<Emphasis Type="Italic">f</Emphasis>-Sensitivity Distance Oracles and Routing Schemes

An f-sensitivity distance oracle for a weighted undirected graph G(V,E) is a data structure capable of answering restricted distance queries between vertex pairs, i.e., calculating distances on a subgraph avoiding some forbidden edges. This paper presents an efficiently constructible f-sensitivity distance oracle that given a triplet (s,t,F), where s and t are vertices and F is a set of forbidden edges such that |F|≤f, returns an estimate of the distance between s and t in G(V,E∖F). For an integer parameter k≥1, the size of the data structure is O(fkn ^1+1/k log (nW)), where W is the heaviest edge in G, the stretch (approximation ratio) of the returned distance is (8k−2)(f+1), and the query time is O(|F|⋅log ² n⋅log log n⋅log log d), where d is the distance between s and t in G(V,E∖F).     

Image Compression and Reconstruction Using pi<Subscript><Emphasis Type="Italic">t</Emphasis></Subscript>-Sigma Neural Networks

A high-order feedforward neural architecture, called pi _t -sigma (π _t σ) neural network, is proposed for lossy digital image compression and reconstruction problems. The π _t σ network architecture is composed of an input layer, a single hidden layer, and an output layer. The hidden layer is composed of classical additive neurons, whereas the output layer is composed of translated multiplicative neurons (π _t -neurons). A two-stage learning algorithm is proposed to adjust the parameters of the π _t σ network: first, a genetic algorithm (GA) is used to avoid premature convergence to poor local minima; in the second stage, a conjugate gradient method is used to fine-tune the solution found by GA. Experiments using the Standard Image Database and infrared satellite images show that the proposed π _t σ network performs better than classical multilayer perceptron, improving the reconstruction precision (measured by the mean squared error) in about 56%, on average.     

The <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-Optimality of the IIPG Method for Odd Degrees of Polynomial Approximation in 1D

The paper deals with a numerical analysis of the incomplete interior penalty Galerkin (IIPG) method applied to one dimensional Poisson problem. Based on a particular choice of the interior penalty parameter σ (order of O(h ⁻¹)), we derive the optimal error estimate in the L ²-norm for odd degrees of polynomial approximation for locally quasi-uniform meshes. Moreover, we show that only the mentioned choice of the penalty parameter leads to optimal orders of convergence. Finally, presented numerical experiments verify the theoretical results.     

Low-rank Kronecker-product Approximation to Multi-dimensional Nonlocal Operators. Part II. HKT Representation of Certain Operators

This article is the second part continuing Part I [16]. We apply the -matrix techniques combined with the Kronecker tensor-product approximation to represent integral operators as well as certain functions F(A) of a discrete elliptic operator A in a hypercube (0,1) ^d ∈ ℝ ^d in the case of a high spatial dimension d. We focus on the approximation of the operator-valued functions A ^{− σ} , σ>0, and sign (A) for a class of finite difference discretisations A ∈ ℝ ^{N × N} . The asymptotic complexity of our data-sparse representations can be estimated by (n ^p log ^q n), p = 1, 2, with q independent of d, where n=N ^{1/ d} is the dimension of the discrete problem in one space direction.     

Michell cantilevers constructed within trapezoidal domains—Part I: geometry of Hencky nets

The present paper is the first part of the four-part work on Michell cantilevers transmitting a given point load to a given segment of a straight-line support, the feasible domain being a part of the half-plane contained between two fixed half-lines. The axial stress σ in the optimal cantilevers is assumed to be bounded by −σ _C ≤σ≤σ _T , where σ _C and σ _T represent the allowable compressive and tensile stresses, respectively. The work provides generalization of the results of the article of Lewiński et al. (Int J Mech Sci 36:375–398, 1994a) to the case of σ _T ≠σ _C . The present, first part of the work concerns the analytical formation of the Hencky nets or the lines of fibres filling up the interior of the optimal cantilevers corresponding to an arbitrary position of the point of application of the given concentrated force.     

Hybrid molecular dynamics-continuum simulation for nano/mesoscale channel flows

The present study deals with multiscale simulation of the fluid flows in nano/mesoscale channels. A hybrid molecular dynamics (MD)-continuum simulation with the principle of crude constrained Lagrangian dynamics for data exchange between continuum and MD regions is performed to resolve the Couette and Poiseuille flows. Unlike the smaller channel heights, H < 50σ (σ is the molecular length scale, σ ≈ 0.34 nm for liquid Ar), considered in the previous works, this study deals with nano/mesoscale channels with height falling into the range of 44σ ≤ H ≤ 400σ, i.e., O(10)–O(10²) nm. The major concerns are: (1) to alleviate statistic fluctuations so as to improve convergence characteristics of the hybrid simulation—a novel treatment for evaluation of force exerted on individual particle is proposed and its effectiveness is demonstrated; (2) to explore the appropriate sizes of the pure MD region and the overlap region for hybrid MD-continuum simulations—the results disclosed that, the pure MD region of at least 12σ and the overlap region of the height 10σ have to be used in this class of hybrid MD-continuum simulations; and (3) to investigate the influences of channel height on the predictions of the flow field and the slip length—a slip length correlation is formulated and the effects of channel size on the flow field and the slip length are discussed. An erratum to this article can be found at     

On signal reconstruction in white noise using dictionaries

Assume that we observe a Gaussian vector Y = Xβ + σζ, where X is a known p × n matrix with p ≥ n, β ∈ ℝ ⁿ is an unknown vector, and ζ ∈ ℝ ⁿ is a standard Gaussian white noise. The problem is to reconstruct Xβ from observations Y, provided that β is a sparse vector.     

Explaining the lazy Krivine machine using explicit substitution and addresses

In a previous paper, Benaissa, Lescanne, and Rose, have extended the weak lambda-calculus of explicit substitution λ σ _w with addresses, so that it gives an account of the sharing implemented by lazy functional language interpreters. We show in this paper that their calculus, called λ σ _w ^a , fits well to the lazy Krivine machine, which describes the core of a lazy (call-by-need) functional programming language implementation. The lazy Krivine machine implements term evaluation sharing, that is essential for efficiency of such languages. The originality of our proof is that it gives a very detailed account of the implemented strategy.     

Generating Combinations by Three Basic Operations

We investigate the problem of listing combinations using a special class of operations, prefix shifts. Combinations are represented as bitstrings of O＇s and l＇s, and prefix shifts are the operations of rotating some prefix of a bitstring by one position to left or right. We give a negative answer to an open problem asked by F. Ruskey and A. Williams （Generating combinations by prefix shifts, In Proc. llth Annual International Computing and Combinatorics Conference 2005, LNCS 3595, Springer, 2005, pp.570-576）, that is whether we can generate combinations by only using three very basic prefix shifts on bitstrings, which are transposition of the first two bits and the rotation of the entire bitstring by one position in either direction （i.e., applying the permutations σ2, σn and σn^-1 to the indices of the bitstrings）.     

Michell cantilevers constructed within a half strip. Tabulation of selected benchmark results

The paper delivers the benchmark results for the Michell cantilevers constructed within a half strip, for selected values of the σ _T /σ _C ratio, σ _T , σ _C being the admissible stresses in tension and compression, respectively.     

Critical Configurations for 1-View in Projections from ℙ<Superscript>k</Superscript> → ℙ<Superscript>2</Superscript>

In this paper we describe, from a theoretical point of view, critical configurations for the projective reconstruction of a set of points, for a single view, i.e. for calibration of a camera, in the case of projections from ℙ^k to ℙ² for k ≥ 4. We give first a general result describing these critical loci in ℙ^k, which, if irreducible, are algebraic varieties of dimension k−2 and degree 3. If k=4 they can be either a smooth ruled surface or a cone and if k = 5 they can be a smooth three dimensional variety, ruled in planes, or a cone. If k≥ 6, the variety is always a cone, the vertex of which has dimension at least k − 6. The reducible cases are studied in Appendix A. These results are then applied to determine explicitly the critical loci for the projections from ℙ^k which arise from the dynamic scenes in ℙ³ considered in [13]. Marina Bertolini is currently Associate Professor of Geometry at the Department of Mathematics at the Università degli Studi di Milano, Italy. Her main field of research is Complex Projective Algebraic Geometry, with particular interest for the classification of projective varieties and for the geometry of Grassmann varieties. On these topics M. Bertolini has published more than twenty reviewed papers on national and international journals. She has been for some years now interested also in applications of Algebraic Geometry to Computer Vision problems. Cristina Turrini is Associate Professor of Geometry at the Department of Mathematics of Università degli Studi di Milano, Italy. Her main research interest is Complex Projective Algebraic Geometry: subvarieties of Grassmannians, special varieties, automorphisms, classification. In the last two years she has started to work on applications of Algebraic Geometry to problems of Computer Vision. She is author or co-author of about thirty reviewed papers. She is also involved in popularization of Mathematics, and on this subject she is co-editor of some books.     

Computing multiple pitchfork bifurcation points

A point (x*,λ*) is called apitchfork bifurcation point of multiplicityp≥1 of the nonlinear systemF(x, λ)=0,F:ℝⁿ×ℝ¹→ℝⁿ, if rank∂ _xF(x*, λ*)=n−1, and if the Ljapunov-Schmidt reduced equation has the normal formg(ξ, μ)=±ξ ²⁺ p±μξ=0. It is shown that such points satisfy a minimally extended systemG(y)=0,G:ℝ ⁿ⁺²→ℝⁿ⁺² the dimensionn+2 of which is independent ofp. For solving this system, a two-stage Newton-type method is proposed. Some numerical tests show the influence of the starting point and of the bordering vectors used in the definition of the extended system on the behavior of the iteration.     

On anisotropic Gauss-Bonnet cosmologies in (<Emphasis Type="Italic">n</Emphasis> + 1) dimensions,governed by an <Emphasis Type="Italic">n</Emphasis>-dimensional Finslerian 4-metric

The (n + 1)-dimensional Einstein-Gauss-Bonnet (EGB) model is considered. For diagonal cosmological metrics, the equations of motion are written as a set of Lagrange equations with the effective Lagrangian containing two “minisuperspace” metrics on ℝ ⁿ : a 2-metric of pseudo-Euclidean signature and a Finslerian 4-metric proportional to the n-dimensional Berwald-Moor 4-metric. For the case of the “pure” Gauss-Bonnet model, two exact solutions are presented, those with power-law and exponential dependences of the scale factors (w.r.t. the synchronous time variable) are presented. (The power-law solution was considered earlier by N. Deruelle, A. Toporensky, P. Tretyakov, and S. Pavluchenko.) In the case of EGB cosmology, it is shown that for any nontrivial solution with an exponential dependence of scale factors, a _i (τ) = A _i exp(v ⁱ τ), there are no more than three different numbers among v ¹, …, v ⁿ .     

Incremental Medians via Online Bidding

In the k-median problem we are given sets of facilities and customers, and distances between them. For a given set F of facilities, the cost of serving a customer u is the minimum distance between u and a facility in F. The goal is to find a set F of k facilities that minimizes the sum, over all customers, of their service costs. Following the work of Mettu and Plaxton, we study the incremental medians problem, where k is not known in advance. An incremental algorithm produces a nested sequence of facility sets F ₁⊆F ₂⊆⋅⋅⋅⊆F _n , where |F _k |=k for each k. Such an algorithm is called c -cost-competitive if the cost of each F _k is at most c times the optimum k-median cost. We give improved incremental algorithms for the metric version of this problem: an 8-cost-competitive deterministic algorithm, a 2e≈5.44-cost-competitive randomized algorithm, a (24+ε)-cost-competitive, polynomial-time deterministic algorithm, and a 6e+ε≈16.31-cost-competitive, polynomial-time randomized algorithm. We also consider the competitive ratio with respect to size. An algorithm is s -size-competitive if the cost of each F _k is at most the minimum cost of any set of k facilities, while the size of F _k is at most sk. We show that the optimal size-competitive ratios for this problem, in the deterministic and randomized cases, are 4 and e. For polynomial-time algorithms, we present the first polynomial-time O(log m)-size-approximation algorithm for the offline problem, as well as a polynomial-time O(log m)-size-competitive algorithm for the incremental problem. Our upper bound proofs reduce the incremental medians problem to the following online bidding problem: faced with some unknown threshold T∈ℝ⁺, an algorithm must submit “bids” b∈ℝ⁺ until it submits a bid b≥T, paying the sum of all its bids. We present folklore algorithms for online bidding and prove that they are optimally competitive. We extend some of the above results for incremental medians to approximately metric distance functions and to incremental fractional medians. Finally, we consider a restricted version of the incremental medians problem where k is restricted to one of two given values, for which we give a deterministic algorithm with a nearly optimal cost-competitive ratio. The conference version of this paper appeared in (Chrobak, M., et al. in Lecture Notes in Computer Science, vol. 3887, pp. 311–322, 2006). Research of M. Chrobak supported by NSF Grant CCR-0208856.     

Steiner Trees in General Nonuniform Orientations

In this paper we consider Steiner minimum trees (SMT) in the plane, where the connections can only be along a given set of fixed but arbitrary (not necessarily uniform) orientations. The orientations define a metric, called the general orientation metric, A _σ, where σ is the number of orientations. We prove that in A _σ metric, there exists an SMT whose Steiner points belong to an (n−2)-level grid. This result generalizes a result by Lee and Shen [11], and a result by Du and Hwang [5]. In the former case, the same result was obtained for the special case when all orientations are uniform, while in the latter case the same result was proven for the special case when there are only three arbitrary orientations. We then modify the proof used in the main result for the special case when σ=3, i.e., only three arbitrary orientations are considered, and obtain a better result, which states that there exists an SMT whose Steiner points belong to an -level grid. The result has also been obtained by Lin and Xue [9] using a different approach. Received September 27, 1999; revised August 14, 2000     

Cutting Planes and the Parameter Cutwidth

We introduce the parameter cutwidth for the Cutting Planes (CP) system of Gomory and Chvátal. We provide linear lower bounds on cutwidth for two simple polytopes. Considering CP as a propositional refutation system, one can see that the cutwidth of a CNF contradiction F is always bound above by the Resolution width of F. We provide an example proving that the converse fails: there is an F which has constant cutwidth, but has Resolution width Ω(n). Following a standard method for converting an FO sentence ψ, without finite models, into a sequence of CNFs, F _ψ,n , we provide a classification theorem for CP based on the sum cutwidth plus rank. Specifically, the cutwidth + rank of F _ψ,n is bound by a constant c (depending on ψ only) iff ψ has no (infinite) models. This result may be seen as a relative of various gap theorems extant in the literature.     

Deep lithography with a low energy synchrotron source: fabrication of X-ray refractive single lenses

 Lithography as deep as 400 μm has been carried out to fabricate X-rays refractive lenses using a low energy synchrotron source (AURORA-2 S, 0.7 GeV). The lens made of PMMA has two parabolic curvatures with radii R=4 μm and apertures A=2(2Rz)^1/2=179 μm, thus the aspect ratio z/R=250 for its curvatures, which is too great for traditional techniques to achieve. Upon fabrication of the lenses, precision of the curvatures has been evaluated by digital imaging analysis. The lens can singly focus a beam of hard X-rays into several microns at a reasonable focal length F=1.5 m. Advantages of using a low energy source for the LIGA process will be discussed regarding problems such as thick absorbers demanded by the LIGA mask and heat-load occurring in thick resist layers. Received: 10 August 2001/Accepted: 24 September 2001     

Generalized multiagent learning with performance bound

We present new Multiagent learning (MAL) algorithms with the general philosophy of policy convergence against some classes of opponents but otherwise ensuring high payoffs. We consider a 3-class breakdown of opponent types: (eventually) stationary, self-play and “other” (see Definition 4) agents. We start with ReDVaLeR that can satisfy policy convergence against the first two types and no-regret against the third, but it needs to know the type of the opponents. This serves as a baseline to delineate the difficulty of achieving these goals. We show that a simple modification on ReDVaLeR yields a new algorithm, RV _σ(t), that achieves no-regret payoffs in all games, and convergence to Nash equilibria in self-play (and to best response against eventually stationary opponents—a corollary of no-regret) simultaneously, without knowing the opponent types, but in a smaller class of games than ReDVaLeR . RV _σ(t) effectively ensures the performance of a learner during the process of learning, as opposed to the performance of a learned behavior. We show that the expression for regret of RV _σ(t) can have a slightly better form than those of other comparable algorithms like GIGA and GIGA-WoLF though, contrastingly, our analysis is in continuous time. Moreover, experiments show that RV _σ(t) can converge to an equilibrium in some cases where GIGA, GIGA-WoLF would fail, and to better equilibria where GIGA, GIGA-WoLF converge to undesirable equilibria (coordination games). This important class of coordination games also highlights the key desirability of policy convergence as a criterion for MAL in self-play instead of high average payoffs. To our knowledge, this is also the first successful (guaranteed) attempt at policy convergence of a no-regret algorithm in the Shapley game.     

Approximating Buy-at-Bulk and Shallow-Light <Emphasis Type="Italic">k</Emphasis>-Steiner Trees

We study two related network design problems with two cost functions. In the buy-at-bulk k-Steiner tree problem we are given a graph G(V,E) with a set of terminals T⊆V including a particular vertex s called the root, and an integer k≤|T|. There are two cost functions on the edges of G, a buy cost b:E→ℝ⁺ and a distance cost r:E→ℝ⁺. The goal is to find a subtree H of G rooted at s with at least k terminals so that the cost ∑ _e∈H b(e)+∑ _t∈T−s dist(t,s) is minimized, where dist(t,s) is the distance from t to s in H with respect to the r cost. We present an O(log ⁴ n)-approximation algorithm for the buy-at-bulk k-Steiner tree problem. The second and closely related one is bicriteria approximation algorithm for Shallow-light k-Steiner trees. In the shallow-light k-Steiner tree problem we are given a graph G with edge costs b(e) and distance costs r(e), and an integer k. Our goal is to find a minimum cost (under b-cost) k-Steiner tree such that the diameter under r-cost is at most some given bound D. We develop an (O(log n),O(log ³ n))-approximation algorithm for a relaxed version of Shallow-light k-Steiner tree where the solution has at least terminals. Using this we obtain an (O(log ² n),O(log ⁴ n))-approximation algorithm for the shallow-light k-Steiner tree and an O(log ⁴ n)-approximation algorithm for the buy-at-bulk k-Steiner tree problem. Our results are recently used to give the first polylogarithmic approximation algorithm for the non-uniform multicommodity buy-at-bulk problem (Chekuri, C., et al. in Proceedings of 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06), pp. 677–686, 2006). A preliminary version of this paper appeared in the Proceedings of 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) 2006, LNCS 4110, pp. 153–163, 2006. M.T. Hajiaghayi supported in part by IPM under grant number CS1383-2-02. M.R. Salavatipour supported by NSERC grant No. G121210990, and a faculty start-up grant from University of Alberta.     

Fault-Tolerant Embedding of Pairwise Independent Hamiltonian Paths on a Faulty Hypercube with Edge Faults

A Hamiltonian path in G is a path which contains every vertex of G exactly once. Two Hamiltonian paths P ₁=〈u ₁,u ₂,…,u _n 〉 and P ₂=〈v ₁,v ₂,…,v _n 〉 of G are said to be independent if u ₁=v ₁, u _n =v _n , and u _i ≠v _i for all 1<i<n; and both are full-independent if u _i ≠v _i for all 1≤i≤n. Moreover, P ₁ and P ₂ are independent starting at u ₁, if u ₁=v ₁ and u _i ≠v _i for all 1<i≤n. A set of Hamiltonian paths {P ₁,P ₂,…,P _k } of G are pairwise independent (respectively, pairwise full-independent, pairwise independent starting at u ₁) if any two different Hamiltonian paths in the set are independent (respectively, full-independent, independent starting at u ₁). A bipartite graph G is Hamiltonian-laceable if there exists a Hamiltonian path between any two vertices from different partite sets. It is well known that an n-dimensional hypercube Q _n is bipartite with two partite sets of equal size. Let F be the set of faulty edges of Q _n . In this paper, we show the following results: