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Channel routing plays an important role in the development of automated layout systems for integrated circuits. Many layout systems first place modules on a chip and then wire together terminals on different modules that should be electrically connected. This wiring problem is often solved by heuristically partitioning the given space into rectangular channels and then assigning to each such channel a set of wires which are to pass through it. This solution reduces a global wiring problem to a set of disjoint (and hopefully easier) local channel routing subproblems. For this reason, the channel routing problem has been intensively studied for over a decade, and numerous heuristics and approximation algorithms have been proposed for its solution. The generic form of the channel routing problem may be described as follows. The channel consists of a rectilinear grid of tracks (or rows) and columns. Along the top and bottom tracks are numbers called terminals, and terminals with the same number form a net. A net with q terminals is called an q-terminal net. The smallest net is a two-terminal net; if q2, we have a multiterminal net. The channel routing problem is to connect all the terminals in each net using horizontal and vertical wires which are routed along the underlying rectilinear grid. The goal is to complete the wiring using the minimum number of tracks; i. e., to minimize the width of the channel.

In this paper, we demonstrate that optimal control algorithms can be used to speed up the implementation of modules of quantum algorithms or quantum simulations in networks of coupled qubits. The gain is most prominent in realistic cases, where the qubits are not all mutually coupled. Thus the shortest times obtained depend on the coupling topology as well as on the characteristicratio of the time scales for local controls {\em vs} non-local ({\em i.e.} coupling) evolutions in the specific experimental setting. Relating these minimal times to the number of qubits gives the tightest known upper bounds to the actual time complexity of the quantum modules. As will be shown, time complexity is a more realistic measure of the experimental cost than the usual gate complexity. In the limit of fast local controls (as {\em e.g.} in NMR), time-optimised realisations are shown for the quantum Fourier transform (QFT) and the multiply controlled {\sc not}-gate ({\sc c$^{n-1}$not}) in various coupling topologies of $n$ qubits. The speed-ups are substantial: in a chain of six qubits the quantum Fourier transform so far obtained by optimal control is more than eight times faster than the standard decomposition into controlled phase, Hadamard and {\sc swap} gates, while the {\sc c$^{n-1}$not}-gate for completely coupled network of six qubits is nearly seven times faster.

Three algorithms of Gram-Schmidt type are given that produce an orthogonal decomposition of finite $d$-dimensional symmetric, alternating, or Hermitian forms over division rings. The first uses $d^3/3+O(d^2)$ ring operations with very simple implementation. Next, that algorithm is adapted in two new directions. One is an optimal sequential algorithm whose complexity matches the complexity of matrix multiplication. The other is a parallel NC algorithm with similar complexity.

This paper presents lossless prefix codes optimized with respect to a pay-off criterion consisting of a convex combination of maximum codeword length and average codeword length. The optimal codeword lengths obtained are based on a new coding algorithm which transforms the initial source probability vector into a new probability vector according to a merging rule. The coding algorithm is equivalent to a partition of the source alphabet into disjoint sets on which a new transformed probability vector is defined as a function of the initial source probability vector and a scalar parameter. The pay-off criterion considered encompasses a trade-off between maximum and average codeword length; it is related to a pay-off criterion consisting of a convex combination of average codeword length and average of an exponential function of the codeword length, and to an average codeword length pay-off criterion subject to a limited length constraint. A special case of the first related pay-off is connected to coding problems involving source probability uncertainty and codeword overflow probability, while the second related pay-off compliments limited length Huffman coding algorithms.

The extended L\"uroth's Theorem says that if the transcendence degree of $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK$ is 1 then there exists $f \in \KK(\underline{X})$ such that $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)$ is equal to $\KK(f)$. In this paper we show how to compute $f$ with a probabilistic algorithm. We also describe a probabilistic and a deterministic algorithm for the decomposition of multivariate rational functions. The probabilistic algorithms proposed in this paper are softly optimal when $n$ is fixed and $d$ tends to infinity. We also give an indecomposability test based on gcd computations and Newton's polytope. In the last section, we show that we get a polynomial time algorithm, with a minor modification in the exponential time decomposition algorithm proposed by Gutierez-Rubio-Sevilla in 2001.

Problems involving the optimal control or ordinary of partial differential equations are infinite dimensional problems which are approximated by discretized problems for their numerical solution. Quasi Newton methods were applied to these finite dimensional problems and it was shown by analysis and numerical tests how the convergence rate could be predicted using information from the underlying infinite dimensional problem. For unconstrained optimal control problems with ordinary differential equations two approaches were studied: In one approach the control functions were used as unknowns whereas for the second route the control, state and costate functions were taken as unknowns. In the latter approach the quasi Newton update made extensive use of the special structure of the control problem and proved to be very effective. These algorithms were also studied for nonlinear elliptic boundary value problems and the optimal control of pseudoparabolic differential equations.

In this paper, we obtain bounds on the probability of convergence to the optimal solution for the compact Genetic Algorithm (cGA) and the Population Based Incremental Learning (PBIL). We also give a sufficient condition for convergence of these algorithms to the optimal solution and compute a range of possible values of the parameters of these algorithms for which they converge to the optimal solution with a confidence level.

We study the fundamental limits on learning latent community structure in dynamic networks. Specifically, we study dynamic stochastic block models where nodes change their community membership over time, but where edges are generated independently at each time step. In this setting (which is a special case of several existing models), we are able to derive the detectability threshold exactly, as a function of the rate of change and the strength of the communities. Below this threshold, we claim that no algorithm can identify the communities better than chance. We then give two algorithms that are optimal in the sense that they succeed all the way down to this limit. The first uses belief propagation (BP), which gives asymptotically optimal accuracy, and the second is a fast spectral clustering algorithm, based on linearizing the BP equations. We verify our analytic and algorithmic results via numerical simulation, and close with a brief discussion of extensions and open questions.

This paper presents a new and efficient approach for capacitor placement in radial distribution systems that determine the optimal locations and size of capacitor with an objective of improving the voltage profile and reduction of power loss. The solution methodology has two parts: in part one the loss sensitivity factors are used to select the candidate locations for the capacitor placement and in part two a new algorithm that employs Shuffle Frog Leaping Algorithm (SFLA) and Particle Swarm Optimization are used to estimate the optimal size of capacitors at the optimal buses determined in part one. The main advantage of the proposed method is that it does not require any external control parameters. The other advantage is that it handles the objective function and the constraints separately, avoiding the trouble to determine the barrier factors. The proposed method is applied to 45-bus radial distribution systems.

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We establish essentially optimal bounds on the complexity of initial-value problems in the randomized and quantum settings. For this purpose we define a sequence of new algorithms whose error/cost properties improve from step to step. These algorithms yield new upper complexity bounds, which differ from known lower bounds by only an arbitrarily small positive parameter in the exponent, and a logarithmic factor. In both the randomized and quantum settings, initial-value problems turn out to be essentially as difficult as scalar integration.

We consider stochastic multi-armed bandit problems where the expected reward is a Lipschitz function of the arm, and where the set of arms is either discrete or continuous. For discrete Lipschitz bandits, we derive asymptotic problem specific lower bounds for the regret satisfied by any algorithm, and propose OSLB and CKL-UCB, two algorithms that efficiently exploit the Lipschitz structure of the problem. In fact, we prove that OSLB is asymptotically optimal, as its asymptotic regret matches the lower bound. The regret analysis of our algorithms relies on a new concentration inequality for weighted sums of KL divergences between the empirical distributions of rewards and their true distributions. For continuous Lipschitz bandits, we propose to first discretize the action space, and then apply OSLB or CKL-UCB, algorithms that provably exploit the structure efficiently. This approach is shown, through numerical experiments, to significantly outperform existing algorithms that directly deal with the continuous set of arms. Finally the results and algorithms are extended to contextual bandits with similarities.

Development of cities based on the economy of the country. Transportation from one city to another plays a major role in the economy of the country. Transportation through roadways is more significant for the development of the country.

We present a study of linear models using the concepts of neutrosophic science, the science that was built on the basis that there is no absolute truth, there is no confirmed data, issues cannot be limited to right and wrong only. There is a third state between error and right, an indeterminate, undetermined, uncertain state. It is indeterminacy. Neutrosophic science gave each issue three dimensions, namely (T, I, F), correctness in degrees, indeterminacy in degrees, and error in degrees. It was founded by the American philosopher and mathematician Florentin Smarandache, in 1995 and came as a generalization of fuzzy logic that was founded by the scientist Lotfi. A. Zadeh, in 1965.

In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of ANOVA-type. The weights model the relative importance of different groups of variables. We present new randomized multilevel algorithms to tackle this integration problem and prove upper bounds for their randomized error. Furthermore, we provide in this setting the first non-trivial lower error bounds for general randomized algorithms, which, in particular, may be adaptive or non-linear. These lower bounds show that our multilevel algorithms are optimal. Our analysis refines and extends the analysis provided in [F. J. Hickernell, T. M\"uller-Gronbach, B. Niu, K. Ritter, J. Complexity 26 (2010), 229-254], and our error bounds improve substantially on the error bounds presented there. As an illustrative example, we discuss the unanchored Sobolev space and employ randomized quasi-Monte Carlo multilevel algorithms based on scrambled polynomial lattice rules.

The described multicoloring problem has direct applications in the context of wireless ad hoc and sensor networks. In order to coordinate the access to the shared wireless medium, the nodes of such a network need to employ some medium access control (MAC) protocol. Typical MAC protocols control the access to the shared channel by time (TDMA), frequency (FDMA), or code division multiple access (CDMA) schemes. Many channel access schemes assign a fixed set of time slots, frequencies, or (orthogonal) codes to the nodes of a network such that nodes that interfere with each other receive disjoint sets of time slots, frequencies, or code sets. Finding a valid assignment of time slots, frequencies, or codes hence directly corresponds to computing a multicoloring of a graph $G$. The scarcity of bandwidth, energy, and computing resources in ad hoc and sensor networks, as well as the often highly dynamic nature of these networks require that the multicoloring can be computed based on as little and as local information as possible.

In this paper, we focus on batch state estimation for linear systems. This problem is important in applications such as environmental field estimation, robotic navigation, and target tracking. Its difficulty lies on that limited operational resources among the sensors, e.g., shared communication bandwidth or battery power, constrain the number of sensors that can be active at each measurement step. As a result, sensor scheduling algorithms must be employed. Notwithstanding, current sensor scheduling algorithms for batch state estimation scale poorly with the system size and the time horizon. In addition, current sensor scheduling algorithms for Kalman filtering, although they scale better, provide no performance guarantees or approximation bounds for the minimization of the batch state estimation error. In this paper, one of our main contributions is to provide an algorithm that enjoys both the estimation accuracy of the batch state scheduling algorithms and the low time complexity of the Kalman filtering scheduling algorithms. In particular: 1) our algorithm is near-optimal: it achieves a solution up to a multiplicative factor 1/2 from the optimal solution, and this factor is close to the best approximation factor 1/e one can achieve in polynomial time for this problem; 2) our algorithm has (polynomial) time complexity that is not only lower than that of the current algorithms for batch state estimation; it is also lower than, or similar to, that of the current algorithms for Kalman filtering. We achieve these results by proving two properties for our batch state estimation error metric, which quantifies the square error of the minimum variance linear estimator of the batch state vector: a) it is supermodular in the choice of the sensors; b) it has a sparsity pattern (it involves matrices that are block tri-diagonal) that facilitates its evaluation at each sensor set.

| Optimal Binary Search Tree | Algorithms | Adda 247 

In this paper, the distributed resource allocation optimization problem is investigated. The allocation decisions are made to minimize the sum of all the agents' local objective functions while satisfying both the global network resource constraint and the local allocation feasibility constraints. Here the data corresponding to each agent in this separable optimization problem, such as the network resources, the local allocation feasibility constraint, and the local objective function, is only accessible to individual agent and cannot be shared with others, which renders new challenges in this distributed optimization problem. Based on either projection or differentiated projection, two classes of continuous-time algorithms are proposed to solve this distributed optimization problem in an initialization-free and scalable manner. Thus, no re-initialization is required even if the operation environment or network configuration is changed, making it possible to achieve a "plug-and-play" optimal operation of networked heterogeneous agents. The algorithm convergence is guaranteed for strictly convex objective functions, and the exponential convergence is proved for strongly convex functions without local constraints. Then the proposed algorithm is applied to the distributed economic dispatch problem in power grids, to demonstrate how it can achieve the global optimum in a scalable way, even when the generation cost, or system load, or network configuration, is changing.

A coflow is a collection of parallel flows belonging to the same job. It has the all-or-nothing property: a coflow is not complete until the completion of all its constituent flows. In this paper, we focus on optimizing \emph{coflow-level delay}, i.e., the time to complete all the flows in a coflow, in the context of an $N\times N$ input-queued switch. In particular, we develop a throughput-optimal scheduling policy that achieves the best scaling of coflow-level delay as $N\rightarrow\infty$. We first derive lower bounds on the coflow-level delay that can be achieved by any scheduling policy. It is observed that these lower bounds critically depend on the variability of flow sizes. Then we analyze the coflow-level performance of some existing coflow-agnostic scheduling policies and show that none of them achieves provably optimal performance with respect to coflow-level delay. Finally, we propose the Coflow-Aware Batching (CAB) policy which achieves the optimal scaling of coflow-level delay under some mild assumptions.

Energy harvesting (EH) has been developed to extend the lifetimes of energy-limited communication systems. In this letter, we consider a single-user EH communication system, in which both of the arrival data and the harvested energy curves are modeled as general functions. Unlike most of the works in the field, we investigate the online algorithms which only acquire the causal information of the arrival data and the harvested energy processes. We study how well the optimal online algorithm works compared with the optimal offline algorithm, and thus our goal is to find the lower and upper bounds for the ratio of the completion time in the optimal online algorithm to the optimal offline algorithm. We propose two online algorithms which achieve the upper bound of 2 on this ratio. Also, we show that this ratio is 2 for the optimal online algorithm.

Dualization is a key discrete enumeration problem. It is not known whether or not this problem is polynomial-time solvable. Asymptotically optimal dualization algorithms are the fastest among the known dualization algorithms, which is supported by new experiments with various data described in this paper. A theoretical justification of the efficiency of these algorithms on the average was given by E.V. Djukova more than 30 years ago. In this paper, new results on the construction of parallel algorithms for intractable enumeration problems are presented. A new static parallelization scheme for asymptotically optimal dualization algorithms is developed and tested. The scheme is based on statistical estimations of subtasks size.

Cubature formulas, asymptotically optimal with respect to accuracy, are derived for calculating multidimensional weakly singular integrals. They are used for developing a universal code for calculating capacitances of conductors of arbitrary shapes.

In this paper, we develop a centralized spectrum sensing and Dynamic Spectrum Access (DSA) scheme for secondary users (SUs) in a Cognitive Radio (CR) network. Assuming that the primary channel occupancy follows a Markovian evolution, the channel sensing problem is modeled as a Partially Observable Markov Decision Process (POMDP). We assume that each SU can sense only one channel at a time by using energy detection, and the sensing outcomes are then reported to a central unit, called the secondary system decision center (SSDC), that determines the channel sensing/accessing policies. We derive both the optimal channel assignment policy for secondary users to sense the primary channels, and the optimal channel access rule. Our proposed optimal sensing and accessing policies alleviate many shortcomings and limitations of existing proposals: (a) ours allows fully utilizing all available primary spectrum white spaces, (b) our model, and thus the proposed solution, exploits the temporal and spatial diversity across different primary channels and (c) is based on realistic local sensing decisions rather than complete knowledge of primary signalling structure. As an alternative to the high complexity of the optimal channel sensing policy, a suboptimal sensing policy is obtained by using the Hungarian algorithm iteratively, which reduces the complexity of the channel assignment from an exponential to a polynomial order. We also propose a heuristic algorithm that reduces the complexity of the sensing policy further to a linear order. The simulation results show that the proposed algorithms achieve a near-optimal performance with a significant reduction in computational time.

A framework is presented for constructing various types of numerical quadrature algorithms that take into account the a-priori known or estimated properties of the signal being processed. This is done by appropriately modeling the signal class to which such a signal belongs. Both linear and nonlinear signal class models are considered and wide use of generalized spline theory is made. For the nonlinear case, a new type of nonlinear generalized spline is defined.

We give a general unified method that can be used for $L_1$ {\em closeness testing} of a wide range of univariate structured distribution families. More specifically, we design a sample optimal and computationally efficient algorithm for testing the equivalence of two unknown (potentially arbitrary) univariate distributions under the $\mathcal{A}_k$-distance metric: Given sample access to distributions with density functions $p, q: I \to \mathbb{R}$, we want to distinguish between the cases that $p=q$ and $\|p-q\|_{\mathcal{A}_k} \ge \epsilon$ with probability at least $2/3$. We show that for any $k \ge 2, \epsilon>0$, the {\em optimal} sample complexity of the $\mathcal{A}_k$-closeness testing problem is $\Theta(\max\{ k^{4/5}/\epsilon^{6/5}, k^{1/2}/\epsilon^2 \})$. This is the first $o(k)$ sample algorithm for this problem, and yields new, simple $L_1$ closeness testers, in most cases with optimal sample complexity, for broad classes of structured distributions.

Cubature formulas, asymptotically optimal with respect to accuracy, are derived for calculating multidimensional weakly singular integrals. They are used for developing a universal code for calculating capacitances of conductors of arbitrary shapes.

We consider the one-to-one Pickup and Delivery Problem (PDP) in Euclidean Space with arbitrary dimension $d$ where $n$ transportation requests are picked i.i.d. with a separate origin-destination pair for each object to be moved. First, we consider the problem from the customer perspective where the objective is to compute a plan for transporting the objects such that the Euclidean distance traveled by the vehicles when carrying objects is minimized. We develop a polynomial time asymptotically optimal algorithm for vehicles with capacity $o(\sqrt[2d]{n})$ for this case. This result also holds imposing LIFO constraints for loading and unloading objects. Secondly, we extend our algorithm to the classical single-vehicle PDP where the objective is to minimize the total distance traveled by the vehicle and present results indicating that the extended algorithm is asymptotically optimal for a fixed vehicle capacity if the origins and destinations are picked i.i.d. using the same distribution.

We approach the hidden subgroup problem by performing the so-called pretty good measurement on hidden subgroup states. For various groups that can be expressed as the semidirect product of an abelian group and a cyclic group, we show that the pretty good measurement is optimal and that its probability of success and unitary implementation are closely related to an average-case algebraic problem. By solving this problem, we find efficient quantum algorithms for a number of nonabelian hidden subgroup problems, including some for which no efficient algorithm was previously known: certain metacyclic groups as well as all groups of the form (Z_p)^r X| Z_p for fixed r (including the Heisenberg group, r=2). In particular, our results show that entangled measurements across multiple copies of hidden subgroup states can be useful for efficiently solving the nonabelian HSP.

We approach the hidden subgroup problem by performing the so-called pretty good measurement on hidden subgroup states. For various groups that can be expressed as the semidirect product of an abelian group and a cyclic group, we show that the pretty good measurement is optimal and that its probability of success and unitary implementation are closely related to an average-case algebraic problem. By solving this problem, we find efficient quantum algorithms for a number of nonabelian hidden subgroup problems, including some for which no efficient algorithm was previously known: certain metacyclic groups as well as all groups of the form (Z_p)^r X| Z_p for fixed r (including the Heisenberg group, r=2). In particular, our results show that entangled measurements across multiple copies of hidden subgroup states can be useful for efficiently solving the nonabelian HSP.

This research project on optimal compression algorithms was concerned with several problems in the area of digital communications and the elimination of redundancy. The importance of the problem arises from the growing use of digital transmission in the military as well as the civilian communications field. There are many advantages enjoyed by digital transmission, the most notable being that there is almost no signal to noise degradation when relayed through a number of repeaters, whereas analog repeaters lose 3 db every time the number of repeaters is doubled. The one marked disadvantage of digital transmissions is that digitization of a basically analog source such as speech or TV results in a bandwidth expansion using conventional techniques. Good quality digitized voice requires on the order of 50,000 bits per second if data compression is not used. With conventional techniques, this requires 30 kHz of bandwidth while the analog voice signal could be transmitted over a 3 kHz channel. The area of bandwidth compression seeks to remove this disadvantage by eliminating the redundancy in the signal.

The positioning of nulls in an antenna array field pattern is essential to the performance of the antenna, in being capable of blocking interference. The null placement must be achieved in such a way that the field pattern in other directions is not adversely affected. One of the most efficient methods of null placement is by perturbing only the phases of the array elements. This document presents two approaches to the placement of nulls by phase perturbation. The first is a least squares method based on exact of approximate null placement, applicable to one-dimensional arrays and extendable to two-dimensional arrays, developed for real quiescent patterns which apparently allows polygonal arrays (in this study, octagonal arrays) to be considered. The second is a minimax method in one or two dimensions based on null placement, which readily permits the omission of failed elements and which involves only the perturbation of selected element phases or amplitudes. Keywords: Great Britain, Ladar antennas.

In a previous work it was shown that the best measure for the efficiency of a single burst-correcting code is obtained using the Gallager bound as opposed to the Reiger bound. In this paper, an efficient algorithm that searches for the best (shortened) cyclic burst-correcting codes is presented. Using this algorithm, extensive tables that either tie existing constructions or improve them are obtained for burst lengths up to b=10.

This is the first of a series of papers constructing an information based general theory of optimal errors and analytic computational complexity. Among the applications are such traditionally diverse areas as approximation, boundary-value problems, quadrature, and nonlinear equations in a finite or infinite dimensional space. Traditionally algorithms are often derived by ad hoc criteria. The information based theory rationalizes the synthesis of algorithms by showing how to construct algorithms which minimize or nearly minimize the error. For certain classes of problems it shows how to construct algorithms (linear optimal error algorithms) which enjoy essentially optimal complexity with respect to all possible algorithms. The existence of strongly non-computable problems is demonstrated. In contrast with the gap theorem of recursively computable functions it is shown that every monotonic real function is the complexity of some problem.

This project considers the dynamic spectrum management (DSM) problem whereby multiple users sharing a common frequency band must choose their transmit power spectra jointly in response to physical channel conditions including the effects of interference. The goal of the users is to maximize a system-wide utility function (e.g., weighted sum-rate of all users), subject to individual power constraints. The proposed work will focus on a general DSM problem formulation which allows correlated signaling rather than being restricted to the conventional independent orthogonal signaling such as OFDM. The general formulation will exploit the concept of 'interference alignment' which is known to provide substantial rate gain over OFDM signalling for general interference channels. We have successfully analyzed the complexity to characterize the optimal spectrum sharing policies and beamforming strategies in interfering broadcast networks and developed efficient computational methods for optimal resource allocations in such networks.

This project considers the dynamic spectrum management (DSM) problem whereby multiple users sharing a common frequency band must choose their transmit power spectra jointly in response to physical channel conditions including the effects of interference. The goal of the users is to maximize a system-wide utility function (e.g., weighted sum-rate of all users), subject to individual power constraints. The proposed work will focus on a general DSM problem formulation which allows correlated signaling rather than being restricted to the conventional independent orthogonal signaling such as OFDM. The general formulation will exploit the concept of 'interference alignment' which is known to provide substantial rate gain over OFDM signalling for general interference channels. We have successfully analyzed the complexity to characterize the optimal spectrum sharing policies and beamforming strategies in interfering broadcast networks and developed efficient computational methods for optimal resource allocations in such networks.

Although Potent purports to use only radial velocities in reconstructing the potential velocity field of galaxies, the derivation of transverse components is implicit in the smoothing procedures adopted. Thus the possibility arises of using nonradial line integrals to derive a smoothed velocity field. For an inhomogeneous galaxy distribution the optimal path for integration need not be radial, and can be obtained using max-flow algorithms. In this paper we describe how one may use Dijkstra's algorithm to obtain this optimal path and velocity field, and present the results of applying the algorithm to a realistic spatial distribution of galaxies. These results show that the method has limited effect due to the large smoothing scales employed in Potent. However, the viability of the technique is demonstrated and, finally, we discuss other possible methods involving averaging over an ensemble of non-radial paths for improving a potential velocity field derived from redshifts.

Constructing a model for data in R2 is a common problem in many scientific fields, including pattern recognition, computer vision, and applied mathematics. Often, little is known about the process which generated the data or its statistical properties. For example, in fitting a piecewise linear model the number of pieces as well as the knot locations may be unknown. Hence the method used to build the statistical model should have few assumptions and yet still provide a model that is optimal in some sense. Such methods can be designed through the use of genetic algorithms. In this paper we examine the use of genetic algorithms to fit piecewise linear functions to data in R2. The number of pieces, the location of the knots, and the underlying distribution of the data are assumed to be unknown. We discuss existing methods which attempt to solve this problem and introduce a new method which employs genetic algorithms to optimize the number and location of the linear pieces. We prove theoretically that our method provides near-optimal functions and present the results of extensive experiments which demonstrate that the proposed method provides better results than existing spline based methods. We conclude that our method represents a valuable tool for fitting both robust and non-robust piecewise linear functions.

In this two-part paper, we address the problem of finding the optimal precoding/multiplexing scheme for a set of non-cooperative links sharing the same physical resources, e.g., time and bandwidth. We consider two alternative optimization problems: P.1) the maximization of mutual information on each link, given constraints on the transmit power and spectral mask; and P.2) the maximization of the transmission rate on each link, using finite order constellations, under the same constraints as in P.1, plus a constraint on the maximum average error probability on each link. Aiming at finding decentralized strategies, we adopted as optimality criterion the achievement of a Nash equilibrium and thus we formulated both problems P.1 and P.2 as strategic noncooperative (matrix-valued) games. In Part I of this two-part paper, after deriving the optimal structure of the linear transceivers for both games, we provided a unified set of sufficient conditions that guarantee the uniqueness of the Nash equilibrium. In this Part II, we focus on the achievement of the equilibrium and propose alternative distributed iterative algorithms that solve both games. Specifically, the new proposed algorithms are the following: 1) the sequential and simultaneous iterative waterfilling based algorithms, incorporating spectral mask constraints; 2) the sequential and simultaneous gradient projection based algorithms, establishing an interesting link with variational inequality problems. Our main contribution is to provide sufficient conditions for the global convergence of all the proposed algorithms which, although derived under stronger constraints, incorporating for example spectral mask constraints, have a broader validity than the convergence conditions known in the current literature for the sequential iterative waterfilling algorithm.

In the (deletion-channel) trace reconstruction problem, there is an unknown $n$-bit source string $x$. An algorithm is given access to independent traces of $x$, where a trace is formed by deleting each bit of~$x$ independently with probability~$\delta$. The goal of the algorithm is to recover~$x$ exactly (with high probability), while minimizing samples (number of traces) and running time. Previously, the best known algorithm for the trace reconstruction problem was due to Holenstein~et~al.; it uses $\exp(\tilde{O}(n^{1/2}))$ samples and running time for any fixed $0 < \delta < 1$. It is also what we call a "mean-based algorithm", meaning that it only uses the empirical means of the individual bits of the traces. Holenstein~et~al.~also gave a lower bound, showing that any mean-based algorithm must use at least $n^{\tilde{\Omega}(\log n)}$ samples. In this paper we improve both of these results, obtaining matching upper and lower bounds for mean-based trace reconstruction. For any constant deletion rate $0 < \delta < 1$, we give a mean-based algorithm that uses $\exp(O(n^{1/3}))$ time and traces; we also prove that any mean-based algorithm must use at least $\exp(\Omega(n^{1/3}))$ traces. In fact, we obtain matching upper and lower bounds even for $\delta$ subconstant and $\rho := 1-\delta$ subconstant: when $(\log^3 n)/n \ll \delta \leq 1/2$ the bound is $\exp(-\Theta(\delta n)^{1/3})$, and when $1/\sqrt{n} \ll \rho \leq 1/2$ the bound is $\exp(-\Theta(n/\rho)^{1/3})$. Our proofs involve estimates for the maxima of Littlewood polynomials on complex disks. We show that these techniques can also be used to perform trace reconstruction with random insertions and bit-flips in addition to deletions. We also find a surprising result: for deletion probabilities $\delta > 1/2$, the presence of insertions can actually help with trace reconstruction.

In this paper, we consider robust control using randomized algorithms. We extend the existing order statistics distribution theory to the general case in which the distribution of population is not assumed to be continuous and the order statistics is associated with certain constraints. In particular, we derive an inequality on distribution for related order statistics. Moreover, we also propose two different approaches in searching reliable solutions to the robust analysis and optimal synthesis problems under constraints. Furthermore, minimum computational effort is investigated and bounds for sample size are derived.

Cooperative control systems are increasingly emerging as significant alternatives to their centralized counterparts. The rising interest in deploying cooperative systems is fueled by the development of decentralized systems with low cost and performance advantages. For example, mobile exploration and information gathering tasks can often be accomplished cheaply and more reliably by swarms of small autonomous robots as opposed to a single more sophisticated one. Cooperative control is also applied in many tasks that can not be performed by a single system, e.g. satellite arrays that enable global communication, geographically remote systems that communicate via network and others. The goal of our research is to investigate optimal control in cooperative systems, using algorithms inspired from biology. We begin with a review of collective behavior in biological systems.

We improve the inequality used in Pronzato [2003. Removing non-optimal support points in D-optimum design algorithms. Statist. Probab. Lett. 63, 223-228] to remove points from the design space during the search for a $D$-optimum design. Let $\xi$ be any design on a compact space $\mathcal{X} \subset \mathbb{R}^m$ with a nonsingular information matrix, and let $m+\epsilon$ be the maximum of the variance function $d(\xi,\mathbf{x})$ over all $\mathbf{x} \in \mathcal{X}$. We prove that any support point $\mathbf{x}_{*}$ of a $D$-optimum design on $\mathcal{X}$ must satisfy the inequality $d(\xi,\mathbf{x}_{*}) \geq m(1+\epsilon/2-\sqrt{\epsilon(4+\epsilon-4/m)}/2)$. We show that this new lower bound on $d(\xi,\mathbf{x}_{*})$ is, in a sense, the best possible, and how it can be used to accelerate algorithms for $D$-optimum design.

In this paper will be presented methodology of encoding information in valuations of discrete lattice with some translational invariant constrains in asymptotically optimal way. The method is based on finding statistical description of such valuations and changing it into statistical algorithm, which allows to construct deterministically valuation with given statistics. Optimal statistics allow to generate valuations with uniform distribution - we get maximum information capacity this way. It will be shown that we can reach the optimum for one-dimensional models using maximal entropy random walk and that for the general case we can practically get as close to the capacity of the model as we want (found numerically: lost 10^{-10} bit/node for Hard Square). There will be also presented simpler alternative to arithmetic coding method which can be used as cryptosystem and data correction method too.

The subject of this work is the patrolling of an environment with the aid of a team of autonomous agents. We consider both the design of open-loop trajectories with optimal properties, and of distributed control laws converging to optimal trajectories. As performance criteria, the refresh time and the latency are considered, i.e., respectively, time gap between any two visits of the same region, and the time necessary to inform every agent about an event occurred in the environment. We associate a graph with the environment, and we study separately the case of a chain, tree, and cyclic graph. For the case of chain graph, we first describe a minimum refresh time and latency team trajectory, and we propose a polynomial time algorithm for its computation. Then, we describe a distributed procedure that steers the robots toward an optimal trajectory. For the case of tree graph, a polynomial time algorithm is developed for the minimum refresh time problem, under the technical assumption of a constant number of robots involved in the patrolling task. Finally, we show that the design of a minimum refresh time trajectory for a cyclic graph is NP-hard, and we develop a constant factor approximation algorithm.

A new Distributed Minimal Residual (DMR) method for the acceleration of explicit iterative algorithms for the numerical solution of systems of partial differential equations has been developed. The method is based on the idea of allowing each partial differential equation in the system to approach the converged solution at its own optimal speed while at the same time communicating with the rest of the equations in the system. The DMR method belongs to a general class of the extrapolation techniques in which the solution is updated using information from a number of consecutive time steps in such a way that the L2 norm of future residual is minimized. Unlike in other similar methods, each component of the solution vector is updated using a separate sequence of acceleration factors. The idea of using different acceleration factors for each component of a solution vector is similar to that of dynamic preconditioning. This allows each equation to evolve at its own optimal convergence rate.

Design of computational algorithms for optical control by Hilbert space methods, and involving cost function

This work concerns a method for identifying an optimal basis for linear programming problems in the setting of interior point methods. To each iterate x-superscript-k generated by a primal interior point algorithm, say, we associate an indicator vector q-superscript-k with the property that if x-superscript-k converges to a nondegenerate vertex x*, then q-superscript-k converges to the 0-1 vector sign(x*). More interestingly, we show that the convergence of q-superscript-k is quadratically faster than that of x-superscript-k in the sense that q-superscript-k - q* = O(x-superscript-k - x*-sq). This clear-cut separation and rapid convergence allow one to infer at an intermediate stage of the iterative process which variables will be zero at optimality and which will not. We also show that under suitable assumptions this method is applicable to dual as well as primal-dual algorithms and can be extended to handle certain types of degeneracy. Numerical examples are included to corroborate the convergence properties of the indicators. The practical limitations of the indicator technique are also discussed.

This work concerns a method for identifying an optimal basis for linear programming problems in the setting of interior point methods. To each iterate x-superscript-k generated by a primal interior point algorithm, say, we associate an indicator vector q-superscript-k with the property that if x-superscript-k converges to a nondegenerate vertex x*, then q-superscript-k converges to the 0-1 vector sign(x*). More interestingly, we show that the convergence of q-superscript-k is quadratically faster than that of x-superscript-k in the sense that q-superscript-k - q* = O(x-superscript-k - x*-sq). This clear-cut separation and rapid convergence allow one to infer at an intermediate stage of the iterative process which variables will be zero at optimality and which will not. We also show that under suitable assumptions this method is applicable to dual as well as primal-dual algorithms and can be extended to handle certain types of degeneracy. Numerical examples are included to corroborate the convergence properties of the indicators. The practical limitations of the indicator technique are also discussed.

We consider stochastic multi-armed bandits where the expected reward is a unimodal function over partially ordered arms. This important class of problems has been recently investigated in (Cope 2009, Yu 2011). The set of arms is either discrete, in which case arms correspond to the vertices of a finite graph whose structure represents similarity in rewards, or continuous, in which case arms belong to a bounded interval. For discrete unimodal bandits, we derive asymptotic lower bounds for the regret achieved under any algorithm, and propose OSUB, an algorithm whose regret matches this lower bound. Our algorithm optimally exploits the unimodal structure of the problem, and surprisingly, its asymptotic regret does not depend on the number of arms. We also provide a regret upper bound for OSUB in non-stationary environments where the expected rewards smoothly evolve over time. The analytical results are supported by numerical experiments showing that OSUB performs significantly better than the state-of-the-art algorithms. For continuous sets of arms, we provide a brief discussion. We show that combining an appropriate discretization of the set of arms with the UCB algorithm yields an order-optimal regret, and in practice, outperforms recently proposed algorithms designed to exploit the unimodal structure.