Mind change efficient learning This paper studies efficient learning with respect to mind changes. Our starting point is the idea that a learner that is efficient with respect to mind changes minimizes mind changes not only globally in the entire learning problem, but also locally in subproblems after receiving some evidence. Formalizing this idea leads to the notion of strong mind change optimality. We characterize the structure of language classes that can be identified with at most mind changes by some learner (not necessarily effective): a language class L is identifiable with mind changes iff the accumulation order of L is at most . Accumulation order is a classic concept from point-set topology. We show that accumulation order is related to other established notions of structural complexity, such as thickness and intrinsic complexity. To aid the construction of learning algorithms, we show that the characteristic property of strongly mind change optimal learners is that they output conjectures (languages) with maximal accumulation order. We illustrate the theory by describing strongly mind change optimal learners for various problems such as identifying linear subspaces, one-variable patterns, and fixed-length patterns. One of the goals of computational learning theory is to design learning algorithms for which we can provide performance guarantees. Identification in the limit is a central performance goal in Gold's language learning paradigm [11] . A well-studied refinement of this notion is identification with bounded mind changes [9, 1] . In this paper, we investigate a further refinement that we term strong mind change optimality (SMC-optimality). Briefly, a learner is SMC-optimal if the learner achieves the best possible mind change bound not only for the entire problem, but also relative to any data sequences that the learner may observe.. The general theory in this paper has two main goals: (1) To provide necessary and sufficient conditions for a language collection to be identifiable with a given (ordinal) mind-change bound by some learner (not necessarily effective). (2) To provide necessary and sufficient conditions for a learner to be SMC-optimal. The results addressing (1) help us determine when an SMC-optimal learning algorithm exists, and the results addressing (2) help us to construct optimal learning algorithms when they do exist.. We situate our study in the framework of point-set topology. Previous work has shown the usefulness of topology for learning theory [34,chapter 10,28,17,4,29] We show how to view a language collection as a topological space; this allows us to apply Cantor's classic concept of accumulation order which assigns an ordinal acc(L) to a language collection, if L has bounded accumulation order. We show that a language collection L is identifiable with mind change bound by a learner if and only if acc(L) = . This result establishes a purely information-theoretic and structural necessary condition for identification with bounded mind changes. Based on the concept of accumulation order, we provide necessary and sufficient conditions for a learner to be SMC-optimal. These results show that SMC-optimality strongly constrains the conjectures of learners. We illustrate these results by analyzing various learning problems, such as identifying a linear subspace and a one-variable pattern.. The paper is organized as follows. Section 2 reviews standard concepts for language identification and presents our definition of mind change optimality. Then we establish the correspondence between mind change complexity and accumulation order. Section 4 gives necessary and sufficient conditions for a learner to be strongly mind change optimal. Next, we describe some general principles for constructing SMC-optimal effective learners and illustrate them with one-variable and fixed-length pattern languages. In Section 6, we show strong relationships between the concept of accumulation order and other structural notions studied in learning theory, such as thickness [41] , elasticity [44, 30] , and intrinsic complexity [10, 13] .

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An independent H-TCP implementation under FreeBSD 7.0: description and observed behaviour A key requirement for IETF recognition of new TCP algorithms is having an independent, interoperable implementation. This paper describes our BSD-licensed implementation of H-TCP within FreeBSD 7.0, publicly available as a dynamically loadable kernel module. Based on our implementation experience we provide a summary description of the H-TCP algorithm to assist other groups build further interoperable implementations. Using data from our live testbed we demonstrate that our version exhibits expected H-TCP behavior, and describe a number of implementation-specific issues that influence H-TCP's dynamic behavior. Finally, we illustrate the actual collateral impact on path latency of using H-TCP instead of NewReno. In particular we illustrate how, compared to NewReno, H-TCP's cwnd growth strategy can cause faster fluctuations in queue sizes at, yet lower median latency through, congestion points. We believe these insights will prove valuable predictors of H-TCP's potential impact if deployed in consumer end-hosts in addition to specialist, high-performance network environments. From the perspective of basic connectivity, the key innovation underlying today's internet is the connectionless, destination-based forwarding of packets. The internet protocol (IP) layer sits at the thin waist of most people's hourglass view of internetworking, providing an abstract layer for common routing and addressing that spans a multitude of different link layer technologies.. However, credit for the internet's wide-spread utility over the past 25+ years must also be given to the transmission control protocol (TCP) [1] . The relatively modern NewReno variant of TCP [2] balances two key goals: Provide reliable transfer of byte-streams across the IP layer's unpredictable packet-based service, and minimise congestion inside end hosts and the underlying IP network(s) while maximising performance [3] . TCP is the dominant transport protocol for internet-based applications [4] , so the latter goal has been an active and challenging area for academic and industry research into congestion control (CC) techniques [5] .. In recent years a number of new CC algorithms have been proposed, simulated and implemented (such as HS-TCP [6] , H-TCP [7] [8], CUBIC [9] , CompoundTCP [10] , and FAST [11] ). Each provide different approaches to inferring the level of, and appropriate dynamic response to, congestion within an IP network. Some have reportedly [12] been deployed operationally prior to full, independent evaluation by the IETF (internet engineering task force) or IRTF (internet research task force).. IETF standards-track progression typically requires multiple independent and interoperable implementations be successfully created from a published specification. In July 2007 the IETF's Transport Area has also requested the assistance of the IRTF's internet congestion control research group (IC-CRG [13] ) to evaluate the dynamic behavior of new TCP CC proposals [14] . The IRTF's transport modeling research group (TMRG [15] ) is developing metrics and test scenarios by which new CC schemes may be compared in a consistent and independent manner.. To hopefully assist IETF and IRTF efforts, this paper reports on our development of a publicly-available, BSDlicensed implementation of H-TCP within FreeBSD 7.0 [16] . H-TCP was initially designed by the Hamilton Institute (Ireland), and implemented in ns-2 [17] and the Linux kernel. Our implementation derives solely from our reading of the published literature on H-TCP and limited email exchanges with the H-TCP authors. We did not look at any of the Linux or ns-2 source code for H-TCP during development of our FreeBSD implementation.. Section 2 begins with an over-view of issues surrounding TCP CC research. Based on our implementation experience section 3 summarises the H-TCP algorithm to assist other groups build further interoperable implementations. Section 5 describes our testbed, instrumented to allow precise tracking of various TCP parameters before, during and af-ter congestion events. In section 6 we discuss a number of lessons and trade-offs, and illustrate the H-TCP behaviour shown by our FreeBSD implementation.. In section 7 we illustrate the interesting collateral impact on RTT of using H-TCP instead of NewReno. In particular we show how unrelated UDP flows sharing a congestion point with a long-lived TCP flow may experience lower nett latency, but higher levels of latency variation, when H-TCP is used instead of NewReno.. We believe these insights will prove valuable predictors of H-TCP's potential impact if deployed in consumer endhosts (rather than simply being limited to specialist, highperformance network environments). This paper's main focus is on providing a useful description of our H-TCP implementation that is shown to behave properly under FreeBSD 7.0. In this section we shall first summarise the broader context surrounding our work.

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Generalized Satisfiability for the Description Logic ALC Abstract. The standard reasoning problem, concept satisfiability, in the basic description logic ALC is PSPACE-complete, and it is EXPTIMEcomplete in the presence of unrestricted axioms. Several fragments of ALC, notably logics in the FL, EL, and DL-Lite families, have an easier satisfiability problem; sometimes it is even tractable. We classify the complexity of the standard satisfiability problems for all possible Boolean and quantifier fragments of ALC in the presence of general axioms. where A ∈ N C , R ∈ N R , and • f is a Boolean operator. For a given set B of Boolean operators, a B-concept is a concept that uses only operators from B. A general concept inclusion (GCI) is an axiom of the form C D where C, D are concepts. We use "C ≡ D" as the usual syntactic sugar for "C D and D C". A TBox is a finite set of GCIs without restrictions. An ABox is a finite set of axioms of the form C(x) or R(x, y), where C is a concept, R ∈ N R and x, y ∈ N I . An ontology is the union of a TBox and an ABox. This simplified view suffices for our purposes.. An interpretation is a pair I = (∆ I , · I ), where ∆ I is a nonempty set and · I is a mapping from N C to P(∆ I ), from N R to P(∆ I × ∆ I ) and from N I to ∆ I that is extended to arbitrary concepts as follows:. • f (C 1 , . . . , C n ) I = {x ∈ ∆ I | f ( x ∈ C Let B be a finite set of Boolean operators and Q ⊆ {∃, ∀}. We use Con Q (B), T Q (B) and O Q (B) to denote the set of all concepts, TBoxes and ontologies that use operators in B only and quantifiers from Q only. The following decision problems are of interest for this paper. With Theorems 8, 14, 21, 27 and 32, we have completely classified the satisfiability problems connected to arbitrary terminologies and concepts for ALC fragments obtained by arbitrary sets of Boolean operators and quantifiers-only the fragments emerging around ontologies with existential quantifier and disjunction as only allowed connective resisted a full classification. In particular we improved and finished the study of [29] . In more detail we achieved a dichotomy for all problems using both quantifiers (EXPTIME-complete vs. trivial fragments), a trichotomy when only one quantifier is allowed (trivial, EXPTIME-, and P-complete fragments), and a quartering for no allowed quantifiers ranging from trivial, NL-complete, P-complete, and NP-complete fragments. Furthermore the connection to well-known logic fragments of ALC, e.g., FL and EL now enriches the landscape of complexity by a generalization of these results. These improve the overall understanding of where the tractability border lies. The most important lesson learnt is that the separation of quantifiers together with the separation of either conjunction and disjunction, or the constants, is the only way to achieve tractability in our setting.. Especially in contrast to similar analyses of logics using Post's lattice, this study shows intractable fragments quite at the bottom of the lattice. This illustrates how expressive the concept of terminologies and assertional boxes is: restricted to only the Boolean function false besides both quantifiers we are still able to encode EXPTIME-hard problems into the decision problems that have a TBox and a concept as input. Thus perhaps the strongest source of intractability can be found in the fact that unrestricted theories already express limited implication and disjunction, and not in the set of allowed Boolean functions alone.. For future work, it would be interesting to see whether the picture changes if the use of general axioms is restricted, for example to cyclic terminologiestheories where axioms are cycle-free definitions A ≡ C with A being atomic.. Theories so restricted are sufficient for establishing taxonomies. Concept satisfiability for ALC w.r.t acyclic terminologies is still PSPACE-complete [27] . Is the tractability border the same under this restriction? One could also look at fragments with unqualified quantifiers, e.g., ALU or the DL-lite family, which are not covered by the current analysis. Furthermore, since the standard reasoning tasks are not always interreducible under restricted Boolean operators, a similar classification for other decision problems such as concept subsumption is pending.

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Expectations of Random Sets and Their Boundaries Using Oriented Distance Functions Abstract Shape estimation and object reconstruction are common problems in image analysis. Mathematically, viewing objects in the image plane as random sets reduces the problem of shape estimation to inference about sets. Currently existing definitions of the expected set rely on different criteria to construct the expectation. This paper introduces new definitions of the expected set and the expected boundary, based on oriented distance functions. The proposed expectations have a number of attractive properties, including inclusion relations, convexity preservation and equivariance with respect to rigid motions. The paper introduces a special class of decomposable oriented distance functions for parametric sets and gives the definition and properties of decomposable random closed sets. Further, the definitions of the empirical mean set and the empirical mean boundary are proposed and empirical evidence of the consistency of the boundary estimator is presented. In addition, the paper discusses loss functions for set inference in frequentist framework and shows how some of the existing expectations arise naturally as optimal estimators. The proposed definitions are illustrated on theoretical examples and real data. Boundary reconstruction and shape estimation are frequently encountered problems in image analysis. For example, it is often of interest to determine a characteristic shape of a cell, to reconstruct tissue boundary in medical images, or to identify fault lines from earthquake data. Mathematically, the objects of interest can be viewed as sets, whilst inherent stochasticity of the acquisition process turns them into random entities. The problems of boundary reconstruction and shape estimation thus reduce to inference about random sets.. More specifically, let F (resp. K) be the family of closed (resp. compact) subsets of R d and let the triple (Ω, A, P) denote the probability space. A random closed set is the mapping A : Ω → F such that for every compact set K ∈ K, {ω : A(ω) ∩ K = ∅} ∈ A.. We use "r.c.s." for a random closed set, though some texts prefer RACS. We write A for an r.c.s. A(ω) and reserve capital roman letters for closed subsets of R d . Conventional notations A c , ∂A, cl A, int(A) denote the complement, boundary, closure, and interior, respectively. We write λ(A) ≡ λ d (A) for Lebesgue measure of A, retaining the dimensionality subscript in some cases for clarity. Throughout this paper, we assume that sets are observed on the domain D ⊆ R d . Also, we assume D to be the working domain and write A = {x : . . .} without specifying x ∈ D explicitly. We write x = (x 1 , . . . , x d ) for point x ∈ R d and |·| for the standard Euclidean norm.. Because the space of closed sets is nonlinear, there is no natural way to define the expected set. Currently, there exist a number of definitions of the expectation, though neither can be spoken of as the best. In fact, the definition of the expectation depends on set features that are important to emphasize [21] .. The selection expectation is perhaps the most studied definition [2, 3, 21] . In particular, a random element ξ is called the selection of A, if it belongs to A with probability one. A selection ξ is integrable, if E |ξ | < ∞. The selection expectation, E S [A] , is the closure of the set of the expectations of all integrable selections of A. The selection expectation depends on the structure of the probability space [21, Example 1.14, Sect. 2]. In addition, if the probability space is nonatomic, E S [A] is necessarily convex, even for nonconvex deterministic sets.. In the context of image analysis, the Vorob'ev expectation is perhaps the most intuitive construction [27, 29] . The definition is given in terms of excursion sets of the coverage function, which heuristically corresponds to the choice of the cut-off threshold for the average of the collection of images.. More precisely, the coverage function of an r.c.s. A with the characteristic function χ A = I A (x) is given by. while the excursions sets of the coverage function are defined as. The Vorob'ev expectation, E V [A], of a random closed set A is the excursion set A q , where q ∈ [0, 1] is such that. The Vorob'ev definition provides a criterion to construct an expected set that is optimal with respect to Lebesgue measure. Note also that the optimality criterion (3) ensures that the Vorob'ev expectation ignores sets of measure zero. The distance-average expectation uses a representation of a set given by some function f A : D → R, which we call a representative function of A [6] . Examples of f include the distance function, the oriented distance function, the characteristic function and others. For an r.c. . Examples of a pseudometric m W include L q distances and their variates, Baddeley's Δ q -distance, or any custom definitions [4, 6] . Note that the pseudometric is computed over the window W, which can be the entire domain of interest or its subset. The Vorob'ev expectation can be viewed as a special case of the distance-average definition with. and L 1 distance m; see Example 5.14 in [6] . The distance-average expectation can be used to average nonconvex and nonconnected random sets and is computationally feasible. However, the expectation strongly depends on the choice of the representative function f , window W, (pseudo-) metric m and parameters of m [6] .. The linearization approach gives a unified view of the existing expectations [21] . In particular, consider a mapping . In general, the preimage rarely exists, and we need to define the criterion so that the resulting expectation is optimal in some sense. This is done as follows. Let d be a pseudometric in Y, then the expectation of an r.c.s. A is given by. Minimizing over the entire family F is often nontrivial. Therefore, we can consider optimizing d over a subfamily H ⊂ F of candidate sets. Clearly, the resulting expectation depends on the choice of Banach space Y, mapping f , metric d, subfamily H and any implicit parameters that enter into the calculation.. For compact convex sets, the selection expectation is an example of a linearization approach, where the embedding into a Banach space is realized by support functions of random sets on the unit sphere. Similarly, the Vorob'ev definition is based on the mapping into the space of characteristic functions, while the distance-average expectation uses the embedding into the space of representative functions.. In this paper, we introduce a new definition of the expected set based on oriented distance functions (ODFs). As opposed to a conventional distance function, an ODF takes into account the set and its complement, providing a more informative representation of a set [10] . The ODFs and the topologies they induce have been used in shape analysis and image segmentation [11] [12] [13] .. Although developed independently, the definition is similar in spirit to the distance-average approach proposed in [6] . The new definition, however, is computationally easier and has a number of attractive theoretical properties, not shared by other expectations. In particular, it satisfies some inclusion relations, preserves convexity and remains equivariant with respect to rigid motions.. Furthermore, we give the definition of the expected boundary in terms of ODFs. This is a notable distinction from currently existing definitions that do not address the question of boundary inference, which is often of primary interest in practice. We also introduce a new class of random parametric sets with a special ODF structure and show the connection between an expected set and the expectation of its parameter. In addition, we outline a general framework for set inference and show how different expectations arise as natural estimators for different loss functions. The proposed definitions are illustrated on a number of examples including simulated and real data. In this paper, we present new definitions of the expected set and the expected boundary, using the ODF representation of a set. In conventional settings, where the parameter of interest is a scalar or a vector, it is straightforward to construct an estimator of the mean from the observed data. However, statistical inference about sets is a more difficult problem. When dealing with sets, one has to determine the features that are important to emphasize and develop inference methods based on them. These features may include the location, the size, or the orientation of a set. If a set has an analytic representation, the most natural approach to set inference is via inference about its parameters. This, however, is not applicable to sets with arbitrary geometry.. A number of existing definitions of the expected set can be generalized using the linearization approach, where inference is made on the space of functions representing a set [21] . The resulting estimator necessarily depends on the type of linearization.. In this paper, we use an approach similar in spirit to the linearization idea, however, we forgo the optimality criterion and define the average as the zero-level set of the expected ODF. Whilst the choice might appear simplistic, it gives an expectation with attractive theoretical traits, including equivariance, convexity-preservation and inclusion properties.. In this paper, we assume that acquired sets or images are aligned at the pre-processing step, otherwise the suggested definition will most likely average out the important features, giving erroneous results. The choice of the alignment method depends on the type of the definition used for inference. For example, the alignment procedure and shape inference with respect to the selection expectation were discussed in [26] .. The proposed definition is particularly appealing for problems in image analysis. In particular, its denoising property implies that random specks are averaged out unless observed with probability one. In comparison, the distanceaverage expectation is not empty by construction, and therefore identifies false positives in any noisy image. In turn, the selection expectation depends on the structure of the probability space, whilst the Lebesgue measure criterion of the Vorob'ev expectation is not well suited for image analysis, although, the optimality of the Vorob'ev median is rather attractive.. The example of flashing discs in Sect. 2 is analogous to the situation when the model assumes the presence of a single object, while in reality there are two objects in the image. We argue that an empty set answer of the proposed method implies that the assumed model is incorrect. In comparison, the Vorob'ev estimators (mean and median) strongly depend on the proximity of the objects in the image. They also favor the more "visible" object, ignoring the second disc all together, while the selection expectation has a position that has little meaning with respect to object identification. In addition to its appealing theoretical properties, the ODF expectation can be efficiently computed, using algorithms for the distance function [7, 14, 24] . The ODF is obtained by combining the distance function of the original image and that of the inverted image, i.e. the black and white colors are interchanged. The examples and results reported in this paper were coded in MATLAB (R2008A, The MathWorks), where the distance function to a set is computed using the bwdist command.. In comparison to the distance-average and Vorob'ev expectations, the ODF average is more efficient, because it requires no optimization since the threshold level is fixed at zero. This difference is particularly prominent in Bayesian reconstruction, where the estimator is based on thousands of samples from the posterior. In addition, the storage requirements for the ODF reconstruction in Bayesian framework are minuscule, as we only need to update the average after every sweep.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Selfish load balancing and atomic congestion games We revisit a classical load balancing problem in the modern context of decentralized systems and self-interested clients. In particular, there is a set of clients, each of whom must choose a server from a permissible set. Each client selfishly wants to minimize its own latency (job completion time). A server's latency is inversely proportional to its speed, but it grows linearly with (or, more generally, as the pth power of) the number of clients matched to it. This interaction is naturally modeled as an atomic congestion game, which we call selfish load balancing. We analyze the Nash equilibria of this game and prove nearly tight bounds on the price of anarchy (worst-case ratio between a Nash solution and the social optimum). In particular, for linear latency functions, we show that if the server speeds are relatively bounded and the number of clients is large compared to the number of servers, then every Nash assignment approaches social optimum. Without any assumptions on the number of clients, servers, and server speeds, the price of anarchy is at most 2.5. If all servers have the same speed, then the price of anarchy further improves to 1 + 2/ √ 3 ≈ 2.15. We also exhibit a lower bound of 2.01. Our proof techniques can also be adapted for the coordinated load balancing problem under L2 norm, where it slightly improves the best previously known upper bound on the competitive ratio of a simple greedy scheme. Consider a set U of n selfish clients, each of whom must choose a server from a set V , in the absence of a coordinating authority. There is a bipartite graph G between U and V and a server j is permissible for client i only if (i, j) is an edge in G. Each client selfishly wants to minimize its latency (job completion time), and rationally prefers a fast server to a slower one. Servers can have different speeds, and a server's latency is inversely proportional to its speed, but it is an increasing function of the server load (the number of clients served by it).. Each client independently trying to minimize its latency is essentially engaged in a game with other selfish clients. We call this the selfish load balancing game. Unlike traditional load balancing, however, the clients are not interested in optimizing the social welfare (e.g., total system-wide latency). Instead, each client has its own private objective. The stable outcomes of these interactions are the Nash Equilibriaoutcomes in which no single client can improve its latency by switching unilaterally. Centralized optimal solutions, in general, are not stable-one or more clients may improve their latency by switching, while worsening the latency for others. On the other hand, the cost of Nash equilibrium solutions can be much worse than that of centralized outcomes, and Papadimitriou [11] has coined the term "price of anarchy" to denote the worst-case ratio between a Nash outcome and the social optimum. In this paper, we give nearly tight bounds for the price of anarchy in the selfish load balancing game.. Nash equilibrium is a compelling solution concept for decentralized systems with self-interested players. Unfortunately, the concept is descriptive, not prescriptive: it does not suggest algorithms for computing an equilibrium and computing Nash equilibria remains a topic of current research. We, therefore, also investigate the following obvious greedy strategy: clients arrive in the system online in an arbitrary order; upon arrival, each client selects a permissible server with the least current latency, and this selection is irrevocable. The greedy is a myopic strategy-each client makes the best choice available to it at the moment, although future choices by other clients may make it regret that selection. While greedy does not generally lead to Nash solutions, it does have the advantage of computational sim-plicity. Thus, a natural question to ask is: how bad is the greedy assignment in the worst case?. The greedy strategy has been analyzed before in the context of centralized L2 norm load balancing [3] -the goal there is to assign clients so as to minimize the L2 norm of the server loads. Because the total latency of all the clients is intimately related to the squared sum of the server loads, our techniques also lead to improved bounds for the competitive ratio of the greedy scheme of [3] . Interestingly, the Nash solutions are strictly better, suggesting the following conclusion: despite lack of central coordination, selfish players find solutions that are better than greedy, which assumes centralized control but non-selfish players.

O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O B I I O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A 3d data intensive tele-immersive grid Networked virtual environments like Second Life enable distant people to meet for leisure as well as work. But users are represented through avatars controlled by keyboards and mouses, leading to a low sense of presence especially regarding body language. Multi-camera real-time 3D modeling offers a way to ensure a significantly higher sense of presence. But producing quality geometries, well textured, and to enable distant user tele-presence in non trivial virtual environments is still a challenge today.In this paper we present a tele-immersive system based on multi-camera 3D modeling. Users from distant sites are immersed in a rich virtual environment served by a parallel terrain rendering engine. Distant users, present through their 3D model, can perform some local interactions while having a strong visual presence. We experimented our system between three large cities a few hundreds kilometers apart from each other. This work demonstrate the feasibility of a rich 3D multimedia environment ensuring users a strong sense of presence. 3D networked virtual environments first targeted the gamer community, but applications like Second Life show they can also effectively support social networks and distributed workplaces. Such environments settle the foundations of what may become a new 3D interactive media. However they still lack many features to enable rich human interactions, amongst them the ability to ensure a strong presence of each user, and the possibility to support data intensive environments.. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. User presence is today limited to an avatar controlled by simple devices, mainly a keyboard and a mouse. Though this avatar can show some likeness with its user, its abilities for body language are almost nonexistent, significantly impairing the user virtual presence. Camera-based 3D modeling is a promising way of computing in real-time a 3D clone of the user. The 3D geometry enables full-body mechanical interactions, triggered for instance when detecting collisions with virtual objects. Once textured with the photometric data extracted from the cameras, the 3D model captures the user appearance and conveys its body language. Difficulties include computing a good quality model, transmitting it at a frame rate and a latency ensuring a good level of telepresence. An adapted display environment is also required to experience a convincing 3D immersion in the virtual environment.. Materials for discussions may involve large and complex data sets. For instance coordinating actions to face a natural disaster requires to have access to multiple data, some pre-existing terrain data complemented by simulation data, information from on-line sensors or observations from people on the field. It raises issues related to data access, sharing and aggregation.. Existing tele-immersive systems achieve various levels of presence, depending on the number and layout of cameras and the 3D modeling algorithm used. Some systems offer a free viewpoint on the observed user, enabling only the visual presence of the user [9] . Other systems compute a partial 3D model based on a depth map [10, 6, 3, 13] . It provides a partial visual and geometrical presence (thickness of modeled objects is unknown for instance). Multi-camera systems can achieve a full-body visual and geometrical presence, some point based algorithms rather focusing on visual presence [4] , while mesh based algorithms are more versatile [11, 12] .. Most existing tele-immersive experiments associate distant users in a simple 3D environment with limited interaction capabilities. To our knowledge, the collaborative geoscience experiment between UC Berkeley and UC Davis is one of the few tele-immersive experiments involving complex data 1 .Two persons are interacting around a scientific data set, one being present through its 3D model computed from a multi-camera system, while the other is immersed in a CAVE. This paper and the attached video show that associating multi-camera 3D modeling, data grid and interactions can lead to a rich 3D multimedia experience. We set-up a distributed application involving 3 cities a few hundreds of kilometers apart from each other. At Orléans, a user has simple tele-presence and interaction capabilities (one camera, one large display). The second site at Grenoble hosts a multi-camera system. The user, recorded by the cameras, is modeled in 3D at the camera frame rate. This provides a textured 3D mesh ensuring the user 3D tele-presence on distant sites. On this site the user is equipped with a tracked head-mounted display, ensuring an intense feel of immersion in the shared 3D scene. This site hosts a terrain server providing data to be rendered by each of the three sites. The last location, Bordeaux, also has a multi-camera system for real-time 3D modeling, while visualization takes place on a large display. The user 3D interactions are performed through a touch pad. In the 3D scene, each user can freely move on the terrain and add some simple objects. Users are present in the scene through their 2D or 3D model. Though the networks between the three sites are heterogeneous, the three users were able to interact with a strong feel of presence.. This application is built from various components assembled with FlowVR, a middleware dedicated to large interactive applications [7] . FlowVR associates a hierarchical component approach with a data flow model. It leads to very modular distributed applications, enabling to keep the application complexity under control. A protocol for transferring 3D primitives for rendering enables to forward and combine various 3D data to the rendering hosts [1] . The 3D modeling on Grenoble and Bordeaux is a parallel application running on a cluster to produce meshes at the camera acquisition rate [11] . The terrain application serves data from a large data set relying on a parallel level-of-details algorithm to ensure an interactive service of the terrain [2] .. We first present the application and its architecture (Sec. 2) before discussing some experimental results (Sec. 3) and concluding. Our application involves 2 multi-camera platforms, one at Bordeaux and one at Grenoble, respectively using 8 0.5M pixel and 8 1M pixel cameras. Video acquisition and computation are parallelized on a local cluster. The 3D modeling algorithm produces a mesh associated with a stream of textures. On average for one user standing in the acquisition space, the mesh produced is about 150 Kbit (about 1100 vertex) in Bordeaux and 300 Kbit (about 2600 vertex) in Grenoble. We also send silhouette masks to decode the textures. To reduce network traffic we reduce texture resolution by half, yielding 5 Mbit of texture data per iteration for 8 cameras for Bordeaux and 10 Mbit for Grenoble. When the cameras are running at 16Hz at Bordeaux and 24Hz at Grenoble, this produces streams of 80 Mbit/s and 240 Mbit/s.. The terrain server manages real data from a LIDAR airborne scan with an average resolution of one meter. The data set consists of hundreds of tiles of about 2 millions triangles each. A given point of view usually requires 9 tiles or about 140 MB, which cannot be transmitted over the network at interactive frame rates. Thus, the server relies on a level of detail parallel algorithm that drastically reduces the volume of data to be transfered. The data are reconstructed on the fly according to their distance from the point of view, their relevance and level of details desired. For example, the level of details for the touch pad is lower than for an image displayed on a wall to match each device capabilities. Data are reduced to 80 KB for standard displays and 3.5 KB for the touch pad. For a frame rate of 50 fps, the bandwidth requirement is about 96 Mbit/s for 3 classical points of views and one supplementary point of view for the touch pad. This of course is a peak value attained when all viewpoints are moving simultaneously. The average is lower as the server does not send already cached data when the viewpoints are standing still.. Orléans has a internal gigabit Ethernet network for its cluster and a 100 Mbits/s connection to the Internet. The Bordeaux cluster also uses a gigabit Ethernet network and a 1 Gbits/s connection to the Internet. The cluster at Grenoble has a DDR Infiniband network (20 Gbits/s) and a 10 Gbits/s Ethernet connection to the Internet. Traffic between Bordeaux and Grenoble moves on the dedicated 10 Gbits/s network of the Grid'5000 experimental grid, while Orléans has only access to the regular Internet network. Table 1 shows the average network traffic measured during execution. Grenoble, benefiting from both a high performance network and powerful processing nodes, handles a higher amount of data than the other sites. The other sites suffer network congestions and CPU overloads, and cannot receive the full data streams, triggering adaptive mechanisms integrated in the application to sub-sample data streams.. We also gathered latency data by measuring the time it takes from video acquisition to the rendering of the corresponding 3D model. The latency measures do not include the actual camera acquisition time, the rendering on GPU and the display. Figure 2 shows that latencies vary significantly between Grenoble and Bordeaux, due to the differences in clusters capabilities and network bandwidth. At Grenoble, the latency remains below 100 ms for the local 3D model, and in the order of 100 ms for the one received from Bordeaux. These values are very acceptable for interactions. At Bordeaux, the latency for the local 3D model is about 140ms due to its lower networking and computing capabilities. The 3D model transmitted from Grenoble to Bordeaux leads to a very high latency of about 500ms. We still do not have a clear understanding of this behavior. As expected, latencies at Orléans were high (above 500 ms) due to its very limited network capabilities.. Globally the users were very satisfied with the feel of presence and discussion capabilities offered by this setup (a conferencing audio system complemented the visual feedback). The best conditions were experienced at Grenoble, where frame rates and latencies where the smallest, but also because the HMD provided the user a strong feel of immersion. He really felt like the user from Bordeaux was right in front of him. 3D modeled users were instinctively relying on their body to transmit information (hand pointing or more unconscious postures).

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Construction algorithm for network error-correcting codes attaining the Singleton bound We give a centralized deterministic algorithm for constructing linear network error-correcting codes that attain the Singleton bound of network error-correcting codes. The proposed algorithm is based on the algorithm by Jaggi et al. We give estimates on the time complexity and the required symbol size of the proposed algorithm. We also estimate the probability of a random choice of local encoding vectors by all intermediate nodes giving a network error-correcting codes attaining the Singleton bound. We also clarify the relationship between the robust network coding and the network error-correcting codes with known locations of errors. Ahlswede et al. [1] proposed the notion of network coding that multicasts data from a single sender to multiple receivers at a rate at which the ordinary store and forward routing cannot multicast the data. Such high rate multicast becomes feasible by allowing intermediate nodes to encode and decode the data. A sender is usually called a source and a receiver is called a sink. A network coding is said to be linear if every intermediate node outputs a linear combination of its inputs [10] .. A study of network coding usually assumes that an error does not occur in networks. Recently, Cai and Yeung [2] , [13] considered errors in network coding, and proposed the network error correcting codes that allow sinks to recover the information even when errors occur on intermediate edges in the network. After formulating the network error correction, they proposed the lower and upper bounds on the number of messages in a network α-error correcting code, and one of their upper bound was a natural generalization of the Singleton bound for the ordinary error-correcting codes. Recently, Zhang [14] and Yang et al. [11] independently observed that the Singleton bound can be refined. We note that the problem formulation in [2] , [13] was later independently presented in [4] . (The proceedings paper of [2] , [13] appeared in 2002. ) Cai and Yeung mostly considered the case that intermediate nodes perform only simple encoding and decoding without delay, such as computing the output of the node as a linear combination of its inputs, and the sinks perform complex decoding computation. The network error correcting codes can avoid introducing decoding computation and delay into intermediate nodes, which is the advantage over use of ordinary error correcting codes between nodes.. Note that a similar type of network failure in a slightly different context was considered in [7, Sect. V] and [6, Sect. VI] in which every sink is assumed to know the set of failed edges and failed edges are assumed to emit zero symbols. Network error correction does not assume the knowledge of edges causing errors, and the problem formulation is different from [6] , [7] . Note also that Kurihara [8] considered the different notion of robustness. In his paper, he considered network coding that allows sinks to recover partial information with edge failures.. For the construction of the network errorcorrecting codes, Jaggi et al. [5] proposed a randomized construction that uses coding among different time intervals. Their method produces codes attains the Singleton bound with high probability with sufficiently long block length, where the block length refers to the number of time intervals among which coding is done. It is desirable to have a network error-correcting code that does not code among different time intervals and thus does not introduce delay. Concurrently to this paper, Yang et al. [11] proposed an explicit construction algorithm that produces codes attaining the refined Singleton bound. The idea in [11] is similar to this paper in the sense that they also regard errors as information from the source and add extra components in the global encoding vectors corresponding to errors.. In this paper, we give a deterministic and centralized algorithm that constructs a network errorcorrecting code that attains the Singleton bound of network error-correcting codes obtained in [13] . We also give a relationship between the success probability and the field size for successful construction of network error-correcting codes when intermediate nodes choose their encoding coefficients randomly and independently. The proposed algorithms are based on [6] . Our network error-correcting codes make multicast robust to errors without introducing delay in the transmission, which is very attractive to delay sensitive multicast applications, such as multicast of video or audio. Our method is also useful for cryptographic applications, because it can tolerate modification and deletion of data by an adversary.. This paper is organized as follows. Section 2 introduces notations and the model of errors. Section 3 proposes an algorithm for constructing network errorcorrecting codes attaining the Singleton bound. Section 4 shows how to modify the algorithm in Sect. 3 to attain the refined Singleton bound, the success probability of the random construction of network error-correcting codes, and the relationship between the robust network coding [6] , [7] and the network error-correcting codes with known locations of errors [12] . Section 5 gives concluding remarks. In this paper, we proposed an algorithm constructing network error-correcting codes attaining the Singleton bound, and clarified its relation to the robust network coding [6, Sect. VI].. There are several research problems that have not been addressed in this paper. Firstly, the proposed deterministic algorithm requires tests of linear independence against |E| 2α sets consisting of k + m F t vectors, which is really time consuming. It is desirable to have a more efficient deterministic construction algorithm.. Secondly, since there seems no structure in the constructed code, the decoding of the original information at a sink t requires the exhaustive search by t for possible information from the source and possible errors. It is desirable to have a code with structure that allows efficient decoding.. Finally, the case |T | = 1 and |E| = n includes the ordinary error correcting codes as a special case. Substituting |T | = 1, |E| = n and 2α = n − k into Eq. (1) gives q > n n−k , which can be regarded as a sufficient condition for the existence of the MDS linear code. On the other hand, a well-known sufficient condition for the existence of the MDS linear code is q > n − 2, which suggests that Eq. (1) is loose and that there is a room for improvement in Eq. (1).

O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


The intelligent water drops algorithm: a nature-inspired swarm-based optimization algorithm Abstract: A natural river often finds good paths among lots of possible paths in its ways from the source to destination. These near optimal or optimal paths are obtained by the actions and reactions that occur among the water drops and the water drops with the riverbeds. The intelligent water drops (IWD) algorithm is a new swarm-based optimisation algorithm inspired from observing natural water drops that flow in rivers. In this paper, the IWD algorithm is tested to find solutions of the n-queen puzzle with a simple local heuristic. The travelling salesman problem (TSP) is also solved with a modified IWD algorithm. Moreover, the IWD algorithm is tested with some more multiple knapsack problems (MKP) in which near-optimal or optimal solutions are obtained.Keywords: swarm intelligence; intelligent water drops; IWDs; travelling salesman problem; TSP; multiple knapsack problem; MKP; n-queen puzzle.Reference to this paper should be made as follows: Shah-Hosseini, H. (2009) 'The intelligent water drops algorithm: a nature-inspired swarm-based optimisation algorithm', Int. J. The natural systems that have developed for so long are one of the rich sources of inspiration for inventing new intelligent systems. Swarm intelligence is one of the scientific fields that are closely related to natural swarms existing in nature, such as ant colonies, bee colonies, brain and rivers. Among the problem solving techniques inspired from nature are evolutionary computation (Eiben and Smith, 2003) , neural networks (Haykin, 1999) , time adaptive self-organising maps (Shah-Hosseini, 2006) , ant colony optimisation (Dorigo and Stutzle, 2004) , bee colony optimisation (Sato and Hagiwara, 1997) , particle swarm optimization (Eberhart and Kennedy, 1995) , DNA computing (Adleman, 1994) , electromagnetism-like optimisation (Birbil and Fang, 2003) and intelligent water drops (Shah-Hosseini, 2007) .. One of the recently proposed algorithms in the field of the swarm intelligence is the intelligent water drops (IWDs) algorithm (Shah-Hosseini, 2007; Shah-Hosseini, 2008) . The IWD algorithm is based on the dynamic of river systems, actions and reactions that happen among the water drops in rivers. The natural water drops are used to develop IWD and the IWDs cooperate together to reach a better solution for a given problem. The IWD algorithm may be used for maximisation or minimisation problems. The solutions are incrementally constructed by the IWD algorithm. Therefore, the IWD algorithm is a population-based constructive optimisation algorithm.. The IWD algorithm has been used for the travelling salesman's problem (TSP) (Shah-Hosseini, 2007) and multiple (or multidimensional) knapsack problem (MKP) (Shah-Hosseini, 2008 ) with promising results. Both the TSP and the MKP are NP-hard combinatorial optimisation problems. In the TSP, a map of cities is given to the salesman and he has to visit all the cities only once to complete a tour such that the length of the tour is the shortest among all possible tours for this map. In the MKP, there are multiple knapsacks and several items and the goal is to include some of the items in the knapsacks to achieve maximum profit with the constraint that none of the knapsacks becomes overflowed. Both the TSP and MKP are often used for testing optimisation algorithms. In this paper, the IWD-TSP algorithm (Shah-Hosseini, 2007 ) is modified to get better results for the TSPs. Moreover, the IWD-MKP algorithm (Shah-Hosseini, 2008 ) is used for some other MKPs to observe its power for getting optimal or near-optimal solutions.. The n-queen problem, an NP-complete problem, is also used here to test the performance of the IWD algorithm. The IWD algorithm that is used for solving the n-queen is called the 'IWD-NQ' algorithm. The n-queen puzzle (problem) needs to place n queens on a chessboard of size n × n such that no two queens attack each other. Therefore, the goal is to obtain the global optimal solution and no near-optimal solution is acceptable.. The next section reviews some processes that occur in a river, which involves the water drops of the river and explains IWDs that have been developed based on the ideas of natural water drops. Section 3 introduces the IWD algorithm, which uses IWDs for problem solving. Section 4 shows how to use the IWD algorithm for solving three different optimisation problems. Experimental results with the proposed IWD algorithm for artificial and standard TSPs, MKPs, and n-queen problems form Section 5. Concluding remarks are the final section of the paper. The IWD algorithm is an optimisation algorithm that uses a swarm of water drops to collectively search for optimal solutions in the environment of the given problem. In fact, each IWD constructs incrementally a solution to the problem by moving on the graph representation of the problem. Then, among the obtained solutions, the best one in terms of quality is chosen and its total path is reinforced by soil removal. During each iteration of the IWD algorithm, an IWD gains some velocity and removes some soil from the path it flows on. After enough iterations of the IWD algorithm, the IWDs find the good paths that are decoded to good solutions of the problem. Three different problems are used to experiment the IWD algorithm, the n-queen puzzle, the TSP and the MKP. Here, for the first time, the IWD algorithm is used for solving n-queen problems having a simple heuristic and a local search algorithm. It is shown that the MKP-NQ algorithm can find the global optimal solutions of the n-queen puzzle. However, by suggesting better heuristics and local search algorithms the number of iterations may be reduced.. The IWD algorithm is modified for the TSP problem and this MIWD-TSP gets better tours with shorter lengths in comparison to the standard MKP-TSP algorithm. Moreover, some new MKPs are tested with the IWD-MKP algorithm and it is observed that the algorithm is able to find optimal or near optimal solutions for the given MKPs. This paper indicates that the IWD algorithm is capable to deal with optimisation problems in finding solutions with good or optimal qualities. However, there is an open space for modifications in the standard algorithm, embedding other mechanisms that exist in natural rivers and/or inventing local heuristics that fit better with the IWD algorithm. It also demonstrates that the nature is an excellent teacher for designing and inventing new swarm-based optimisation algorithms.

O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I I B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B I B I I B B I I B B I I B O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O B I O O B I I B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O B O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B I O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Study on the mathematical model of the effects of NPK on winter cauliflower In order to study the effects of nitrogen (N), phosphorus (P) and potassium (K) on the yield, the appearance eligibility ratio and the contents of total glucosinolates and glucoraphanin of cauliflower, the 311-A optimization regression design was used to conduct field experiments of N, P, K fertilizer treatments using winter cauliflower variety ''HB16''. The models of the balanced application for N, P and K fertilizers were set up. According to the different production targets, the different optimal combinations of applying NPK fertilizers could be carried out. The results showed that the optimal combination of applying fertilizer was 372.83 kg/hm 2 N, 89.65 kg/hm 2 P 2 O 5 and 201.41 kg/hm 2 K 2 O, the highest yield was 19089.25 kg/hm 2 . When the optimal combination of applying fertilizers was 434.20 kg/hm 2 N, 90.11 kg/hm 2 P 2 O 5 , 213.92 kg/hm 2 K 2 O, the eligibility products ratio could reach the highest at 84.95%. It was found that there were significant effects on the contents of total glucosinolates and glucoraphanin by applying N fertilizer, but not by P and K fertilizers. When N fertilizer application reached 408.78 kg/hm 2 and 394.62 kg/hm 2 , total glucosinolates and glucoraphanin content would get to maximum, respectively. Cauliflower (Brassica oleracea var. botrytis) is an important field vegetable in China. Nitrogen (N), phosphorus (P) and potassium (K) were vital factors which affected the yield and quality of cauliflower. It has been shown that increasing N, K fertilizer application could increase cauliflower yield [1] [2] [3] [4] . However overcommitting fertilizer would reduce yield and bring negative effect on soil. Developing scientific fertilizer effect function is the base of prescribing fertilizer recommendations for cauliflower.. In the past, lots of research have been done on yield response models for balanced fertilization in various arable crops [5, 6] , but relatively little work on this subject has been done for field vegetables including cauliflower. These different forms of fertilizer response models reflect the relations between fertilizer rates and yield or quality [7, 8] . Fertilizer response function could be obtained by multi-factor quadrature, regression design method, etc. In the method of regression design, based on the multiple levels experimental design, the yield of different treatments was statistically analyzed, the function relation between fertilizer and yield could be set up [9] . According to the functional equation, the single and interaction effects of NPK could be directly deduced, and the appropriate fertilization recommendations with optimum fertilizer rate and optimum NPK proportion could be obtained. In addition, the yield, quality and profit could be analyzed by means of marginal effect [10, 11] .

O O O B I O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Global optimization of polynomials restricted to a smooth variety using sums of squares Let f1, . . . , fp be in Q [X], where X = (X1, . . . , Xn) t , that generate a radical ideal and let V be their complex zero-set. Assume that V is smooth and equidimensional. Given f ∈ Q[X] bounded below, consider the optimization problem of computing f = inf x∈V ∩R n f (x). For A ∈ GLn(C), we denote by f A the polynomial f (AX) and by V A the complex zero-set of f Hence, we can obtain algebraic certificates for lower bounds on f using semidefinite programs. Some numerical experiments are given. We also discuss how to decrease the number of polynomials in M A i . Motivation and Problem statement. Consider the global constrained optimization problem. where f ∈ Q[X 1 , . . . , X n ] is bounded below and V ⊂ C n is an algebraic variety given by a set of defining equations f 1 = · · · = f p = 0 in Q[X 1 , . . . , X n ].. Given a ∈ R, providing algebraic certificates of positivity for f − a over V ∩ R n allowing certification of lower bounds on f (i.e. a ≤ f ) is a question of first importance since it arises in several applications of engineering sciences (e.g. control theory Henrion and Garulli (2005) ; or static analysis of programs Cousot (2005) ; Monniaux (2010) ). This problem can be solved in theory through the Positivstellensatz (Bochnak et al., 1998, Chapter 4) . The issue is that computing such an algebraic certificate of positivity is empirically known to be computationally expensive. Our approach fits in the framework of sums of squares decompositions of multivariate polynomials through a relaxation to semi-definite programming (see Shor (1987) ; Parrilo (2000) ; Lasserre (2001) ; Parrilo and Sturmfels (2003) for the semi-definite relaxations methods). The goal is to obtain algebraic certificates of positivity by means of sums-of-squares decompositions which could be easier to compute.. In this context, the issue is to provide results ensuring the existence of algebraic certificates of positivity by means of sums of squares decompositions. For instance, it is well-known that not all positive polynomials are sums-of-squares of polynomials. Nevertheless, in the univariate case, positive polynomials are sums-of-squares (see Hilbert (1888) ). This gives the intuition that over regions of "small dimension" positive polynomials can be written as sums-of-squares of polynomials.. Thus, the idea is to consider additional constraints to define subsets of V ∩ R n of smaller dimension so that one can ensure two properties:. • if f − a is positive over these subsets then a ≤ f ;. • There exist sum-of-squares certificates for the positivity of f − a on these subsets. Under these conditions, one can certify that a is a lower bound for f .. Prior works. This approach has been previously developed in the case where f is reached. We denote by ∇f the ideal ∂f ∂X1 , . . . , ∂f ∂Xn . Nie et al. (2006) prove that either f is positive over V ∂f ∂X1 , . . . , ∂f ∂Xn , or f is non-negative over V ∂f ∂X1 , . . . , ∂f ∂Xn and ∇f is radical, then f is a sum of squares of polynomials modulo ∇f . Note that if the infimum is reached, it is reached over V ∂f ∂X1 , . . . , ∂f ∂Xn ∩ R n . Then over the gradient variety, f − f can be written as a sum of squares and outside the gradient variety, it is necessarily greater than 0. Here the local certificate is actually a global certificate of non-negativity. These results have been recently generalized for the constrained case that we are considering in this paper in Nie (2010) but still with the assumption that the global infimum f is reached. When one does not know a priori if f attains a minimum, one has to take into account asymptotic phenomena. To do that, Schweighofer (2006) replaces the gradient variety with its gradient tentacle. Over the gradient tentacle, a positive polynomial for which its values "at infinity" is a finite subset of R >0 , (see point (3) in our Proposition 1.3 for a formal definition) belongs to the preordering generated by the polynomials defining the gradient tentacle. Hà and Pha . m (2009) follow the approach initiated by Schweighofer with their truncated tangency variety, which are subsets of the region defined by the constraints of smaller dimension and on which the target function f has a finite number of values "at infinity". These truncated tangency varieties are related to critical loci of the square of distance functions to a given point, say (a 1 , . . . , a n ). They are defined by considering (n− d + 2, n − d + 2) minors of the Jacobian matrix associated to f 1 , . . . , f p , f,. 2 . Considering simpler critical loci of linear projections leads to consider only (n − d + 1, n − d + 1)-minors of the Jacobian matrix associated to f 1 , . . . , f p , f . This may lead to simpler algebraic certificates and a better numerical behavior of programs computing numerical approximations of sums-of-squares decompositions via semi-definite programming.. In Guo et al. (2010) , we successfully reached this goal in the unconstrained case. In this paper, we go further and investigate the constrained case which is conceptually harder.. The subsets of V that we consider are related to critical loci of linear projections. This is related to the notion of polar variety already investigated for the real root finding problem in the solution of polynomial systems using Computer Algebra techniques (see e.g. Bank et al. (1997) ; Safey El Din and Schost (2003); Bank et al. (2005 Bank et al. ( , 2010 ). We provide several numerical experiments showing the relevance of our approach. Before describing in detail our contributions we need to introduce some definitions.. Basic definitions, assumptions and notations. We need a few definitions and refer to Zariski and Samuel (1958) ; Mumford (1976) ; Shafarevich (1977) ; Eisenbud (1995) for standard notions which are not recalled here. An algebraic variety V ⊂ C n is the set of common zeros of some polynomial equations f 1 , . . . , f p in variables X 1 , . . . , X n ; we write V = V (f 1 , . . . , f p ) and d its dimension. Moreover, we assume in the sequel that the ideal f 1 , . . . , f p is radical.. The Zariski-tangent space to V at x ∈ V is the vector space T x V defined by the equations ∂f ∂X1 (x)v 1 + · · · + ∂f ∂Xn (x)v n = 0, for all polynomials f that vanish on V . We will only consider equidimensional algebraic varieties. In this context, the regular points on V are those points x where dim(T x V ) = dim(V ); the singular points are all other points. The set of singular points is defined as the set of points on V where all (n−d, n−d)-minors of the Jacobian matrix ∂fi ∂Xj 1≤i≤p,1≤j≤n. vanish. An equidimensional variety V such that its set of singular points is empty will be said to be smooth.. For A ∈ GL n (Q) and g ∈ Q[X 1 , . . . , X n ], we denote by g A the polynomial g(AX) where X = (X 1 , . . . , X n ) t . In the sequel, the algebraic variety. . . , X n ] and a non-negative integer k ≤ n, jac(F, [X k , . . . , X n ]) denotes the truncated Jacobian matrix ∂fi ∂Xj 1≤i≤p,k≤j≤n . Given a matrix M and an integer r, we denote by Minors(M, r) the set of (r, r)-minors of M.. In the sequel, we suppose that the set of polynomials. . . , X n ] satisfies the following regularity assumptions R: R 1 : the ideal f 1 , . . . , f p is radical and equidimensional; we denote its dimension by d;. we denote by M A i the set of polynomials which is the union of. • the sequence of variables X 1 , . . . , X i−1 .

O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O


Approximate

Probabilistic

Optimization

Using

Exact-Capacity-Approximate-Response-Distribution

(ECARD)

There

are

two

major

barriers

in

front

of

probabilistic

structural

design.

First,

uncertainties

associated

with

errors

in

structural

and

aerodynamic

modeling

and

quality

of

construction

are

not

well

characterized

as

statistical

distributions

and

insufficient

information

may

lead

to

large

errors

in

probability

calculations.

Second,

probabilistic

design

is

computationally

expensive

because

repeated

stress

calculations

(typically

finite

element

analysis)

are

required

for

updating

probability

calculation

as

the

structure

is

being

re-designed.

Targeting

these

two

barriers,

we

propose

a

probabilistic

design

optimization

method,

where

the

probability

of

failure

calculation

is

confined

to

failure

stresses,

to

approximate

probability

of

failure

of

probabilistic

design

P

f

p

probability

of

failure

at

previous

design

P

f

actual

probability

of

failure

P

fd

probability

of

failure

at

given

deterministic

design

W

d

weight

of

deterministic

design

W

weight

of

probabilistic

design

here

are

two

major

barriers

in

front

of

probabilistic

(or

reliability-based)

structural

design.

First,

uncertainties

associated

with

errors

in

structural

and

aerodynamic

modeling

and

quality

of

construction

are

not

well

characterized

as

statistical

distributions,

and

it

has

been

shown

that

insufficient

information

may

lead

to

large

errors

in

probability

calculations

(e.g.,

Ben-Haim

and

Elishakoff

Neal,

et

al.

Due

to

this

fact,

many

engineers

are

reluctant

to

pursue

probabilistic

design.

The

second

barrier

to

the

application

of

probabilistic

structural

optimization

is

computational

expense.

Probabilistic

structural

optimization

is

expensive

because

repeated

stress

calculations

(typically

FEA)

are

required

for

updating

probability

calculation

as

the

structure

is

being

re-designed.

Targeting

these

two

main

barriers,

we

propose

an

approximate

method

that

dispenses

with

expensive

probabilistic

stress

calculations.

In

the

proposed

method,

the

probabilistic

calculation

is

confined

only

to

failure

stress,

which

is

often

well

characterized..

Traditionally,

reliability-based

design

optimization

(RBDO)

is

performed

based

on

a

double-loop

optimization

scheme,

where

the

outer

loop

is

used

for

design

optimization

while

the

inner

loop

performs

a

sub-optimization

for

reliability

analysis,

using

methods

such

as

First-Order

Reliability

Method

(FORM).

Since

this

traditional

approach

is

computationally

expensive,

even

prohibitive

for

problems

that

require

complex

finite

element

analysis

(FEA),

alternative

methods

have

been

proposed

by

many

researchers

(e.g.,

Lee

and

Kwak

Kiureghian

et

al.

Tu

et

al.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Instantaneous frequency estimation using higher order L-Wigner distributions with data-driven order and window length The L-Wigner distributions are defined in order to improve the concentration of signal's time-frequency representation. For a finite distribution order and nonlinear instantaneous frequency (IF) it gives biased IF estimates. In the case of noisy signals the optimal window length and distribution order depend on the noise variance and unknown IF. In this article an adaptive IF estimator with the time-varying and data-driven window length and distribution order is developed. Based on the analysis that has been done here, lower order time-frequency distributions are introduced. Since there is no distribution from the Cohen class (with signal-independent kernel) which can produce the complete concentration along the instantaneous frequency (IF) when it is a nonlinear function of time, [5] - [7] , [21] , various higher order distributions have been derived. For the analysis of signals with polynomial phase the polynomial Wigner-Ville distributions are proposed by Boashash et al. in [2] - [4] . The same class of signals may be treated by the local polynomial distributions, defined in [10] and [11] . The L-Wigner distribution, introduced and described in [19] , [20] , [22] , and [23] , significantly reduces influence of higher order terms in the phase function when it is a nonlinear function of time. The polynomial Wigner-Ville distribution, and the L-Wigner distributions are closely related to the time-varying higher order spectra [3] , [17] .. In this correspondence we analyze the IF estimator, in the case of additive noisy signals, using the L-Wigner distribution. The estimator's variance and bias are highly dependent on the window length and distribution order. Provided that the signal and noise parameters are known, the optimal window length and/or distribution order may be determined by minimizing the estimation mean squared error. However, those parameters are not available in advance. Especially it is true for the IF derivatives that determine the estimation bias. Here, we will present an adaptive algorithm which does not require a priori knowledge of the estimation bias. The basic idea of the method that we use for selection of the data-driven window length and order has originated from [8] and [9] , where it was proposed for the nonparametric regression. The idea of this method was exploited in [12] for development of the adaptive local polynomial periodogram, giving estimates of the IF and its derivative. It was subsequently used in [13] for development of the nonparametric estimator of the IF, based on the Wigner distribution with the data-driven adaptive window size. A discrete nature of optimization parameters of the L-Wigner distribution (window length and distribution order), along with a small number of their possible values, resulted in a modification of the original algorithm. Also, the analysis presented in this correspondence leads to an interesting conclusion that not only higher order distributions may improve time-frequency presentation, but also "lower order" distributions can be the best choice in some cases.. The correspondence is organized as follows. A review of the L-Wigner distribution definition, along with a noise modeling, is done in Section II. In Section III the variance and bias of the IF estimate, using the L-Wigner distributions, are derived. The optimal window length and distribution order are discussed in this section. A numerical implementation of the L-Wigner distribution is presented in Section IV. An adaptive IF estimator, with the data-driven window size and distribution order is described in Section V. Numerical examples are presented in Section VI.. A technique has been proposed recently [1] which provides several benefits for orthogonal frequency division multiplexing (OFDM). In particular, this technique, termed polynomial cancellation coding (PCC), reduces the intercarrier interference (ICI) due to frequency shift between transmitter and receiver [2] . Further, PCC has been discussed in the context of reduction to out-of-band power and intersymbol interference in OFDM systems [3] . Used in its simplest form, PCC achieves these advantages at the cost of bandwidth efficiency. However, the advantages can be retained while maintaining, or even increasing, bandwidth efficiency if the symbol periods of the PCC coded data are overlapped [3] and an equalizer is used at the receiver to recover the transmitted data from the overlapped symbols.. The main idea of PCC is to map each complex number which is to be transmitted onto a group of k subcarriers, with appropriate weightings, rather than to a single subcarrier. In a previous article [2] , the weightings have been given as the coefficients of the polynomial (1 0 x) k01 . It has been claimed that if the same weightings are applied in decoding the received signal, polynomial variation of order (2k 0 3) in the ICI is canceled.. In this correspondence, we explain how ICI cancellation is achieved by relating PCC to finite-difference techniques, well known in numerical analysis. In Section II, we give mathematical expressions for polynomial cancellation coding as described in [2] . In Section III we list some standard results in finite difference theory, and recast our equations in this language. Finally, in Section IV we discuss the reduced ICI obtained by use of PCC. The L-Wigner distribution with the data-driven and time-varying window length and order is presented, as an adaptive estimator of the IF. The choice of the window length and the distribution order is based on the intersection of the confidence intervals of the IF estimates. The developed algorithm uses only the formula for the asymptotic variance of the IF estimates. Simulations show a significant accuracy improvement of the adaptive algorithm.

B I B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O