure $\frac{1}{n}\mu_n$ converges in probability to some deterministic measure $\mu$ which is characterized in terms of the Legendre-Fenchel transform of $u$. The limiting measure $\mu$ does not depend on the distribution of the $\xi_k$'s. This result is applied to several ensembles of random analytic functions including the ensembles corresponding to the three two-dimensional geometries of constant curvature. As another application, we prove a random polynomial analogue of the circular law for random matrices.

Probabilistic Algorithm for Polynomial Optimization over a Real Algebraic Set

Let $f, f_1, \ldots, f_{s}$ be $n$-variate polynomials with rational coefficients of maximum degree $D$ and let $V$ be the set of common complex solutions of $\mathbf{F}=(f_1,\ldots, f_{s})$. We give an algorithm which, up to some regularity assumptions on $\mathbf{F}$, computes an exact representation of the global infimum $f^\star$ of the restriction of the map $x\to f(x)$ to ${V\cap\mathbb{R}^n}$, i.e., a univariate polynomial vanishing at $f^\star$ and an isolating interval for $f^\star$. Furthermore, it decides whether $f^\star$ is reached, and if so, it returns $x^\star\in V\cap\mathbb{R}^n$ such that $f(x^\star)=f^\star$. This algorithm is probabilistic. It makes use of the notion of polar varieties. Its complexity is essentially cubic in $(s D)^n$ and linear in the complexity of evaluating the input. This fits within the best known deterministic complexity class $D^{O(n)}$. We report on some practical experiments of a first implementation that is available as a Maple package. It appears that it can tackle global optimization problems that were unreachable by previous exact algorithms and can manage instances that are hard to solve with purely numeric techniques. As far as we know, even under the extra genericity assumptions on the input, it is the first probabilistic algorithm that combines practical efficiency with good control of complexity for this problem.