A tight lower bound for k-set agreement
Minimizing the Stabbing Number of Matchings, Trees, and
1
objective functions: for example, one can ask for the total turn cost between between adjacent line segments; e.g., see [3]. When dealing with structural or algorithmic properties, a different objective function of interest is called the stabbing number: for a given set of line segments, this is the maximum number of segments that are encountered (in their interior or at an endpoint) by any line. If we consider only axis-parallel lines, we get the axis-parallel stabbing number. A closely related measure defined by Matouˇ sek [20] is the crossing number, which is the number of connected components of the intersection of a line with the union of line segments1 In the absence of connected components of collinear segments (which is the case for matchings), the crossing number coincides with the stabbing number. When considering structures like triangulations, the crossing number is precisely one more than the maximum number of triangles intersected by any one line. Stabbing problems have been considered for several years. The complexity of many algorithms in computational geometry is directly dependent on the complexity of ray shooting; as described by Agarwal [1], the later can be improved by making use of spanning trees of low stabbing number. We will sketch some related results further down. However, most previous work on stabbing and crossing problems has focused on extremal properties, and little has been known about the computational complexity of actually finding structures of low stabbing number, or possible approximation algorithms. In fact, settling the complexity of Minimum Stabbing Number for spanning trees has been one of the original 30 outstanding open problems of computational geometry on the list by Mitchell and O’Rourke [21]. (An up-to-date list is maintained online by Demaine, Mitchell, and O’Rourke [8].) Our Contributions. We describe a general proof technique that shows N P -hardness of minimizing the stabbing number of perfect matchings, triangulations, and spanning trees. For the case of matchings we show that it is also hard to approximate the minimum stabbing number within a factor below 6/5. On the other hand, we present a mathematical programming framework for actually finding structures with small stabbing number. Our approach characterizes solutions to stabbing problems as integer programs with an exponential number of cut constraints. We describe how the corresponding linear programming (LP) relaxations can be solved in polynomial time, providing empirically excellent lower bounds. Furthermore, we show that there always is an optimal fractional matching (or spanning tree) that contains an edge of weight above a lower bound of 1/3 (or 1/5 for spanning trees), allowing an iterated rounding scheme similar to the one developed by Jain for the generalized Steiner network problem [17]: compute a heuristic solution by solving a polynomial sequence of LPs. We conjecture that this heuristic solution is within a constant factor of the optimum. Our mathematical programming approach is also practically useful: as described in detail in our experimental study [12], we can optimally solve stabbing problems for instances (taken from well-known benchmark sets of other geometric optimization problems) of vertex sets up to several hundred vertices. Our results in detail. • We prove that deciding whether a vertex set has a perfect matching of axis-parallel stabbing number 5 is an N P -complete problem; we also extend this result to general stabbing number. • We prove that finding a triangulation of minimum axis-parallel stabbing number is an N P -hard problem; we extend also extend this result to general stabbing number.
INTRINSIC VOLUMES AND SUCCESSIVE RADII
´ MARTIN HENK AND MAR´ IA A. HERNANDEZ CIFRE Abstract. Motivated by a problem of Teissier to bound the intrinsic volumes of a convex body in terms of the inradius and the circumradius of the body, we give upper and lower bounds for the intrinsic volumes of a convex body in terms of the elementary symmetric functions of the so called successive inner and outer radii. These results improve on former bounds and, in particular, they also provide bounds for the elementary symmetric functions of the roots of Steiner polynomials in terms of the elementary symmetric functions of these radii.
n
(1.2)
i=0
κn−i Vi (K )ρn−i .
Vi (K ) is called the i-th intrinsic volume of K since, if K is i-dimensional, then Vi (K ) is the ordinary i-dimensional volume of K . In particular, we have that Vn (K ) is the volume of K , 2Vn−1 (K ) is the surface area of K , 2κn−1 /(nκn )V1 (K ) is the mean width of K (see [19, p. 42]) and V0 (K ) = 1 is the Euler characteristic.
Curvelets a surprisingly effective nonadaptive representation of objects with
Curvelets–A Surprisingly EffectiveNon adaptive Representation For Objects with Edges Emmanuel J.Cand`e s and David L.DonohoAbstract.It is widely believed that to efficiently represent an otherwisesmooth object with discontinuities along edges,one must use an adaptiverepresentation that in some sense‘tracks’the shape of the discontinuityset.This folk-belief—some would say folk-theorem—is incorrect.Atthe very least,the possible quantitative advantage of such adaptation isvastly smaller than commonly believed.We have recently constructed atight frame of curvelets which provides stable,efficient,and near-optimalrepresentation of otherwise smooth objects having discontinuities alongsmooth curves.By applying naive thresholding to the curvelet transformof such an object,one can form m-term approximations with rate of L2approximation rivaling the rate obtainable by complex adaptive schemeswhich attempt to‘track’the discontinuity set.In this article we explainthe basic issues of efficient m-term approximation,the construction ofefficient adaptive representation,the construction of the curvelet frame,and a crude analysis of the performance of curvelet schemes.§1.IntroductionIn many important imaging applications,images exhibit edges–discontinu-ities across curves.In traditional photographic imaging,for example,this occurs whenever one object occludes another,causing the luminance to un-dergo step discontinuities at boundaries.In biological imagery,this occurs whenever two different organs or tissue structures meet.In image synthesis applications,such as CAD,there is no problem in deal-ing with such discontinuities,because one knows where they are and builds the discontinuities into the representation by specially adapting the representation —for example,inserting free knots,or adaptive refinement rules.In image analysis applications,the situation is different.When working with real rather than synthetic data,one of course doesn’t‘know’where these edges are;one only has a digitized pixel array,with potential imperfections caused by noise,by blurring,and of course by the unnatural pixelization of the underlying continuous scene.Hence the typical image analyst onlySaint-Malo Proceedings1 XXX,XXX,and Larry L.Schumaker(eds.),pp.1–10.Copyright o c2000by Vanderbilt University Press,Nashville,TN.ISBN1-xxxxx-xxx-x.All rights of reproduction in any form reserved.2 E.J.Cand`e s and D.L.Donoho has recourse to representations which don’t‘know’about the existence andgeometry of the discontinuities in the image.The success of discontinuity-adapting methods in CAD and related imagesynthesisfields creates a temptation for an image analyst–a temptation tospend a great deal of time and effort importing such ideas into image analysis.Almost everyone we know has yielded to this temptation in some form,whichcreates a possibility for surprise.Oracles and Ideally-Adapted RepresentationOne could imagine an ideally-privileged image analyst who has recourse toan oracle able to reveal the positions of all the discontinuities underlying theimage formation.It seems natural that this ideally-privileged analyst coulddo far better than the normally-endowed analyst who knows nothing aboutthe position of the discontinuities in the image.To elaborate this distinction,we introduce terminology borrowed fromfluid dynamics,where‘edges’arise in the form of fronts or shock fronts.A Lagrangian representation is constructed using full knowledge of theintrinsic structure of the object and adapting perfectly to that structure.•Influid dynamics this means that thefluidflow pattern is known,and one constructs a coordinate system which‘flows along with the particles’,with coordinates mimicking the shape of theflow streamlines.•In image representation this could mean that the edge curves are known, and one constructs an image representation adapted to the structure of the edge curves.For example,one might construct a basis with disconti-nuities exactly where the underlying object has discontinuities.An Eulerian representation isfixed,constructed once and for all.It isnonadaptive–having nothing to do with the known or hypothesized detailsof the underlying object.•Influid dynamics,this would mean a usual euclidean coordinate system, one that does not depend in any way on thefluid motion.•In image representation,this could mean that the representation is some fixed coordinate representation,such as wavelets or sinusoids,which does not change depending on the positions of edges in the image.It is quite natural to suppose that the Lagrangian perspective,whenit is available,is much more powerful that the Eulerian one.Having theprivilege of‘inside information’about the position of important geometriccharacteristics of the solution seems a priori rather valuable.In fact,thisposition has rather a large following.Much recent work in computationalharmonic analysis(CHA)attempts tofind bases which are optimally adaptedto the specific object in question[7,10,11];in this sense much of the ongoingwork in CHA is based on the presumption that the Lagrangian viewpoint isbest.In the setting of edges in images,there has,in fact,been considerableinterest in the problem of developing representations which are adapted tothe structure of discontinuities in the object being studied.The(equivalent)Curvelets3 concepts of probing and minimum entropy segmentation are old examples of this: wavelet systems which are specifically constructed to allow discontinuities in the basis elements at specific locations[8,9].More recently,we are aware of much informal unpublished or preliminary work attempting to build2D edge-adapted schemes;we give two examples.•Adaptive triangulation aims to represent a smooth function by partition-ing the plane into a sequence of triangular meshes,refining the meshes at one stage to createfiner meshes at the next stage.One represents the underlying object using piecewise linear functions supported on individ-ual triangles.It is easy to see how,in an image synthesis setting,one can in principle develop a triangulation where the triangles are arranged to track a discontinuity very faithfully,with the bulk of refinement steps allocated to refinements near the discontinuity,and one obtains very ef-fective representation of the object.It is not easy to see how to do this in an image analysis setting,but one can easily be persuaded that the development of adaptive triangulation schemes for noisy,blurred data is an important and interesting project.•In an adaptively warped wavelet representation,one deforms the under-lying image so that the object being analyzed has all its discontinuities aligned purely horizontal or vertical.Then one analyzes the warped ob-ject in a basis of tensor-product wavelets where elements take the form ψj,k(x1)·ψj ,k (x2).This is very effective for objects which are smooth apart from purely horizontal and purely vertical discontinuities.Hence, the warping deforms the singularities to render the the tensor product scheme very effective.It is again not easy to see how adaptive warping could work in an image analysis setting,but one is easily persuaded that development of adaptively warped representations for noisy,blurred data is an important and interesting project.Activity to build such adaptive representations is based on an article of faith:namely,that Eulerian approaches are inferior,that oracle-driven Lagrangian approaches are ideal,and that one should,in an image analysis setting,mimic Lagrangian approaches,attempting empirically to estimate from noisy,blurred data the information that an oracle would supply,and build an adaptive representation based on that information.Quantifying Rates of ApproximationIn order to get away from articles of faith,we now quantify performance,using an asymptotic viewpoint.Suppose we have an object supported in[0,1]2which has a discontinuity across a nice curveΓ,and which is otherwise smooth.Then using a standard Fourier representation,and approximating with˜f F m built from the best m nonzero Fourier terms,we havef−˜f F m 22 m−1/2,m→∞.(1)4 E.J.Cand`e s and D.L.Donoho This rather slow rate of approximation is improved upon by wavelets.The approximant˜f W m built from the best m nonzero wavelet terms satisfiesf−˜f W m 22 m−1,m→∞.(2) This is better than the rate of Fourier approximation,and,until now,is the best published rate for afixed non-adaptive method(i.e.best published result for an‘Eulerian viewpoint’).On the other hand,we will discuss below a method which is adapted to the object at hand,and which achieves a much better approximation rate than previously known‘nonadaptive’or‘Eulerian’approaches.This adaptive method selects terms from an overcomplete dictionary and is able to achievef−˜f A m 22 m−2,m→∞.(3) Roughly speaking,the terms in this dictionary amount to triangular wedges, ideallyfitted to approximate the shape of the discontinuity.Owing to the apparent trend indicated by(1)-(3)and the prevalence of the puritanical belief that‘you can’t get something for nothing’,one might suppose that inevitably would follow theFolk-Conjecture/[Folk-Theorem].The result(3)for adaptive representa-tions far exceeds the rate of m-term approximation achievable byfixed non-adaptive representations.This conjecture appeals to a number of widespread beliefs:•the belief that adaptation is very powerful,•the belief that the way to represent discontinuities in image analysis is to mimic the approach in image synthesis•the belief that wavelets give the bestfixed nonadaptive representation.In private discussions with many respected researchers we have many times heard expressed views equivalent to the purported Folk-Theorem.The SurpriseIt turns out that performance almost equivalent to(3)can be achieved by a non adaptive scheme.In other words,the Folk-Theorem is effectively false.There is a tight frame,fixed once and for all nonadaptively,which we call a frame of curvelets,which competes surprisingly well with the ideal adaptive rate(3).A very simple m-term approximation–summing the m biggest terms in the curvelet frame expansion–can achievef−˜f C m 22≤C·m−2(log m)3,m→∞,(4) which is nearly as good as(3)as regards asymptotic order.In short,in a problem of considerable applied relevance,where one would have thought that adaptive representation was essentially more powerful than fixed nonadaptive representation,it turns out that a newfixed nonadaptive representation is essentially as good as adaptive representation,from the point of view of asymptotic m-term approximation errors.As one might expect, the new nonadaptive representation has several very subtle and distinctive features.Curvelets5 ContentsIn this article,we would like to give the reader an idea of why(3)represents the ideal behavior of an adaptive representation,of how the curvelet frame is constructed,and of the key elements responsible for(4).We will also attempt to indicate why curvelets perform for singularities along curves the task that wavelets perform for singularities at points.§2.A Precedent:Wavelets and Point SingularitiesWe mention an important precedent–a case where a nonadaptive scheme is roughly competitive with an ideal adaptive scheme.Suppose we have a piecewise polynomial function f on the interval[0,1], with jump discontinuities at several points.An obvious adaptive representation is tofit a piecewise polynomial with breakpoints at the discontinuities.If there are P pieces and each polynomial is of degree≤D,then we need only keep P·(D+1)coefficients and P−1 breakpoints to exactly represent this mon sense tells us that this is the natural,and even,the ideal representation for such a function.To build this representation,we need to know locations of the discontinu-ities.If the measurements are noisy or blurred,and if we don’t have recourse to an oracle,then we can’t necessarily build this representation.A less obvious but much more robust representation is to take a nice wavelet transform of the object,and keep the few resulting nonzero wavelet coefficients.If we have an N-point digital signal f(i/N),1≤i≤N,and we use Daubechies wavelets of compact support,then there are no more than C·log2(N)·P·(D+1)nonzero wavelet coefficients for the digital signal.In short,the nonadaptive representation needs only to keep a factor C log2(N)more data to give an equally faithful representation.We claim that this phenomenon is at least partially responsible for the widespread success of wavelet methods in data compression settings.One can build a single fast transform and deal with a wide range of different f,with different discontinuity sets,without recourse to an oracle.In particular,since one almost never has access to an oracle,the nat-uralfirst impulse of one committed to the adaptive viewpoint would be to ‘estimate’the break points–i.e.to perform some sort of edge detection.Un-fortunately this is problematic when one is dealing with noisy blurred data. Edge detection is a whole topic in itself which has thousands of proposed so-lutions and(evidently,as one can see from the continuing rate of publication in this area)no convincing solution.In using wavelets,one does not need edge detectors or any other prob-lematic schemes,one simply extracts the big coefficients from the transform domain,and records their values and positions in an organized fashion.We can lend a useful perspective to this phenomenon by noticing that the discontinuities in the underlying f are point singularities,and we are saying that wavelets need in some sense at most log(n)coefficients to represent a point singularity out to scale1/n.6 E.J.Cand`e s and D.L.DonohoIt turns out that even in higher dimensions wavelets have a near-ideal ability to represent objects with point singularities.The two-dimensional object fβ(x1,x2)=1/((x1−1/2)2+(x2−1/2)2)βhas,forβ<1/2,a square-integrable singularity at the point(1/2,1/2)and is otherwise smooth.At each level of the2D wavelet pyramid,there are effec-tively only a few wavelets which‘feel’the point singularity,other coefficients being effectively negligible.In approximation out to scale1/n,only about O(log(n))coefficients are required.Another approach to understanding the representation of singularities, which is not limited by scale,is to consider rates of decay of the countable coefficient sequence.Analysis of wavelet coefficients of fβshows that for any desired rateρ,the N-th largest coefficient can be bounded by CρN−ρfor all N.In short,the wavelet coefficients of such an object are very sparse.Thus we have a slogan:wavelets perform very well for objects with point singularities in dimensions1and2.§3.Failure of Wavelets on EdgesWe now briefly sketch why wavelets,which worked surprisingly well in repre-senting point discontinuities in dimension1,are less successful dealing with ‘edge’discontinuities in dimension2.Suppose we have an object f on the square[0,1]2and that f is smooth away from a discontinuity along a C2curveΓ.Let’s look at the number of substantial wavelet coefficients.A grid of squares of side2−j by2−j has order2j squares intersectingΓ. At level j of the two-dimensional wavelet pyramid,each wavelet is localized near a corresponding square of side2−j by2−j.There are therefore O(2j) wavelets which‘feel’the discontinuity alongΓ.Such a wavelet coefficient is controlled by| f,ψj,k1,k2 |≤ f ∞· ψj,k1,k2 1≤C·2−j;and in effect no better control is available,since the object f is not smoothwithin the support ofψj,k1,k2[14].Therefore there are about2j coefficients ofsize about2−j.In short,the N-th largest wavelet coefficient is of size about 1/N.The result(2)follows.We can summarize this by saying that in dimension2,discontinuities across edges are spatially distributed;because of this they can interact rather extensively with many terms in the wavelet expansion,and so the wavelet representation is not sparse.In short,wavelets do well for point singularities,and not for singularities along curves.The success of wavelets in dimension1derived from the fact that all singularities in dimension1are point singularities,so wavelets have a certain universality there.In higher dimensions there are more types of singularities,and wavelets lose their universality.For balance,we need to say that wavelets do outperform classical meth-ods.If we used sinusoids to represent an object of the above type,then weCurvelets7 have the result(1),which is far worse than that provided by wavelets.For completeness,we sketch the argument.Suppose we use for‘sinusoids’the complex exponentials on[−π,π]2,and that the object f tends smoothly to zero at the boundary of the square[0,1]2,so that we may naturally extend it to a function living on[−π,π]2.Now typically the Fourier coefficients of an otherwise smooth object with a discontinuity along a curve decay with wavenumber as|k|−3/2(the very well-known example is f=indicator of a disk,which has a Fourier transform described by Bessel functions).Thus there are about R2coefficients of size≥c·R−3/2,meaning that the N-th largest is of size≥c·N−3/4,from which(1)follows.In short:neither wavelets nor sinusoids really sparsify two-dimensional objects with edges(although wavelets are better than sinusoids).§4.Ideal Representation of Objects with EdgesWe now consider the optimality result(3),which is really two assertions.On the one hand,no reasonable scheme can do better than this rate.On the other hand,a certain adaptive scheme,with intimate connections to adaptive triangulation,which achieves it.For more extensive discussion see[10,11,13].In talking about adaptive representations,we need to define terms care-fully,for the following reason.For any f,there is always an adaptive repre-sentation of f that does very well:namely the orthobasisΨ={ψ0,ψ1,...} withfirst elementψ0=f/ f 2!This is,in a certain conception,an‘ideal representation’where each object requires only one nonzero coefficient.In a certain sense it is a useless one,since all information about f has been hidden in the definition of representation,so actually we haven’t learned anything. Most of our work in this section is in setting up a notion of adaptation that will free us from fear of being trapped at this level of triviality. Dictionaries of AtomsSuppose we are interested in approximating a function in L2(T),and we have a countable collection D={φ}of atoms in L2(T);this could be a basis,a frame, afinite concatenation of bases or frames,or something even less structured.We consider the problem of m-term approximation from this dictionary, where we are allowed to select m termsφ1,...,φm from D and we approximate f from the L2-closest member of the subspace they span:˜f=P roj{f|span(φ1,...,φm)}.mWe are interested in the behavior of the m-term approximation errore m(f;D)= f−˜f m 22,where in this provisional definition,we assume˜f m is a best approximation of this form after optimizing over the selection of m terms from the dictionary.However,to avoid a trivial result,we impose regularity on the selection process.Indeed,we allow rather arbitrary dictionaries,including ones which8 E.J.Cand`e s and D.L.Donoho enumerate a dense subset of L2(T),so that in some sense the trivial result φ1=f/ f 2e m=0,∀m is always a lurking possibility.To avoid this possibility we forbid arbitrary selection rules.Following[10]we proposeDefinition.A sequence of selection rules(σm(·))choosing m terms from a dictionary D,σm(f)=(φ1,...,φm),is said to implement polynomial depth search if there is a singlefixed enumeration of the dictionary elements and afixed polynomialπ(t)such that terms inσm(f)come from thefirstπ(m)elements in the dictionary.Under this definition,the trivial representation based on a countable dense dictionary is not generally available,since in anyfixed enumeration, a decent1-term approximation to typical f will typically be so deep in the enumeration as to be unavailable for polynomial-depth selection.(Of course, one can make this statement quantitative,using information-theoretic ideas).More fundamentally,our definition not only forbids trivialities,but it allows us to speak of optimal dictionaries and get meaningful results.Starting now,we think of dictionaries as ordered,having afirst element,second element, etc.,so that different enumerations of the same collection of functions are different dictionaries.We define the m-optimal approximation number for dictionary D and limit polynomialπase m(f;D;π)= f−˜f m 22,where˜f m is constructed by optimizing the choice of m atoms among thefirst π(m)in thefixed enumeration.Note that we use squared error for comparison with(1)-(3)in the Introduction.Approximating Classes of FunctionsSuppose we now have a class F of functions whose members we wish to ap-proximate.Suppose we are given a countable dictionary D and polynomial depth search delimited by polynomialπ(·).Define the error of approximation by this dictionary over this class bye m(F;D,π)=maxe m(f;D,π).f∈FWe mayfind,in certain examples,that we can establish boundse m(F;D,π)=O(m−ρ),m→∞,for allρ<ρ∗.At the same time,we may have available an argument showing that for every dictionary and every polynomial depth search rule delimited by π(·),e m(F;D,π)≥cm−ρ∗,m≥m0(π).Then it seems natural to say thatρ∗is the optimal rate of m-term approxi-mation from any dictionary when polynomial depth search delimited byπ(·).Curvelets9Starshaped Objects with C 2Boundaries We define Star-Set 2(C ),a class of star-shaped sets with C 2-smooth bound-aries,by imposing regularity on the boundaries using a kind of polar coor-dinate system.Let ρ(θ):[0,2π)→[0,1]be a radius function and b 0=(x 1,0,x 2,0)be an origin with respect to which the set of interest is star-shaped.With δi (x )=x i −x i,0,i =1,2,define functions θ(x 1,x 2)and r (x 1,x 2)byθ=arctan(−δ2/δ1);r =((δ1)2+(δ2)2)1/2.For a starshaped set,we have (x 1,x 2)∈B iff0≤r ≤ρ(θ).Define the class Star-Set 2(C )of sets by{B :B ⊂[110,910]2,110≤ρ(θ)≤12θ∈[0,2π),ρ∈C 2,|¨ρ(θ)|≤C },and consider the corresponding functional class Star 2(C )= f =1B :B ∈Star-Set 2(C ) .The following lower rate bound should be compared with (3).Lemma.Let the polynomial π(·)be given.There is a constant c so that,for every dictionary D ,e m (Star 2(C );D ,π)≥c 1m 2log(m ),m →∞.This is proved in [10]by the technique of hypercube embedding.Inside the class Star 2(C )one can embed very high-dimensional hypercubes,and the ability of a dictionary to represent all members of a hypercube of dimension n by selecting m n terms from a subdictionary of size π(m )is highly limited if π(m )grows only polynomially.To show that the rate (3)can be achieved,[13]adaptively constructs,for each f ,a corresponding orthobasis which achieves it.It tracks the boundary of B at increasing accuracy using a sequence of polygons;in fact these are n -gons connecting equispaced points along the boundary of B ,for n =2j .The difference between n -gons for n =2j and n =2j +1is a collection of thin triangular regions obeying width ≈length 2;taking the indicators of each region as a term in a basis,one gets an orthonormal basis whose terms at fine scales are thin triangular pieces.Estimating the coefficient sizes by simple geometric analysis leads to the result (3).In fact,[13]shows how to do this under the constraint of polynomial-depth selection,with polynomial Cm 7.Although space constraints prohibit a full explanation,our polynomial-depth search formalism also makes perfect sense in discussing the warped wavelet representations of the Introduction.Consider the noncountable ‘dic-tionary’of all wavelets in a given basis,with all continuum warpings applied.Notice that for wavelets at a given fixed scale,warpings can be quantized with a certain finite accuracy.Carefully specifying the quantization of the warping,one obtains a countable collection of warped wavelets,for which polynomial depth search constraints make sense,and which is as effective as adaptive triangulation,but not more so .Hence (3)applies to (properly interpreted)deformation methods as well.10 E.J.Cand`e s and D.L.Donoho§5.Curvelet ConstructionWe now briefly describe the curvelet construction.It is based on combining several ideas,which we briefly review•Ridgelets,a method of analysis suitable for objects with discontinuities across straight lines.•Multiscale Ridgelets,a pyramid of windowed ridgelets,renormalized and transported to a wide range of scales and locations.•Bandpass Filtering,a method of separating an object out into a series of disjoint scales.We briefly describe each idea in turn,and then their combination.RidgeletsThe theory of ridgelets was developed in the Ph.D.Thesis of Emmanuel Cand`e s(1998).In that work,Cand`e s showed that one could develop a system of analysis based on ridge functionsψa,b,θ(x1,x2)=a−1/2ψ((x1cos(θ)+x2sin(θ)−b)/a).(5)He introduced a continuous ridgelet transform R f(a,b,θ)= ψa,b,θ(x),f with a reproducing formula and a Parseval relation.He also constructed frames, giving stable series expansions in terms of a special discrete collection of ridge functions.The approach was general,and gave ridgelet frames for functions in L2[0,1]d in all dimensions d≥2–For further developments,see[3,5].Donoho[12]showed that in two dimensions,by heeding the sampling pat-tern underlying the ridgelet frame,one could develop an orthonormal set for L2(I R2)having the same applications as the original ridgelets.The orthonor-mal ridgelets are convenient to use for the curvelet construction,although it seems clear that the original ridgelet frames could also be used.The ortho-ridgelets are indexed usingλ=(j,k,i, , ),where j indexes the ridge scale,k the ridge location,i the angular scale,and the angular location; is a gender token.Roughly speaking,the ortho-ridgelets look like pieces of ridgelets(5) which are windowed to lie in discs of radius about2i;θi, = /2i is roughly the orientation parameter,and2−j is roughly the thickness.A formula for ortho-ridgelets can be given in the frequency domainˆρλ(ξ)=|ξ|−12(ˆψj,k(|ξ|)w i, (θ)+ˆψj,k(−|ξ|)w i, (θ+π))/2.are periodic wavelets for[−π,π), Here theψj,k are Meyer wavelets for I R,wi,and indices run as follows:j,k∈Z Z, =0,...,2i−1−1;i≥1,and,if =0, i=max(1,j),while if =1,i≥max(1,j).We letΛbe the set of suchλ.The formula is an operationalization of the ridgelet sampling principle:•Divide the frequency domain in dyadic coronae|ξ|∈[2j,2j+1].•In the angular direction,sample the j-th corona at least2j times.•In the radial frequency direction,sample behavior using local cosines.The sampling principle can be motivated by the behavior of Fourier trans-forms of functions with singularities along lines.Such functions have Fourier transforms which decay slowly along associated lines through the origin in the frequency domain.As one traverses a constant radius arc in Fourier space,one encounters a ‘Fourier ridge’when crossing the line of slow decay.The ridgelet sampling scheme tries to represent such Fourier transforms by using wavelets in the angular direction,so that the ‘Fourier ridge’is captured neatly by one or a few wavelets.In the radial direction,the Fourier ridge is actu-ally oscillatory,and this is captured by local cosines.A precise quantitative treatment is given in [4].Multiscale RidgeletsThink of ortho-ridgelets as objects which have a “length”of about 1and a “width”which can be arbitrarily fine.The multiscale ridgelet system renor-malizes and transports such objects,so that one has a system of elements at all lengths and all finer widths.In a light mood,we may describe the system impressionistically as “brush strokes”with a variety of lengths,thicknesses,orientations and locations.The construction employs a nonnegative,smooth partition of energyfunction w ,obeying k 1,k 2w 2(x 1−k 1,x 2−k 2)≡1.Define a transportoperator,so that with index Q indicating a dyadic square Q =(s,k 1,k 2)of the form [k 1/2s ,(k 1+1)/2s )×[k 2/2s ,(k 2+1)/2s ),by (T Q f )(x 1,x 2)=f (2s x 1−k 1,2s x 2−k 2).The Multiscale Ridgelet with index µ=(Q,λ)is thenψµ=2s ·T Q (w ·ρλ).In short,one transports the normalized,windowed ortho-ridgelet.Letting Q s denote the dyadic squares of side 2−s ,we can define the subcollection of Monoscale Ridgelets at scale s :M s ={(Q,λ):Q ∈Q s ,λ∈Λ}.Orthonormality of the ridgelets implies that each system of monoscale ridgelets makes a tight frame,in particular obeying the Parseval relationµ∈M s ψµ,f 2= f 2L 2.It follows that the dictionary of multiscale ridgelets at all scales,indexed byM =∪s ≥1M s ,is not frameable,as we have energy blow-up:µ∈M ψµ,f 2=∞.(6)。
On the power of Ambainis’s lower bounds
a rX iv:q ua nt -ph /0311060v2 8 F e b 2004On the power of Ambainis’s lower bounds ∗Shengyu Zhang Computer Science Department,Princeton University,NJ 08544,USA.Email:szhang@ Abstract The polynomial method and the Ambainis’s lower bound (or Alb ,for short)method are two main quantum lower bound techniques.While recently Ambainis showed that the polynomial method is not tight,the present paper aims at studying the power and limitation of Alb ’s.We first use known Alb ’s to derive Ω(n 1.5)lower bounds for Bipartiteness ,Bipartiteness Matching and Graph Matching ,in which the lower bound for Bipartiteness improves the previous Ω(n )one.We then show that all the three known Ambainis’s lower bounds have a limitation C 0(f )C 1(f ),∗This research was supported in part by NSF grant CCR-0310466.A preliminary report of the present paper is at arXiv:quant-ph/0311060is interesting to know how tight they are.In a recent work[5],Ambainis proved that polynomial method is not tight,by showing a function with polynomial degree M and quantum query complexity Ω(M1.321...).So a natural question is the power of Ambainis’s lower bounds.We show that all known Ambainis’s lower bounds are not tight either,among other results.There are several known versions of Ambainis’s lower bounds,among which the three Ambainis’s theorems are widely used partly because they have simple forms and are thus easy to use.Thefirst Alb’s two are given in[3]as follows.Theorem1(Ambainis,[3])Let f:{0,1}N→{0,1}be a function and X,Y be two sets of inputs s.t.f(x)=f(y)if x∈X and y∈Y.Let R⊆X×Y be a relation s.t.1.∀x∈X,there are at least m different y∈Y s.t.(x,y)∈R.2.∀y∈Y,there are at least m′different x∈X s.t.(x,y)∈R.3.∀x∈X,∀i∈[N],there are at most l different y∈Y s.t.(x,y)∈R and x i=y i.4.∀y∈Y,∀i∈[N],there are at most l′different x∈X s.t.(x,y)∈R and x i=y i. Then Q2(f)=Ω( ll′).Theorem2(Ambainis,[3])Let f:I N→{0,1}be a Boolean function where I is afinite set, and X,Y be two sets of inputs s.t.f(x)=f(y)if x∈X and y∈Y.Let R⊆X×Y satisfy1.∀x∈X,there are at least m different y∈Y s.t.(x,y)∈R.2.∀y∈Y,there are at least m′different x∈X s.t.(x,y)∈R.Denotel x,i=|{y:(x,y)∈R,x i=y i}|,l y,i=|{x:(x,y)∈R,x i=y i}|l x,i l y,i.l max=maxx,y,i:(x,y)∈R,i∈[N],x i=y iThen Q2(f)=Ω( l max).Obviously,Theorem2generalizes Theorem1.In[5],Ambainis gave another(weighted)way to generalize Theorem1.We restate it in a form similar to Theorem1.Definition1Let f:I N→{0,1}be a Boolean function where I is afinite set.Let X,Y be two sets of inputs s.t.f(x)=f(y)if x∈X and y∈Y.Let R⊆X×Y be a relation.A weight scheme for X,Y,R consists three weight functions w(x,y)>0,u(x,y,i)>0and v(x,y,i)>0satisfyingu(x,y,i)v(x,y,i)≥w2(x,y)(1) for all(x,y)∈R and i∈[N]with x i=y i.We further denotew x= y:(x,y)∈R w(x,y),w y= x:(x,y)∈R w(x,y)u x,i= y:(x,y)∈R,x i=y i u(x,y,i),v y,i= x:(x,y)∈R,x i=y i v(x,y,i).Theorem3(Ambainis,[5])Let f:I N→{0,1}where I is afinite set,and X⊆f−1(0), Y⊆f−1(1)and R⊆X×Y.Let w,u,v be a weight scheme for X,Y,R.ThenQ2(f)=Ω( u x,i·min y∈Y,j∈[N]w yLet us denote by Alb1(f),Alb2(f)and Alb3(f)the best lower bound for function f achieved by Theorem1,2and3,respectively1.Note that in the four Alb’s,there are many parameters(X,Y,R,u,v,w)to be set.By setting these parameters in an appropriate way,one can get lowerbounds of quantum query complexity for many problems.In particular,we consider the followingthree graph properties2.1.Bipartiteness:Given an undirected graph,decide whether it is a bipartite graph.2.Graph Matching:Given an undirected graph,decide whether it has a perfect matching.3.Bipartite Matching:Given an undirected bipartite graph,decide whether it has a perfectmatching.We show by using Alb2that all these three graph properties have aΩ(n1.5)lower bound,where n isthe number of vertices.For Bipartiteness,this improves the previous result ofΩ(n)lower boundby Laplante and Magniez[21]and Durr et al[17].Since Alb2and Alb3generalizes Alb1in different ways,it is interesting to compare them and see which is more powerful.It turns out that Alb2(f)≤Alb3(f).However,even Alb3has a limitation:we show that Alb3(f)≤√N·min{C0(f),C1(f)}=N·CI(f),where CI(f)is the the size of the largest intersection of a0-certificate set and a1-certificate set,so CI(f)≤C−(f).The second approach leads to another result Alb3(f)≤ N)lower boundcannot be further improved by using Ambainis’s lower bounds.It is also natural to consider combining the different approaches that Alb2and Alb3use to generalize Alb1,and get a further general one.Based on this idea,we give a new and more generalizedlower bound theorem,which we call pared with Alb3,this may be easier to use.Related workRecently,Szegedy independently shows that Alb3(f)≤C0(f)C1(f)in a different way[30].He also shows in[30]that Alb3by Ambainis[5],Alb4in thepresent paper[32],and another quantum adversary method proposed in[13]are equivalent.The theorem Alb3(f)≤1To make the later results more precise,we actually use Alb i(f)to denote the value inside theΩ()notation.For example,Alb1(f)=max(X,Y,R) ll′.2In this paper,all the graph property problems are given by adjacency matrix input.2Old Ambainis’s lower boundsIn this section we first use Alb 2to derive Ω(n 1.5)lower bounds for Bipartiteness ,Bipartite Matching and Graph Matching ,then show that Alb 3has actually at least the same power as Alb 2.Theorem 4All the three graph properties Bipartiteness ,Bipartite Matching and Graph Matching have Q 2(f )=Ω(n 1.5).Proof 1.Bipartiteness .Theproof is very similar to the one for proving Ω(n 1.5)lower bound of Graph Connectivity by Durr et al [15].Without loss of generality,we assume n is even,because otherwise we can use the following argument on arbitrary n −1(out of total n )nodes and leave the n th node isolated.LetX ={G :G is composed of a single n-length cycle },Y ={G :G is composed of two cycles each with length being an odd number between n/3and 2n/3},andR ={(G,G ′)∈X ×Y :∃four nodes v 1,v 2,v 3,v 4s.t.the only difference between graphs G and G ′is that (v 1,v 2),(v 3,v 4)are edges in G but not in G ′and (v 1,v 3),(v 2,v 4)are edges in G ′but not in G }.Note that a graph is bipartite if and only if it contains no cycle with odd length.Therefore,any graph in X is a bipartite graph because n is even,and any graph in Y is not bipartite graph because it contains two odd-length cycles.Then all the remaining analysis is the same as calculation in the proof for Graph Connectivity (undirect graph and matrix input)in [15],and finally Alb 2(Bipartiteness )=Ω(n 1.5).2.Bipartite Matching .Let X be the set of the bipartite graphs like Figure 1(a)where τand σare two permutations of {1,...,n },and n3.Let Y be the set of the bipartite graphs like Figure 1(b),where τ′and σ′are two permutations of {1,...,n },and also n 3.It is easy to see that all graphs in X have no matching,while all graphs in Y have one.σ(1)......(n)σ...σ(k’)...(1)’τ’τ’’’(n)σ’(k’+1)τ’(k’)τ’(k’+1)(b): Y (a): X τ(1)σ(1)...τ(k)(k+1)τ...τ(n)σ(n)...σ(k−1)σ(k)...Figure 1:X and YLet R be the set of all pairs of (x,y )∈X ×Y as in Figure 2,where graph y is obtained from x by choosing two horizonal edges (τ(i ),σ(i )),(τ(j ),σ(j )),removing them,and adding two edges (τ(i ),σ(j )),(τ(j ),σ(i )).Now it is not hard to calculate the m,m ′,l max in Alb 2.For example,to get m we study x in two cases.When n 2,any edge (τ(i ),σ(i ))where i ∈[k −n/3,k ]has at least n/6choices for edge (τ(j ),σ(j ))because the only requirement for choosing is that k ′∈[n/3,2n/3]and k ′=j −i .The case when n 3can be handled symmetrically.Thus m =Θ(n 2).Same argument yields m ′=Θ(n 2).Finally,for l max ,we note that if the edge e =(τ(i ),σ(i ))for some i ,then l x,e =O (n )τ(1)σ(1)τ(k)τ(n)σσ(k)τ(i)τ(j)σσ(i)(j)(n)τ(1)σ(1)τ(n)σστ(i)τ(j)σσ(k)ττ(k+1)σ(i)σ(k−1)(j)(k)(n)(j−1)xyFigure 2:R ⊆X ×Y and l y,e =1;if the edge e =(τ(i ),σ(j ))for some i,j ,then l x,e =1and l y,e =O (n ).For all other edges e ,l x,e =l y,e =0.Putting all together,we have l max =O (n ).Thus by Theorem 2,we know that Alb 2(Bipartite Matching )=Ω(n 1.5).3.Graph Matching .This can be easily shown either by using the same (X,Y,R )as the proof for Bipartiteness ,because a cycle with odd length has no matching,or by noting that Bipartite Matching is a special case of Graph Matching .It is interesting to note that we can also prove the above theorem by Alb 3.For example,for Bipartite Matching ,We choose X,Y,R in the same way,and let w (x,y )=1for all (x,y )∈R .Let u (x,y,e )=1/√n if e =(τ(i ),σ(j ))or e =(τ(j ),σ(i ))in x .Thus u x,e =Θ(√l max /l x,i and v (x,y,i )=√l x,i l y,i≥1=w (x,y )Now that u (x,y,i )is independent on y ,so we have u x,i =l x,i u (x,y,i )=√l max .Thus,by denoting m x =|{y :(x,y )∈R }|and m y =|{x :(x,y )∈R }|,we have min x,i w x v y,i =min x,i m x l max min y,i m y l max =m l max m ′l max=mm ′Definition 2For an N -ary Boolean function f :I N →{0,1}and an input x ∈I N ,a certificate set CS x of f on x is a set of indices such that f (x )=f (y )whenever y i =x i for all i ∈CS x .The certificate complexity C (f,x )of f on x is the size of a smallest certificate set of f on x .The b -certificate complexity of f is C b (f )=max x :f (x )=b C (f,x ).The certificate complexity of f is C (f )=max {C 0(f ),C 1(f )}.We further denote C −(f )=min {C 0(f ),C 1(f )}.3.1A general limitation for Ambainis’s lower boundsIn this subsection,we give an upper bound for Alb 4(f ),which implies a limitation of all the three known Ambainis’s lower bound techniques.Theorem 6Alb 3(f )≤u x,i v y,i ≤NC −(f ).With out loss of generality,we assume that C −(f )=C 0(f ),and X ⊆f −1(0)and Y ⊆f −1(1).We can actually further assume that R =X ×Y ,because otherwise we just let R ′=X ×Y ,and set new weight functions as follows.u ′(x,y,i )= u (x,y,i )(x,y )∈R 0otherwise ,v ′(x,y,i )= v (x,y,i )(x,y )∈R 0otherwise,w ′(x,y )= w (x,y )(x,y )∈R 0otherwise.Then it is easy to see that it satisfies (1)so it is also a weight scheme.And for these new weight functions,we have u ′x,i = y :(x,y )∈R ′,x i =y i u ′(x,y,i )= y :(x,y )∈R,x i =y i u (x,y,i )=u x,i and sim-ilarly v ′y,i =v y,i and w ′x =w x ,w ′y =w y .3It follows that w x w yu ′x,i v ′y,i ,thus we can use(X ′,Y ′,R ′,u ′,v ′,w ′)to derive the same lower bound as we use (X,Y,R,u,v,w ).So now we suppose R =X ×Y and We prove that ∃x ∈X,y ∈Y,i ∈[N ],s.t.w x w y ≤N ·C 0(f )u x,i v y,i ,Suppose the claim is not true.Then for all x ∈X,y ∈Y,i ∈[N ],we havew x w y >N ·C 0(f )u x,i v y,i .(2)We first fix i for the moment.And for each x ∈X ,we fix a smallest certificate set CS x of f on x .Clearly |CS x |≤C 0(f ).We sum (2)over {x ∈X :i ∈CS x }and {y ∈Y }.Then we get x ∈X :i ∈CS x w x y ∈Y w y >N ·C 0(f ) x ∈X :i ∈CS x u x,i y ∈Yv y,i .(3)Note that y ∈Y w y = x ∈X,y ∈Y w (x,y )= x ∈X w x ,and that y ∈Y v y,i = x ∈X,y ∈Y :x i =y i v (x,y,i )= x ∈X v x,i where v x,i = y ∈Y :x i =y i v (x,y,i ).Inequality (3)turns to x ∈X :i ∈CS x w x x ∈X w x >N ·C 0(f ) x ∈X :i ∈CS x u x,i x ∈X v x,i ≥N ·C 0(f ) x ∈X :i ∈CS x u x,i x ∈X :i ∈CS x v x,i ≥N ·C 0(f )( x ∈X :i ∈CS x √3Note that the function values of u ′,v ′,w ′are zero when (x,y )=R ,which does not conform to the definition of weight scheme.But actually Theorem 3also holds for u ≥0,v ≥0,w ≥0as long as u x,i ,v y,i ,w x ,w y are all strictly positive for any x,y,i .This can be seen from the proof of Alb 4in Section 4.due to Cauchy-Schwartz Inequality.We further note thatu x,i v x,i= y∈Y:x i=y i u(x,y,i) y∈Y:x i=y i v(x,y,i)≥( y∈Y:x i=y iN factor seems too large.But in fact it is necessary at least for partialfunctions.Consider the problem of Invert A Permutation[3]4,where C0(f)=C1(f)=1but√even the Alb2(f)=Ω((f)=Θ(N),NCwe derive the following interesting corollary from the above theorem.Corollary7Alb3is not tight.We make some remarks on the quantityN(N−Γ(f)))by using Paturi’s result deg(f)=Θ(N·C−(f)).N)lower bound still holds.3.2Two better upper bounds for total functionsIt turns out that if the function is total,then the upper bound can be further tightened.We introduce a new measure which basically characterizes the size of intersection of a0-and1-certificate sets. Definition3For any function f,if there is a certificate set assignment CS:{0,1}N→2[N] such that for any inputs x,y with f(x)=f(y),|CS x∩CS y|≤k,then k is called a candidate certificate intersection complexity of f.The minimal candidate certificate intersection complexity of f is called the certificate intersection complexity of f,denoted by CI(f).In other words,CI(f)= min CS max x,y:f(x)=f(y)|CS x∩CS y|.Now the improved theorem is as follows.Note that by the above definition we know CI(f)≤C−(f), thus the the following theorem really improves Theorem7for total functions. Theorem8Alb3(f)≤u x,i v y,i≤N·CI(f).Similar to the proof for Theorem6,we assume with out loss of generality that R=X×Y and for all x∈X,y∈Y,we havew x w y>N·CI(f)u x,i v y,i.(5) and we shall show a contradiction.Nowfirstfix i and sum(5)over{x∈X:i∈CS x}and {y∈Y:i∈CS y}.We getx∈X,y∈Y:i∈CS x∩CS y w x w y>N·CI(f) x∈X:i∈CS x u x,i y∈Y:i∈CS y v y,i=N·CI(f) x∈X,y∈Y:i∈CS x,x i=y i u(x,y,i)· x∈X,y∈Y:i∈CS y,x i=y i v(x,y,i)≥N·CI(f) x∈X,y∈Y:i∈CS x∩CS y,x i=y i u(x,y,i)· x∈X,y∈Y:i∈CS x∩CS y,x i=y i v(x,y,i)≥N·CI(f)( x∈X,y∈Y:i∈CS x∩CS y,x i=y ias desired.In[5],Ambainis proposed the open problem And-Or Tree.In the problem,there is a complete binary tree with height2n.Any node in odd levels is labelled with AND and any node in even levels is labelled with OR.The N=4n leaves are the input variables,and the value of the function is the value that we get at the root,with value of each internal node calculated from the values of√its two children in the common AND/OR interpretation.The best quantum lower bound isΩ(33N and thusN.Proof It is sufficient to prove that there is a certificate assignment CS s.t.|CS x∩CS y|=1for any f(x)=f(y).In fact,by a simple induction,we can prove that the standard certificate assignment satisfies this property.The base case is trivial.For the induction step,we note that for an AND connection of two subtrees,the0-certificate set of the new larger tree can be chosen as any one of the two0-certificate sets of the two subtrees,and the1-certificate set of the new larger tree can be chosen as the union of the two1-certificate sets of the two subtrees.As a result,the intersection of the two new certificate sets is not enlarged.The OR connection of two subtrees is analyzed in the same way.Thus the intersection of thefinal0-and1-certificate sets is of size1.We can also tighten theC0(f)C1(f),for any total Boolean function f.Proof For any(X,Y,R,u,v,w)in Theorem3,we assume without loss of generality that X⊆f−1(0),Y⊆f−1(1)and R=X×Y.We are to prove∃x,y,i,j s.t.w x w y≤C0(f)C1(f)u x,i v y,j. Suppose this is not true,i.e.for all x∈X,y∈Y,i,j∈[N],w x w y>C0(f)C1(f)u x,i v y,j.Firstfix x,y and sum over i∈CS x and j∈CS y.Since|CS x|≤C0(f),|CS y|≤C1(f),we havew x w y> i∈CS x u x,i j∈CS y v y,jNow we sum over x∈X and y∈Y,x∈X w x y∈Y w y> x∈X,i∈CS x u x,i y∈Y,j∈CS y v y,j= x∈X,y∈Y,i∈[N]:x i=y i,i∈CS x u(x,y,i) x∈X,y∈Y,j∈[N]:x j=y j,j∈CS y v(x,y,j) Since f is total,there is at least one i0∈CS x∩CS y s.t.x i0=y i0.Thusx∈X w x y∈Y w y> x∈X,y∈Y u(x,y,i0) x∈X,y∈Y v(x,y,i0)≥ x∈X,y∈Y u(x,y,i0)v(x,y,i0)≥( x∈X,y∈Yn). But both N·CI(f)areΘ(n).4A further generalized Ambainis’s lower boundWhile Alb2and Alb3use different ideas to generalize Alb1,it is natural to combine both and get a further generalization.The following theorem is a result in this direction.This theorem to Theorem3is as Theorem2to Theorem1.The proof is similar to the ones in[3,5],with inner products substituted for density operators to make it look easier5.Theorem11Let f:I N→{0,1}where I is afinite set,and X,Y be two sets of inputs s.t.f(x)=f(y)if x∈X and y∈Y.Let R⊆X×Y.Let w,u,v be a weight scheme for X,Y,R.ThenQ2(f)=Ω( u x,i v y,i)Proof The query computation is a sequence of operations U0→O x→U1→...→U T on somefixed initial state,say|0 .Note that here T is the number of queries.Denote|ψk x =U k−1O x...U1O x U0|0 . Note that|ψ1x =|0 for all input x.Because the computation is correct with high probability(1−ǫ),for any(x,y)∈R,the twofinal states have to have some distance to let the measurement distinguish them.In other words,we can assume that| ψT x|ψT y |≤c for some constant c<1.Now supposethat|ψk−1x = i,a,zαi,a,z|i,a,z ,|ψk−1y = i,a,zβi,a,z|i,a,zwhere i is for the index address,a is for the answer,and z is the workspace.Then the oracle worksas follows.O x|ψk−1x = i,a,zαi,a,z|i,a⊕x i,z = i,a,zαi,a⊕x i,z|i,a,zO y|ψk−1y = i,a,zβi,a,z|i,a⊕y i,z = i,a,zβi,a⊕y i,z|i,a,zSo we haveψk x|ψk y = i,a,zα∗i,a⊕x i,zβi,a⊕y i,z= i,a,z:x i=y iα∗i,a⊕x i,zβi,a⊕y i,z+ i,a,z:x i=y iα∗i,a⊕x i,zβi,a⊕y i,z= ψk−1x|ψk−1y + i,a,z:x i=y iα∗i,a⊕x i,zβi,a⊕y i,z− i,a,z:x i=y iα∗i,a,zβi,a,zThus1−c=1−| ψT x|ψT y |= T k=1(| ψk−1x|ψk−1y |−| ψk x|ψk y |)≤ T k=1| ψk−1x|ψk−1y − ψk x|ψk y |= T k=1| i,a,z:x i=y i(α∗i,a⊕x i,zβi,a⊕y i,z−α∗i,a,zβi,a,z)|≤ T k=1 i,a,z:x i=y i(|αi,a⊕x i,z||βi,a⊕y i,z|+|αi,a,z||βi,a,z|)Summing up the inequalities for all(x,y)∈R,with weight w(x,y)multiplied,yields(1−c) (x,y)∈R w(x,y)≤ T k=1 (x,y)∈R i,a,z:x i=y i w(x,y)(|αi,a⊕x i,z||βi,a⊕y i,z|+|αi,a,z||βi,a,z|)≤ T k=1 (x,y)∈R i,a,z:x i=y i u(x,y,i)v(x,y,i)(|αi,a⊕x i,z||βi,a⊕y i,z|+|αi,a,z||βi,a,z|)by(1).We then use inequality2AB≤A2+B2to get2(u(x,y,i) u x,i w x u x,i w x|βi,a⊕y i,z|2),2(u(x,y,i) u x,i w x u x,i w x|βi,a,z|2),w x w yDenote A=min x,y,i:(x,y)∈R,xi=y i2 T k=1 i,a,z[ x∈X w x w y w x(|αi,a⊕x i,z|2+|αi,a,z|2)+ y∈Y w x w y w y(|βi,a⊕y i,z|2+|βi,a,z|2)]≤11/Aw x i,a,z(|αi,a⊕x i,z|2+|αi,a,z|2)+ y∈Y1/A T k=1( x∈X w x+ y∈Y w y)=2T A).We denote by Alb4(f)the best possible lower bound for function f achieved by this theorem.It is easy to see that Alb4generalizes Alb3.Alb4may be easier to use than Alb3.However,according to Szegedy’s recent result[30],Alb3,Alb4and the quantum adversary method proposed by Barnum, Saks and Szegedy in[13]are all equivalent.AcknowledgementThe author would like to thank Andrew Yao who introduced the quantum algorithm and quan-tum query complexity area to me,and made invaluable comments to this paper.Yaoyun Shi and Xiaoming Sun also read a preliminary version of the present paper and both,esp.Yaoyun Shi,gave many invaluable comments and corrections.Thanks also to Andris Ambainis for telling that it is still open whether Alb2(f)≤[9]R.Beals,H.Buhrman,R.Cleve,M.Mosca,R.deWolf.Quantum lower bounds by polynomials. Journal of ACM,48:778-797,2001.Earlier versions at FOCS’98and quant-ph/9802049 [10]H.Buhrman,R.Cleve,A.Wigderson,Quantum vs.classical communication and computation, STOC’98,63-68.quant-ph/9802040[11]H.Buhrman,Durr,Ch.,M.Heiligman,P.Hoyer,F.Magniez,M.Santha,R.De Wolf,Quantum algorithms for element distinctness,Complexity’01131-137,quant-ph/0007016.[12]H.Barnum,M.Saks.A lower bound on the quantum query complexity of read-once functions, ECCC2002[13]H.Barnum,M.Saks,M.Szegedy.Quantum query complextiy and semidefinite programming, IEEE Conference on Computational Complexity2003[14]H.Buhrman,R.de plexity measures and decision tree complexity:a survey.Theo-retical Computer Science,Volume288,Issue1,9October2002,Pages21-43[15]C.Durr,M.Mhalla,Y.Lei.Quantum query complexity of graph connectivity, quant-ph/0303169[16]W.van Dam,Quantum oracle interrogation:Getting all information for almost half the price, FOCS’98362-367,quant-ph/9805006[17]C.Durr,M.Heiligman,P.Hyer,M.Mhalla,and Y.Lei.Quantum query complexity of some graph problems.Manuscript,2003,cited by[21][18]D.Deutsch,R.Jozsa.Rapid solution of problems by quantum computation.Proceeding of the Royal Society of London,A439(1992),553-558[19]L.Grover.A fast quantum mechaqnical algorithm for database search,STOC’96,212-219[20]P.Hoyer,J.Neerbek,Y.Shi.Quantu lower bounds of ordered searching,sorting and element distinctness.Algorithmica,34:429-448,2002.Eearlier versions at ICALP’01and quant-ph/0102078[21]plante,F.Magniez.Lower bounds for randomized and quantum query complexity using Kolmogorov arguments.quant-ph/0311189[22]F.Magniez,M.Santha,M.Szegedy.An O(n1.3)quantum algorithm for the Triangle Problem, quant-ph/0310134[23]A.Nayak,F.Wu.The quantum query complexity of approximating the median and related statistics.Proceedings of STOC99,pp.384-393,also quant-ph/9804066.[24]R.Paturi.On the Degree of Polynomials that Approximate Symmetric Boolean Functions, STOC’92,468-474.[25]A.Razborov.Quantum communication complexity of symmetric predicates,Izvestiya of the Russian Academy of Science,mathematics,2002.also quant-ph/0204025.[26]D.Simon.On the power of quantum computation,SIAM Journal on Computing,26:1474-1483, 1997.[27]P.Shor.Polynomial time algorithms for prime factorization and discrete logarithms on a quan-tum computer.SIAM Journal on Computing,26:1484-1509,1997[28]Y.Shi.Quantum lower bounds for the collision and the element distinctness problems. FOCS’02,513-519,Earlier version at quant-ph/0112086[29]Mario Szegedy,On the Quantum Query Complexity of Detecting triangles in Graphs, quant-ph/0310107[30]Mario Szegedy,A note on the limitations of quantum adversary method,manuscript[31]Andrew Yao,Shengyu Zhang,Graph property and circular functions:how low can quantum query complexity go?[32]Shengyu Zhang,On the power of Ambainis’s lower bounds,quant-ph/031106012。
An explicit lower bound for TSP with distances one and two
An Explicit Lower Bound forTSP with Distances One and TwoLars EngebretsenDept.of Numerical Analysis and Computing ScienceRoyal Institute of TechnologySE-10044Stockholm,SWEDENE-mail:enge@nada.kth.seAbstract.We show that it is for anyǫ>0NP-hard to approximate thetraveling salesman problem with distances one and two within4709/4708−ǫ.Our proof is a reduction from systems of linear equations mod2with twounknowns in each equation and at most three occurrences of each variable.1IntroductionA common special case of the traveling salesman problem is the metric trav-eling salesman problem,where the distances between the cities satisfy the triangle inequality.In this paper,we study a further specialization:The traveling salesman problem with distances one and two between the cities. This problem was shown to be NP-complete by Karp[7].Since this means that we have little hope of computing exact solutions,it is interesting to try tofind approximate solutions,i.e.,tours with a weight close to the optimum weight.Christofides[2]has constructed an elegant algorithm approximat-ing the metric traveling salesman problem within3/2.This algorithm also applies to the traveling salesman problem with distances one and two,but it is possible to better;Papadimitriou and Yannakakis[8]have shown that it is possible to approximate the latter problem within7/6.They also show a lower bound;that there exists some constant,which is never given explicitly in the paper,such that it is NP-hard to approximate the problem within that constant.Recently,there has been a renewed interest in the hardness of approxi-mating the traveling salesman problem with distances one and two.Fernan-dez de la Vega and Karpinski[3]and,independently,Fotakis and Spirakis[4] have shown that the hardness result of Papadimitriou and Yannakakis holds also for dense instances.We contribute to this line of research by show-ing an explicit lower bound on the approximability.More specifically,we construct a reduction from linear equations mod2with three occurrences1of each variable to show that it is NP-hard to approximate the traveling salesman problem with distances one and two within4709/4708−ǫ.2PreliminariesDefinition1.We denote by E2-Lin mod2the following maximization problem:Given a system of linear equations mod2with exactly two vari-ables in each equation,maximize the number of satisfied equations.We denote by E2-Lin(3)mod2the special case of E2-Lin mod2where there are exactly three occurrences of each variable.Definition2.We denote by(1,2)-TSP the traveling salesman problem where the distance matrix is symmetric and the entries are either one or two,and by△-TSP the traveling salesman problem where the distance ma-trix is symmetric and obeys the triangle inequality.Remark3.Since(1,2)-TSP is a special case of△-TSP,a lower bound on the approximability of(1,2)-TSP is also a lower bound on the approxima-bility of△-TSP.Remark4.To describe a(1,2)-TSP instance,it is enough to specify the edges of weight one.We will do this by constructing a graph G,and then let the(1,2)-TSP instance have the nodes of G as cities.The distance between two cities u and v is defined to be one if(u,v)is an edge in G and two otherwise.To compute the weight of a tour,it is enough to study the parts of the tour traversing edges of G.Definition5.We will call a node where the tour leaves or enters G an endpoint.A city with the property that the tour both enters and leaves G in that particular city is called a double endpoint,and counts as two endpoints. Remark6.If c is the number of cities and2e is the total number of end-points,the weight of the tour is c+e,since every edge of weight two corre-sponds to two endpoints.3Our constructionTo obtain our hardness result we reduce from E2-Lin(3)mod2.Previous reductions from integer programming[6]and satisfiability[8]to(1,2)-TSP make heavy use of the so called xor gadget.This gadget is used both to link variable gadgets with equation gadgets and to obtain a consistent as-signment to the variables in the original instance.The xor gadget contains twelve nodes,which means that a gadget containing some twenty xor gad-gets for each variable—which is actually the case in the previously known2A B C D E F GH I JKL M N OA B C DEF GHIJKLMNO(a)(b)Figure1.The equation gadget is connected to other gadgets through thevertices A and G and through the edges shown above from the vertices Kand O.Gadget a corresponds to an equation of the form x+y=1andgadget b to an equation of the form x+y=0.reductions—will produce a very poor lower bound.To obtain a reason-able inapproximability result,we modify the previously used xor gadget to construct an equation gadget.A specific node in the equation gadget cor-responds to one occurrence of a variable.Since each variable occurs three times,there are three nodes corresponding to each variable.These nodes are linked together in a variable cluster.The idea behind this is that the extra edges in the cluster should force the nodes to represent the same value for all three occurrences of the variable.This construction contains21nodes for each variable,which is a vast improvement compared to earlier construc-tions.We give our construction in greater detail below.In a sequel of lemmas, we show that the tour produced by any polynomial time algorithm can be assumed to have a certain structure.We do this by showing that we can transform,by local transformations which do not increase the length of the tour,any tour into a tour with the sought structure.This new tour, obtained after the local transformations,can then be used to construct an assignment to the variables in the original E2-Lin(3)mod2instance.Our main result follows from a recent hardness result of Berman and Karpinski[1] together with a correspondence between the length of the tour in the(1,2)-TSP instance and the number of unsatisfied equations in the E2-Lin(3) mod2instance.3.1The equation gadgetThe equation gadget is shown in Fig.1.It is connected to other gadgets in four places.The vertices A and G actually coincide with similar vertices at other gadgets to form a long chain,which is described in more detail in Sec.3.3.The edges from the vertices K and O join the equation gadget with other equation gadgets.We will study this closely in Sec.3.2.No other vertex in the gadget is joined with vertices not belonging to the gadget. Definition7.We will from now on call the vertices K and O in Fig.1the lambda-vertices of the gadget and the boundary edges connected to these3(a)(b)(c)(d)Figure2.Given that the lambda-edges are traversed as shown above,it ispossible to construct a tour through the equation gadget such that there areno endpoints in the gadget.vertices lambda-edges.For short,we will often refer to the pair of lambda-edges linked to a particular lambda-vertex as a lambda.Definition8.We say that a lambda is traversed if both lambda-edges are traversed by the tour,untraversed if none of the lambda-edges are traversed, and semitraversed otherwise.Lemma9.Suppose that we have a tour traversing an equation gadget in such a way that there are no semitraversed lambdas in it.Then it is possible to modify this tour,without increasing its length and without changing the tour on the lambdas,in such a way that the following holds:If the gadget is of the type shown in Fig.1a,there are no endpoints in the gadget if there is exactly one traversed lambda.Otherwise,it is possible to construct a tour with two endpoints in the gadget and impossible to construct a tour with less than two endpoints in the gadget.If the gadget is of the type shown in Fig.1b,there are no endpoints in the gadget if there are either zero or two traversed lambdas.Otherwise,it is possible to construct a tour with two endpoints in the gadget and impossible to construct a tour with less than two endpoints in the gadget.Proof.Figures2and3show that there exists tours with the number of endpoints stated in the lemma.To complete the proof,we must show that it is impossible to construct better tours in the cases where the tour has two endpoints in the gadget.It can never be suboptimal to let the tour traverse the edge AB and the edge FG in Fig.1.Thus we can assume that one tour enters the gadget through the vertex A and that another,or possibly the same,tour enters through the vertex G.Since there are no semitraversed lambdas in the gadget,the only way,other than through the above described vertices,a tour can leave the gadget is through an endpoint,which in turn implies that there is an even number of endpoints in the gadget.4(a)(b)(c)(d)Figure3.Given that the lambda-edges are traversed as shown above,theremust be at least two endpoints in the gadget.There are in fact many tourswith this property,we show only a few above.If there is to be no endpoint in the gadget,all of the edges CHL,DIM and EJN must be traversed by the tour.Also,the edges CD and LM cannot be traversed simultaneously,neither can DE and MN.The only way we can avoid making the vertices D and M endpoints is to traverse either the edges CD and MN or the edges DE and LM.Let us suppose that the lambdas in the gadget are traversed as shown in Fig.3a.By our reasoning above and the symmetry of the gadget,we can assume that the edges AB,CHLMIDEJN,and FG are traversed by the tour.To avoid making the vertex C an endpoint,the tour must traverse the edges BC.To avoid making the vertex N an endpoint the tour must tra-verse the edge NO.But this is impossible,since the right lambda is already traversed.Thus,there is no tour with zero endpoints in the gadget,which implies that is impossible to construct a tour with less than two endpoints in the gadget.With a similar argument,we conclude that the same holds for the other cases shown in Fig.3.Lemma10.Suppose that we have a tour traversing an equation gadget in such a way that there is exactly one semitraversed lambda in it.Then it is possible to modify this tour,without increasing its length and without changing the tour on the lambdas,in such a way that there is one endpoint in the gadget,and it is impossible to construct a tour with less than one endpoint in the gadget.Proof.From Fig.4,we see that we can always construct tours such that there is one endpoint in the gadget.We now show that it is impossible to construct a tour with fewer endpoints.As in the proof of Lemma9,we can assume that one tour enters the gadget at A and that another,or the same, enters at G.Since there is one semitraversed lambda in the gadget,one tour enters the gadget at that lambda,which implies that there must be an odd number of endpoints in the gadget.5(a)(b)(c)(d)Figure 4.If one lambda is semi-traversed,there must be at least oneendpoint in the gadget.(a)(b)(c)(d)Figure5.If both lambdas in a gadget are semi-traversed,there must be atleast two endpoints in the gadget.Lemma11.Suppose that we have a tour traversing an equation gadget in such a way that there are two semitraversed lambdas in it.Then it is possible to modify this tour,without increasing its length and without changing the tour on the lambdas,in such a way that there are two endpoints in the gadget, and it is impossible to construct a tour with less than two endpoints in the gadget.Proof.From Fig.5,we see that we can always construct tours such that there are two endpoints in the gadget.In order to prove last part of the lemma we must argue that it is impossible to traverse the gadget in such a way that there are no endpoints in it.By an argument similar to that present in the proofs of Lemmas9and10there will be four tours entering the gadget,which implies that there is an even number of endpoints in the gadget.If there is to be no endpoints in the graph,the four tours entering it must actually be connected to each other.Since the tours cannot cross each other and the gadget is planar we have two possible cases.Thefirst case is that the tour entering the gadget at A leaves it at G and the tour entering at K leaves at O.These two tours cannot,however,6traverse any of the edges CHL,DIM and EJN without crossing or touching each other.As noted in the proof of Lemma9,all three abovementioned edges must be traversed for the gadget to contain no endpoints.Thus,we can rule this case out.The second case is that the tour entering the gadget at A leaves it at K and the tour entering at G leaves at O.Since these two tours cannot traverse all three edges CHL,DIM and EJN without crossing or touching each other, we conclude that at least two endpoints must occur within the equation gadget.Lemma12.It is always possible to change a semitraversed lambda to ei-ther a traversed or an untraversed lambda without increasing the number of endpoints in the tour.Proof.First suppose that only one of the lambdas in the equation gadget is semitraversed.By Lemma10we can assume that the gadgets are traversed according to Fig.4.Let us study the tour shown in Fig.4a.By replacing it with the tour shown in Fig.2a,we remove one endpoint from the equation gadget,but we may in that process introduce one endpoint somewhere else in the graph.In proof,letλbe the left lambda-vertex in Fig.4a and v be the vertex adjacent toλthrough the untraversed lambda edge.If v is an endpoint,we simply let the tour ending at v continue toλ,thereby saving one endpoint.If v is not an endpoint,we have to reroute the tour at v toλ. This introduces an endpoint at a neighbor of v,but that endpoint is set offagainst the endpoint removed from the equation gadget.To sum up,we have shown that it is possible to convert the tour in Fig.4a to the one in Fig.2a without increasing the total number of endpoints in the graph.In a similar way,we can convert the tour in Fig.4b to the one in Fig2b,the tour in Fig.4c to the one in Fig2c,and the tour in Fig.4d to the one in Fig2d,respectively.Finally,suppose that both lambdas are semitraversed.By Lemma11 we can assume that the gadgets are traversed according to Fig.5.By the method described in the previous paragraph we can convert the tour in Fig.5a to the one in Fig4a,the tour in Fig.5b to the one in Fig4b,the tour in Fig.5c to the one in Fig4c,and the tour in Fig.5d to the one in Fig4d,respectively.3.2The variable clusterThe variable cluster is shown in Fig.6.The vertices A and B coincide with similar vertices at other gadgets to form a long chain,as described in Sec.3.3.Suppose that the variable cluster corresponds to some variable x. Then,the upper three vertices in the cluster are lambda-vertices in the equation gadgets corresponding to equations where x occurs.The remaining two vertices in the cluster are not joined with vertices outside the cluster.7A BFigure6.There is one lambda-vertex for each occurrence of each variable.The three lambda-vertices corresponding to one variable in the system of lin-ear equations are joined together in a variable cluster.The three uppermostvertices in thefigure are the lambda-vertices.(a)(b)Figure7.If thefirst or the last lambda in a variable cluster is semitraversed,the cluster will be traversed as shown above.Lemma13.Suppose that we have a tour traversing a cluster in such a way that there are some semitraversed lambdas in it.Then,it is possible to modify the tour,without making it longer,in such way that there are no semitraversed lambdas in the cluster.Proof.Suppose that there is one semitraversed lambda.If the semitraversed lambda is the middle lambda of the variable cluster it can,by Lemma12, be transformed into either a traversed or an untraversed lambda.This will move the semitraversed lambda to the end of the cluster.Then the cluster will look as in Fig.7.By moving the endpoint in the variable cluster to the equation gadget corresponding to the semitraversed lambda,we can make the last semitraversed lambda traversed or untraversed without changing the number of endpoints.Suppose now that there are two semitraversed lambdas.By Lemma12, they can be transformed into either a traversed or an untraversed lambda without changing the number of endpoints in the tour.This implies that we can transform the tour in such a way that there is only one semitraversed lambda without changing the number of endpoints in the tour.Then we can use the method from the above paragraph to transform that tour in such a way that there are no semitraversed lambdas.Finally,suppose that all three lambdas are semitraversed.Then the variable cluster would be traversed as in Fig.8.By Lemma12,the tour can be transformed in such a way that the two outer lambdas in the variable Figure8.If all lambdas in a variable cluster are semitraversed,the clusterwill be traversed as shown above.8Figure9.All equation gadgets and all variable clusters are linked togetherin a circular chain as shown schematically above.The equation gadgets areat the top of thefigure and the variable clusters at the bottom.The preciseorder of the gadgets is not important.For clarity,we have omitted ninevertices from each equation gadget.cluster are either traversed or untraversed without changing the weight of the tour.If the center lambda is not semitraversed after the transformation, the proof is complete.Otherwise we can apply thefirst paragraph of this proof.3.3The entire(1,2)-TSP instanceTo produce the(1,2)-TSP instance,all equation gadgets are linked together in series,followed by all variable clusters.Thefirst equation gadget is also linked to the last variable cluster.The construction is shown schematically in Fig.9.The precise order of the individual gadgets in the chain is not important.Our aim in the analysis of our construction is to show that we can con-struct from a tour containing e edges of weight two an assignment to the variables such that at most e equations are not satisfied.When we combine this with Berman’s and Karpinski’s recent hardness results for E2-Lin(3) mod2[1],we obtain Theorem15,our main result.Theorem14.Given a tour with2e endpoints,we can construct an assign-ment leaving at most e equations unsatisfied.Proof.Given a tour,we can by Lemmas9–13construct a new tour,without increasing its length,such that for each variable cluster either all or no lambda-edges are traversed.Then we can construct an assignment as follows: If the lambda-edges in a cluster are traversed by the tour,the corresponding variable is assigned the value one;otherwise it is assigned zero.By Lemma9, this assignment has the property that there are two endpoints in the equation gadgets corresponding to unsatisfied equations.Thus,the assignment will leave at most e equations unsatisfied if there are2e endpoints.Theorem15.It is NP-hard to decide whether an instance of the travel-ing salesman problem with distances one and two with4704n nodes has an optimum tour with length above(4709−ǫ1)n or below(4708+ǫ2)n.9Corollary16.It is for anyǫ>0NP-hard to approximate the traveling salesman problem with distances one and two within4709/4708−ǫ.Proof of Theorem15.The result of Berman and Karpinski[1]states that it is NP-hard to determine if an instance of E2-Lin(3)mod2with336n equa-tions has its optimum above(332−ǫ2)n or below(331+ǫ1)n.If we construct from an instance of E2-Lin(3)mod2an instance of(1,2)-TSP as described above,the graph will contain42n nodes if the E2-Lin(3)mod2instance contains2n variables and3n equations.Thus,Theorem14and the above hardness result together imply that it is NP-hard to decide whether an instance of(1,2)-TSP with4704n nodes has an optimum tour with length above(4709−ǫ1)n or below(4708+ǫ2)n.4Concluding remarksWe have shown in this paper that it is for anyǫ>0NP-hard to approximate (1,2)-TSP within4709/4708−ǫ.Since the best known upper bound on the approximability is7/6,there is certainly room for improvements.Our lower bound follows from a sequence of reductions,which makes it unlikely to be optimal.The sequence starts with E3-Lin mod2,systems of linear equations mod2with exactly three variables in each equation.Then follows reductions to,in turn,E2-Lin mod2,E2-Lin(3)mod2,and(1,2)-TSP. Thus,our hardness result ultimately follows from H˚astad’s optimal lower bound on E3-Lin mod2[5].Obvious ways to improve the lower bound is to improve the reductions used in each step,in particular our construction in this paper and the construction of Berman and Karpinski[1].It is probably harder to improve the lower bound on E2-Lin mod2,since the gadgets used in the reduction from E3-Lin mod2to E2-Lin mod2are optimal,in the sense that better gadgets do not exists for that particular reduction[5,9]. Even better of course,would be to obtain a direct proof of a lower bound on(1,2)-TSP.It would also be interesting to study the approximability of △-TSP in general,and try to determine if△-TSP is harder to approximate than(1,2)-TSP.5AcknowledgmentsViggo Kann contributed with fruitful discussions on this subject.Also,Gun-nar Andersson and Viggo Kann gave valuable comments on early versions of the paper.References1.Piotr Berman and Marek Karpinski.On some tighter inapproximability results.Tech-nical Report29,Electronic Colloquium on Computational Complexity,June1998.102.Nicos Christofides.Worst-case analysis of a new heuristic for the travelling salesman problem.Technical Report CS-93-13,Graduate School of Industrial Administration,Carnegie Mellon University,Pittsburgh,1976.3.Wenceslas Fernandez de la Vega and Marek Karpinski.On approximation hardness of dense TSP and other path problems.Technical Report 24,Electronic Colloquium on Computational Complexity,April 1998.4.Dimitris A.Fotakis and Paul G.Spirakis.Graph properties that faciliate travelling.Technical Report 31,Electronic Colloquium on Computational Complexity,June 1998.5.Johan H ˚astad.Some optimal inapproximability results.In Proceedings Twenty-ninth Annual ACM Symposium on Theory of Computing ,pages 1–10.ACM,New York,1997.6.David S.Johnson and Christos putational complexity.In Eugene wler,Jan K.Lenstra,Alexander H.G.Rinnoy Kan,and David B.Shmoys,editors,The Traveling Salesman Problem ,chapter 3,pages 37–85.John Wiley &Sons,New York,1985.7.Richard M.Karp.Reducibility among combinatorial problems.In Raymond ler and James W.Thatcher,editors,Complexity of Computer Computations ,pages 85–103.Plenum Press,New York,1972.8.Christos H.Papadimitriou and Mihalis Yannakakis.The traveling salesman problem with distances one and two.Mathematics of Operations Research ,18(1):1–11,February 1993.9.Luca Trevisan,Gregory B.Sorkin,Madhu Sudan,and David P.Williamson.Gadgets,approximation,and linear programming.In Proceedings of 37th Annual IEEE Sympo-sium on Foundations of Computer Science ,pages 617–626.IEEE Computer Society,Los Alamitos,1996.11ftpmail@ftp.eccc.uni-trier.de, subject ’help eccc’ftp://ftp.eccc.uni-trier.de/pub/eccchttp://www.eccc.uni-trier.de/ecccECCC ISSN 1433-8092。
An Algorithm that Computes a Lower Bound on the Distance Between a Segment and Z
Ecole Normale Supérieure de Lyon Unité de recherche associée au CNRS n°1398
An Algorithm that Computes a Lower Bound on the Distance Between a Segment and Z 2
An Algorithm that Computes a Lower Bound on the Distance Between a Segment and Z 2
Vincent Lefevre June 1997
Abstract
We give a fast algorithm for computing a lower bound on the distance between a straight line and the points of a regular grid. This algorithm is used to nd worst cases when trying to round the elementary functions correctly in oating-point arithmetic, which consists in returning the machine number that is the closest (there are other rounding modes) to the exact result.
Vincent Lefevre
June 1997
Research Report No 97-18
Ecole Normale Supérieure de Lyon
Lower Bounds to the Size of the Minimum Feedback Vertex Sets in Split-stars
(中英文对照)2014秋季算法分析与设计模拟试卷
北京工业大学2014 ——2015 学年第二学期算法设计与分析期末考试模拟试卷 A卷考试说明：承诺：本人已学习了《北京工业大学考场规则》和《北京工业大学学生违纪处分条例》，承诺在考试过程中自觉遵守有关规定，服从监考教师管理，诚信考试，做到不违纪、不作弊、不替考。
若有违反，愿接受相应的处分。
承诺人：学号：班号：。
注：本试卷共三大题，共 6 页，满分100分，考试时答案请写在试卷空白处。
一、算法时间复杂性问题（共30分）Part 1. The Time Complexity Of the Algorithm Test1、试证明下面的定理：[12分](1) 如果f(n)=O(s(n))并且g(n)=O(r(n))，则f(n)+g(n)=O(s(n)+r(n))(2) 如果f(n)=O(s(n))并且g(n)=O(r(n))，则f(n)*g(n)=O(s(n)*r(n)) 1. Prove the following Theorem [12 marks](1) if f(n)=O(s(n)) and g(n)=O(r(n)), to prove f(n)+g(n)=O(s(n)+r(n))(2) if f(n)=O(s(n)) and g(n)=O(r(n))，to prove f(n)*g(n)=O(s(n)*r(n))2、已知有如下的断言：f(n)=O(s(n))并且g(n)=O(r(n))蕴含f(n)-g(n)=O(s(n)-r(n)) 请你举出一个反例。
[8分]2. Known as the following assertionIf f(n)=O(s(n)) and g(n)=O(r(n))，then f(n)-g(n)=O(s(n)-r(n)) 。
Please cite a counter-example [8 marks]3、假设某算法在输入规模为n时的计算时间为：T（n）=3*2n，在A型计算机上实现并完成该算法的时间为t秒，现有更先进的B型计算机，其运算速度为A 型计算机的256倍。
A theory of weak bisimulation for core cml
J.Functional Programming1(1):1–000,January1993c1993Cambridge University Press1A Theory of Weak Bisimulation for Core CMLWILLIAM FERREIRA†Computing LaboratoryUniversity of CambridgeMATTHEW HENNESSY AND ALAN JEFFREY‡School of Cognitive and Computing SciencesUniversity of Sussex1IntroductionThere have been various attempts to extend standard programming languages with con-current or distributed features,(Giacalone et al.,1989;Holmstr¨o m,1983;Nikhil,1990). Concurrent ML(CML)(Reppy,1991a;Reppy,1992;Panangaden&Reppy,1996)is a practical and elegant example.The language Standard ML is extended with two new type constructors,one for generating communication channels,and the other for delayed com-putations,and a new function for spawning concurrent threads of computation.Thus the language has all the functional and higher-order features of ML,but in addition pro-grams also have the ability to communicate with each other by transmitting values along communication channels.In(Reppy,1992),a reduction style operational semantics is given for a subset of CML calledλcv,which may be viewed as a concurrent version of the call-by-valueλ-calculus of(Plotkin,1975).Reppy’s semantics gives reduction rules for whole programs,not for program fragments.It is not compositional,in that the semantics of a program is not defined in terms of the semantics of its subterms.Reppy’s semantics is designed to prove properties about programs(for example type safety),and not about program fragments(for example equational reasoning).In this paper we construct a compositional operational semantics in terms of a labelled †William Ferreira was funded by a CASE studentship from British Telecom.‡This work is carried out in the context of EC BRA7166CONCUR2.2W.Ferreira,M.Hennessy and A.S.A.Jeffreytransition system,for a core subset of CML which we callµCML.This semantics not only describes the evaluation steps of programs,as in(Reppy,1992),but also their communi-cation potentials in terms of their ability to input and output values along communication channels.This semantics extends the semantics of higher-order processes(Thomsen,1995) with types andfirst-class functions.We then proceed to demonstrate the usefulness of this semantics by using it to define a version of weak bisimulation,(Milner,1989),suitable forµCML.We prove that,modulo the usual problems associated with the choice operator of CCS,our chosen equivalence is preserved by allµCML contexts and therefore may be used as the basis for reasoning about CML programs.In this paper we do not investigate in detail the resulting theory but confine ourselves to pointing out some of its salient features;for example standard identities one would expect of a call-by-valueλ-calculus are given and we also show that certain algebraic laws common to process algebras,(Milner,1989),hold.We now explain in more detail the contents of the remainder of the paper.In Section2we describeµCML,a monomorphically typed core subset of CML,which nonetheless includes base types for channel names,booleans and integers,and type con-structors for pairs,functions,and delayed computations which are known as events.µCML also includes a selection of the constructs and constants for manipulating event types,such as and for constructing basic events for sending and receiving values, for combining delayed computations,for selecting between delayed compu-tations,and a function for launching new concurrent threads of computation within a program.The major omission is thatµCML has no facility for generating new channel names.However we believe that this can be remedied by using techniques common to the π-calculus,(Milner,1991;Milner et al.,1992;Sangiorgi,1992).In the remainder of this section we present the operational semantics ofµCML in terms of a labelled transition system.In order to describe all possible states which can arise dur-ing the computation of a well-typedµCML program we need to extend the language.This extension is twofold.Thefirst consists in adding the constants of event type used by Reppy in(Reppy,1992)to defineλcv,i.e.constants to denote certain delayed computations.This extended language,which we callµCML cv,essentially coincides with theλcv,the lan-guage used in(Reppy,1992),except for the omissions cited above.However to obtain a compositional semantics we make further extensions toµCML cv.We add a parallel oper-ator,commonly used in process algebras,which allows us to use programs in place of the multisets of programs of(Reppy,1992).Thefinal addition is more subtle;we include inµCML cv expressions which correspond to the ed versions of Reppy’s constants for representing delayed computations.Thus the labelled transition system uses as states programs from a language which we call µCML.This language is a superset ofµCML cv,which is our version of Reppy’sλcv, which in turn is a superset ofµCML,our mini-version of CML.The following diagramA Theory of Weak Bisimulation for Core CML3indicates the relationships between these languages:µCMLλcvCMLIn Section3we discuss semantic equivalences defined on the labelled transition of Sec-tion2.We demonstrate the inadequacies of the obvious adaptations of strong and weak bisimulation equivalence,(Milner,1989),and then consider adaptations of higher-order and irreflexive bisimulations from(Thomsen,1995).Finally we suggest a new variation called hereditary bisimulation equivalence which overcomes some of the problems en-countered with using higher-order and irreflexive bisimulations.In Section4we show that hereditary bisimulation is preserved by allµCML contexts.This is an application of the proof method originally suggested in(Howe,1989)but the proof is further complicated by the fact that hereditary bisimulations are defined in terms of pairs of relations satisfying mutually dependent properties.In Section5we briefly discuss the resulting algebraic theory ofµCML expressions.This paper is intended only to lay the foundations of this theory and so here we simply indicate that our theory extends both that of call-by-valueλ-calculus(Plotkin,1975)and process algebras(Milner,1989).In Section6we show that,up to weak bisimulation equivalence,our semantics coincides with the reduction semantics forλcv presented in(Reppy,1992).This technical result ap-plies only to the common sub-language,namelyµCML cv.In Section7we briefly consider other approaches to the semantics of CML and related languages and we end with some suggestions for further work.2The LanguageIn this section we introduce our languageµCML,a subset of Concurrent ML(Reppy, 1991a;Reppy,1992;Panangaden&Reppy,1996).We describe the syntax,including a typing system,and an operational semantics in terms of a labelled transition system. Unfortunately,there is not enough space in this paper to provide an introduction to pro-gramming in CML:see(Panangaden&Reppy,1996)for a discussion of the design and philosophy of CML.The type expressions for our language are given by:A::A A A A A AThus we have three base types,,and;the latter two are simply examples of useful base types and one could easily include more.These types are closed under four con-structors:pairing,function space,and the less common and type constructors.4W.Ferreira,M.Hennessy and A.S.A.JeffreyOur language may be viewed as a typedλ-calculus augmented with the type constructors A for communication channels sending and receiving data of type A,and A for constructing delayed computations of type A.Let Chan A be a type-indexed family of disjoint sets of channel names,ranged over by k, and let Var denote a set of variables ranged over by x,y and z.The expressions ofµCML are given by the following abstract syntax:e f g Exp::v ce e e e e e x e e eev w Val::x y e x k01c Const::The main syntactic category is that of Exp which look very much like the set of expressions for an applied call-by-value version of theλ-calculus.There are the usual pairing,let-binding and branching constructors,and two forms of application:the application of one expression to another,ee,the application of a constant to an expression,ce.There is also a syntactic category of value expressions Val,used in giving a semantics to call-by-value functions and communicate-by-value channels.They are restricted in form: either a variable,a recursively defined function,x y e,or a predefined literal value for the base types.We will use some syntax sugar,writing y e for x y e when x does not occur in e,and e;f for x e f when x does not occur in f. Finally there are a small collection of constant functions.These consist of a representa-tive sample of constants for manipulating objects of base type,,which could easily be extended,the projection functions and,together with the set of constants for manipulating delayed computations taken directly from(Reppy,1992):and,for constructing delayed computations which can send and receive values,,for constructing alternatives between delayed computations,,for spawning new computational threads,,for launching delayed computations,,for combining delayed computations,,for a delayed computation which always deadlocks,and,for a delayed computation which immediately terminates with a value. Note that there is no method for generating channel names other than using the predefined set of names Chan A.There are two constructs in the language which bind occurrences of variables,xe1e2where free occurrences of x in e2are bound and x y e where free oc-currences of both x and y in e are bound.We will not dwell on the precise definitions of free and bound variables but simply use f v e to denote the set of variables which have free occurrences in e.If f v e/0then e is said to be a closed expression,which we sometimes refer to as a program.We also use the standard notation of e v x to denote the substitution of the value v for all free occurrences of x in e where bound names may be changed in order to avoid the capture of free variables in v.(Since we are modelling aA Theory of Weak Bisimulation for Core CML5:A B A:A A:A B B:A A::A A A::::A A B B:A A:A:A AFigure1a.Type rules forµCML constant functionsx yΓx:A y:BΓ:Γ:Γx y e:A BΓe:AΓf:BΓe:AΓe f:BΓe:AΓx:A f:BΓe f g:A6W.Ferreira,M.Hennessy and A.S.A.Jeffreythis reduction semantics are of the form:CτCwhere C C are configurations which combine a closed expression with a run-time envi-ronment necessary for its evaluation,andτis Milner’s notation for a silent action.However this semantics is not compositional as the reductions of an expression can not be deduced directly from the reductions of it constituent components.Here we give a compositional operational semantics with four kinds of judgements:eτe,representing a one step evaluation or reduction,e v e,representing the production of the value v,with a side effect e,e k?x e,representing the potential to input a value x along the channel k,ande k!v e,representing the output of the value v along the channel k.These are formally defined in Figure2,but wefirst give an informal overview.In order to define these relations we introduce extra syntactic constructs.These are introduced as required in the overview but are summarized in Figure3.The rules for one step evaluation or reduction have much in common with those for a standard call-by-valueλ-calculus.But in addition a closed expression e of type A should evaluate to a value of type A and it is this production of values which is the subject of the second kind of judgement.HoweverµCML expressions can spawn subprocesses before returning a value,so we have to allow expressions to continue evaluation even after they have returned a result.For example in the expression:0a;aone possible reduction is(whereτindicates a sequence ofτ-reductions):0a;aτa?11a!0where the process returns the value1before outputting0.For this reason we need a reduc-tion e v e rather than the more usual termination e v.The following diagram illustrates all of the possible transitions from this expression:0a;aτa!0τa?vva!0vA Theory of Weak Bisimulation for Core CML7 judgements of the operational semantics apply to these configurations.The second,more common in work on process algebras,(Bergstra&Klop,1985;Milner,1989),extends the syntax of the language being interpreted to encompass configurations.We choose the latter approach and one extra construct we add to the language is a parallel operator,e f.This has the same operational rules as in CCS,allowing reduction of both processes:eαee fαe fand communication between the processes:e k!v ef k?x fe fτe v x fThe assymetry is introduced by termination(a feature missing from CCS).A CML process has a main thread of control,and only the main thread can return a value.By convention, we write the main thread on the right,so the rule is:f v feαeSecondly,e may have spawned some concurrent processes before returning a function,and these should carry on evaluation,so we use the silent rule for constant application:e v e8W.Ferreira,M.Hennessy and A.S.A.JeffreyThe well-typedness of the operational semantics will ensure that v is a function of the appropriate type,.With this method of representing newly created computation threads more of the rules corresponding toβ-reduction in a call-by-valueλ-calculus may now be given.To evaluate an application expression e f,first e is evaluated to a value of functional form and then the evaluation of f is initiated.This is represented by the rules:eαee fτe yf g(In fact we use a slightly more complicated version of the latter rule as functions are al-lowed to be recursive.)Continuing with the evaluation of e f,we now evaluate f to a value which is then substituted into g for y.This is represented by the two rules:fτfx f gτf g v xThe evaluation of the application expression c f is similar;f is evaluated to a value and then the constant c is applied to the resulting value.This is represented by the two rulesfτfc fτfδc vHere,borrowing the notation of(Reppy,1992),we use the functionδto represent the effect of applying the constant c to the value v.This effect depends on the constant in question and we have already seen one instance of this rule,for the constant,which result from the fact thatδv v.The definition ofδfor all constants in the language is given in Figure2f.For the constants associated with the base types this is self-explanatory; the others will be explained below as the constant in question is considered.Note that because of the introduction of into the language we can treat all constants uniformly, unlike(Reppy,1992)where and have to considered in a special manner.In order to implement the standard left-to-right evaluation of pairs of expressions we introduce a new value v w representing a pair which has been fully evaluated.Then to evaluate e f:first allow e to evaluate:eαee fτe xf v xThese value pairs may then be used by being applied to functions of type A B.For example the following inferences result from the definition of the functionδfor the constants and:e v w eeτe m nIt remains to explain how delayed computations,i.e.programs of type A,are han-dled.It is important to realise that expressions of type A represent potential rather than actual computations and this potential can only be activated by an application of theA Theory of Weak Bisimulation for Core CML9eαee fαe feαee f gαe f geαee fαef fαfe f v e fFigure2a.Operational semantics:static rules ge1αege1ge2αegeαeceτeδc v e ee f gτe ge v ee fτe yfg v xv x y g e v ee fτef v xe k?x ef k!v fv vΛk?k?x x10W.Ferreira,M.Hennessy and A.S.A.Jeffreye f g Exp::v ce e e e e e x e e eev w Val::x y e x k01c Const::Figure3a.Syntax ofµCMLv w Val::v v gege GExp::v!v v?ge v ge geΛA vFigure3b.Syntax ofµCML cve f g Exp::ge e eFigure3c.Syntax ofµCMLconstant,of type A A.Thus for example the expression k is of type A and represents a delayed computation which has the potential to receive a value of type A along the channel k.The expression k can actually receive such a value v along channel k,or more accurately can evaluate to such a value,provided some other computation thread can send the value along channel k.The semantics of is handled by introducing a new constructor for values.For certain kinds of expressions ge of type A,which we call guarded expressions,let ge be a value of type A;this represents a delayed computation which when launched initiates a new computation thread which evaluates the expression ge.Then the expression ge reduces in one step to the expression ge.More generally the evaluation of the expressione proceeds as follows:First evaluate e until it can produce a value:eτeeτe geNote that here,as always,the production of a value may have as a side-effect the generation of a new computation thread e and this is launched concurrently with the delayed compu-tation ge.Also both of these rules are instances of more general rules already considered. Thefirst is obtained from the rule for the evaluation of applications of the form ce and the second by definingδge to be ge.The precise syntax for guarded expressions will emerge by considering what types of values of the form e can result from the evaluation of expressions of type from the basic languageµCML.The constant is of type A A and thereforethe evaluation of the expression e proceeds byfirst evaluating e to a value of type A until it returns a value k,and then returning a delayed computation consisting of an event which can receive any value of type A on the channel k.To represent this event we extend the syntax further by letting k?be a guarded expression for any k and A,with the associated rule:e k eeτe k!vIt is these two new expressions k?and k!v which perform communication between compu-tation threads.Formally k!v is of type and we have the axiom:k?k?x xTherefore in general input moves are of the form e k?x f where e:B and x:A f:B. Communication can now be modelled as in CCS by the simultaneous occurrence of input and output actions:e k?x ef k!v feτeΛobtained,once more,by definingδto beΛ.The constant is of type A A B B.The evaluation of e proceeds in the standard way by evaluating e until it produces a value,which must be of the form ge v,where ge is a guarded expression of type A and v has type A B.Then the evaluation of e continues by the construction of the new delayed computation ge v.Bearing in mind the fact that the production of values can generate new computation threads,this is formally represented by the inference rule:e ge v ege vαveThe construct,of type A A,evaluates its argument to a value v,and thenreturns a trivial a delayed computation;this computation,when activated,immediately evaluates to the value v.In order to represent these trivial computations we introduce a new constructor for guarded expressions,A and the semantics of is then captured by the rule:e v eA vτvThe choice construct e is a choice between delayed computations as has the type A A A.To interpret it we introduce a new choice constructor ge1ge2where ge1and ge2are guarded expressions of the same type.Then e pro-ceeds by evaluating e until it can produce a value,which must be of the form ge1ge2, and the evaluation continues by constructing the delayed computation ge1ge2.This is represented by the rule:e ge1ge2ege2αege1ge2αeΓv:AΓw:BΓge:AΓv:AΓw:AΓv?:AΓge:AΓv:A BΓge1ge2:AΓA v:AΓe:AΓf:BΓτ:A Γv:AΓk?x:AΓw:Bof the form e k ?xf where f may be an open expression we need to consider relations over open expressions.Let an open type-indexed relation R be a family of relations R ΓA such that if e R ΓA f then Γe :A and Γf :A .We will often elide the subscripts from relations,for example writing e R f for e R ΓA f when context makes the type obvious.Let a closed type-indexed relation R be an open type-indexed relation where Γis everywhere the empty context,and can therefore be elided.For any closed type-indexed relation R ,let its open extension R be defined as:e R x :A Bf iff e v x R B f v x for allv :AA closed type-indexed relation R is structure preserving iff:if v R A w and A is a base type then v w ,if v 1v 2R A 1A 2w 1w 2then v i R A i w i ,if ge 1R A ge 2then ge 1R A ge 2,andif v R A B v then for all w :A we have vw R B v w .With this notation we can now define strong bisimulations over µCML expressions.A closed type-indexed relation R is a first-order strong simulation iff it is structure-preserving and the following diagram can be completed:e 1R e 2e 1R e 2ase 1lRe 2lsince the definition of strong bisimulation demands that the actions performed by expres-sions match up to syntactic identity.This counter-example can also be reproduced using only µCML contexts:kx121kx21since the left hand side can perform the move:kx12τk !x12but this can only be matched by the right hand side up to strong bisimulation:kx21τk !x21In fact,it is easy to verify that the only first-order strong bisimulation which is a congruence for µCML is the identity relation.To find a satisfactory treatment of bisimulation for µCML,we need to look to higher-order bisimulation ,where the structure of the labels is accounted for.To this end,given a closed type-indexed relation R ,define its extension to labels R l as:v R l A wk !v R l A k !wkChan BThen R is a higher-order strong simulation iff it is structure-preserving and the followingdiagram can be completed:e 1R e 2e 1R e 2aswhere l 1R l l 2e 1l 1Re 2l 2lotherwise.Then R is a first-order weak simulation iff it is structure-preserving and the following diagram can be completed:e 1R e 2e 1R e 2ase 1lRe 2ˆl Let1be the largest first-order weak bisimulation.Proposition 3.31is an equivalence.ProofSimilar to the proof of Proposition 3.1.Unfortunately,1is not a congruence,for the same reason as 1,and so we can attempt the same modification.R is a higher-order weak simulation iff it is structure-preserving and the following diagram can be completed:e 1R e 2e 1R e 2aswhere l 1R ll 2e 1l 1Re 2ˆl 2Lethbe the largest higher-order weak bisimulation.Proposition 3.4h is an equivalence.ProofSimilar to the proof of Proposition 3.1.However,h is still not a congruence,for the usual reason that weak bisimulation equiva-lence is not a congruence for CCS summation.Recall from (Milner,1989)that in CCS 0τ0but a 00a 0τ0.We can duplicate this counter-example in µCML since the CCS operator corresponds to the µCML operator and 0corresponds to Λ.However may only be applied to guarded expressions and therefore we need a guarded expressionwhich behaves like τ0;the required expression is A Λ.Thus:ΛhA Λsince the right hand side has only one reduction:A ΛτΛτΛbut:Λk !0hA Λk !0because the only reduction of Λk !0is Λk !0k !0ΛΛand:A Λk !0τΛτΛThis counter-example can also be replicated using the restricted syntax of µCML.We have:hsince the left hand side has only one reduction:ΛΛand the right hand side can match this with:A ΛΛand we have seen:ΛhA ΛHowever:k 0hk 0since the left hand side has only one reduction:k 0τΛk !0whereas the right hand side has the reduction:k 0τA Λk !0A first attempt to rectify this is to adapt Milner’s observational equivalence for µCML,and to define h as the smallest symmetric relation such that the following diagram can be completed:e 1he 2e 1he 2aswhere l 1h ll 2e 1l 1he 2l 2Proposition 3.5h is an equivalence.ProofSimilar to the proof of Proposition 3.1.This attempt fails,however,since it only looks at the first move of a process,and not at thefirst moves of any processes in its transitions.Thus,the above µCML counter-example for h being a congruence also applies to h ;i.e.hbut:k 0hk 0This failure was first noted in (Thomsen,1995)for CHOCS.Thomsen’s solution to this problem is to require that τ-moves can always be matched by at least one τ-move,which produces his definition of an irreflexive simulation as a structure-preserving relation where the following diagram can be completed:e 1R e 2e 1R e 2aswhere l 1R l l 2e 1l 1Re 2l 2Letibe the largest irreflexive bisimulation.Proposition 3.6iis a congruence.ProofThe proof that i is an equivalence is similar to the proof of Proposition 3.1.The proof that it is a congruence is similar to the proof of Theorem 4.7in the next section.However this relation is rather too strong for many purposes,for example 12i111since the right hand side can perform more τ-moves than the left hand side.This is similar to the problem in CHOCS where a τP i a P .In order to find an appropriate definition of bisimulation for µCML,we observe that µCML only allows to be used on guarded expressions ,and not on arbitrary expressions.We can thus ignore the initial τ-moves of all expressions except for guarded expressions.For this reason,we have to provide two equivalences:one on terms where we are not interested in initial τ-moves,and one on terms where we are.A pair of closed type-indexed relations R R n R s form a hereditary simulation (we call R n an insensitive simulation and R s a sensitive simulation )iff R s is structure-preserving and we can complete the following diagrams:e 1R ne 2e 1R ne 2aswhere l 1R sll 2e 1l 1R ne 2ˆl 2and:e 1R se 2e 1R se 2aswhere l 1R s l l 2e 1l 1R ne 2l 2Let n sbe the largest hereditary bisimulation.Note that we require R s to be structure-preserving because it is used to compare the labels in transitions,which may contain ab-stractions or guarded events.In the operational semantics of µCML expressions,guarded expressions can only appear in labels,and not as the residuals of transitions.This explains why in the definition of n labels are compared with respect to the sensitive relation s whereas the insensitive relation is used for the residuals.For example,if ge 1n s ge 2then we have:xge 1nxge 2since once either side is applied to an argument,their first action will be a τ-step.On the other hand:ge 1nge 2sinceis precisely the construct which allows us to embed ge 1and ge 2in acontext.Theorem 3.7s is a congruence for µCML ,andnis a congruence for µCML.ProofThe proof that s and n are equivalences is similar to the proof of Proposition 3.1.The proof that they form congruences is the subject of the next section.Proposition 3.8The equivalences on µCML have the following strict inclusions:1shh111x1xk k i h k12s i1111n s x1xh n1h h x1xwhere:x x(Note that this settles an open question(Thomsen,1995)as to whether i is the largestcongruence contained in h.)It is the operator which differentiates between the two equivalences n and h.Howeverin order to demonstrate the difference we need to be able to apply to guarded expressionswhich can spontaneously evolve,i.e.performτ-moves.The onlyµCML constructor for guarded expressions which allows this is A,and in turn occurrences of this can only begenerated by theµCML constructor.Therefore:Proposition3.9For the subset ofµCML without and A,n is the same as h,and s is the same as h.ProofFrom Proposition3.8n h.For the subset ofµCML without and A,define R s as:v w v h w ge1ge2ge1h ge2v1w v2w v1h v2Then since no event without A can perform aτ-move,and since the only initial moves ofv i w areβ-reductions,we can show that h R s forms an hereditary bisimulation,and so h n.From this it is routine to show that s h.Unfortunately we have not been able to show that n is the largestµCML congruence con-tained in weak higher-order bisimulation equivalence.However we do have the following characterisation:Theorem3.10n is the largest higher-order weak bisimulation which respectsµCML contexts.ProofBy definition,n is a higher-order weak bisimulation,and we have shown that it respectsµCML contexts.All that remains is to show that it is the largest such.Let R be a higher-order weak bisimulation which respectsµCML contexts.Then define: R n R v1w e2v1R v2v2wτe2e1v2w v1R v2v1wτe1R s v w v R w ge1ge2ge1R ge2v1w v2w v1R v2We will now show that R n R s forms an hereditary simulation,from which we can de-duce R R n n.。
MaxDEA
Detailed Contents
Chapter 1: Main Features of MaxDEA ..................................................8
1.1 Main Features ............................................................................................... 8 1.2 Models in MaxDEA...................................................................................... 9 1.3 What’s NEW ............................................................................................... 12 1.4 Compare MaxDEA Editions ..................................................................... 17
3.1 Import Data ................................................................................................ 19 3.2 Define Data ................................................................................................. 24 3.3 Set and Run Model..................................................................................... 25 3.4 Export Results ............................................................................................ 77
pwscf说明书
User’s Guide for Quantum ESPRESSO(version4.2.0)Contents1Introduction31.1What can Quantum ESPRESSO do (4)1.2People (6)1.3Contacts (8)1.4Terms of use (9)2Installation92.1Download (9)2.2Prerequisites (10)2.3configure (11)2.3.1Manual configuration (13)2.4Libraries (13)2.4.1If optimized libraries are not found (14)2.5Compilation (15)2.6Running examples (17)2.7Installation tricks and problems (19)2.7.1All architectures (19)2.7.2Cray XT machines (19)2.7.3IBM AIX (20)2.7.4Linux PC (20)2.7.5Linux PC clusters with MPI (22)2.7.6Intel Mac OS X (23)2.7.7SGI,Alpha (24)3Parallelism253.1Understanding Parallelism (25)3.2Running on parallel machines (25)3.3Parallelization levels (26)3.3.1Understanding parallel I/O (28)3.4Tricks and problems (29)4Using Quantum ESPRESSO314.1Input data (31)4.2Datafiles (32)4.3Format of arrays containing charge density,potential,etc (32)5Using PWscf335.1Electronic structure calculations (33)5.2Optimization and dynamics (35)5.3Nudged Elastic Band calculation (35)6Phonon calculations376.1Single-q calculation (37)6.2Calculation of interatomic force constants in real space (37)6.3Calculation of electron-phonon interaction coefficients (38)6.4Distributed Phonon calculations (38)7Post-processing397.1Plotting selected quantities (39)7.2Band structure,Fermi surface (39)7.3Projection over atomic states,DOS (39)7.4Wannier functions (40)7.5Other tools (40)8Using CP408.1Reaching the electronic ground state (42)8.2Relax the system (43)8.3CP dynamics (45)8.4Advanced usage (47)8.4.1Self-interaction Correction (47)8.4.2ensemble-DFT (48)8.4.3Treatment of USPPs (50)9Performances519.1Execution time (51)9.2Memory requirements (52)9.3File space requirements (52)9.4Parallelization issues (52)10Troubleshooting5410.1pw.x problems (54)10.2PostProc (61)10.3ph.x errors (62)11Frequently Asked Questions(F AQ)6311.1General (63)11.2Installation (63)11.3Pseudopotentials (64)11.4Input data (65)11.5Parallel execution (66)11.6Frequent errors during execution (66)11.7Self Consistency (67)11.8Phonons (69)1IntroductionThis guide covers the installation and usage of Quantum ESPRESSO(opEn-Source Package for Research in Electronic Structure,Simulation,and Optimization),version4.2.0.The Quantum ESPRESSO distribution contains the following core packages for the cal-culation of electronic-structure properties within Density-Functional Theory(DFT),using a Plane-Wave(PW)basis set and pseudopotentials(PP):•PWscf(Plane-Wave Self-Consistent Field).•CP(Car-Parrinello).It also includes the following more specialized packages:•PHonon:phonons with Density-Functional Perturbation Theory.•PostProc:various utilities for data postprocessing.•PWcond:ballistic conductance.•GIPAW(Gauge-Independent Projector Augmented Waves):EPR g-tensor and NMR chem-ical shifts.•XSPECTRA:K-edge X-ray adsorption spectra.•vdW:(experimental)dynamic polarizability.•GWW:(experimental)GW calculation using Wannier functions.The following auxiliary codes are included as well:•PWgui:a Graphical User Interface,producing input datafiles for PWscf.•atomic:a program for atomic calculations and generation of pseudopotentials.•QHA:utilities for the calculation of projected density of states(PDOS)and of the free energy in the Quasi-Harmonic Approximation(to be used in conjunction with PHonon).•PlotPhon:phonon dispersion plotting utility(to be used in conjunction with PHonon).A copy of required external libraries are included:•iotk:an Input-Output ToolKit.•PMG:Multigrid solver for Poisson equation.•BLAS and LAPACKFinally,several additional packages that exploit data produced by Quantum ESPRESSO can be installed as plug-ins:•Wannier90:maximally localized Wannier functions(/),writ-ten by A.Mostofi,J.Yates,Y.-S Lee.•WanT:quantum transport properties with Wannier functions.•YAMBO:optical excitations with Many-Body Perturbation Theory.This guide documents PWscf,CP,PHonon,PostProc.The remaining packages have separate documentation.The Quantum ESPRESSO codes work on many different types of Unix machines,in-cluding parallel machines using both OpenMP and MPI(Message Passing Interface).Running Quantum ESPRESSO on Mac OS X and MS-Windows is also possible:see section2.2.Further documentation,beyond what is provided in this guide,can be found in:•the pw forum mailing list(pw forum@).You can subscribe to this list,browse and search its archives(links in /contacts.php).Only subscribed users can post.Please search the archives before posting:your question may have already been answered.•the Doc/directory of the Quantum ESPRESSO distribution,containing a detailed de-scription of input data for most codes infiles INPUT*.txt and INPUT*.html,plus and a few additional pdf documents;people who want to contribute to Quantum ESPRESSO should read the Developer Manual,developer man.pdf.•the Quantum ESPRESSO Wiki:/wiki/index.php/Main Page.This guide does not explain solid state physics and its computational methods.If you want to learn that,you should read a good textbook,such as e.g.the book by Richard Martin: Electronic Structure:Basic Theory and Practical Methods,Cambridge University Press(2004). See also the Reference Paper section in the Wiki.This guide assume that you know the basic Unix concepts(shell,execution path,directories etc.)and utilities.If you don’t,you will have a hard time running Quantum ESPRESSO.All trademarks mentioned in this guide belong to their respective owners.1.1What can Quantum ESPRESSO doPWscf can currently perform the following kinds of calculations:•ground-state energy and one-electron(Kohn-Sham)orbitals;•atomic forces,stresses,and structural optimization;•molecular dynamics on the ground-state Born-Oppenheimer surface,also with variable cell;•Nudged Elastic Band(NEB)and Fourier String Method Dynamics(SMD)for energy barriers and reaction paths;•macroscopic polarization andfinite electricfields via the modern theory of polarization (Berry Phases).All of the above works for both insulators and metals,in any crystal structure,for many exchange-correlation(XC)functionals(including spin polarization,DFT+U,hybrid function-als),for norm-conserving(Hamann-Schluter-Chiang)PPs(NCPPs)in separable form or Ultra-soft(Vanderbilt)PPs(USPPs)or Projector Augmented Waves(PAW)method.Non-collinear magnetism and spin-orbit interactions are also implemented.An implementation offinite elec-tricfields with a sawtooth potential in a supercell is also available.PHonon can perform the following types of calculations:•phonon frequencies and eigenvectors at a generic wave vector,using Density-Functional Perturbation Theory;•effective charges and dielectric tensors;•electron-phonon interaction coefficients for metals;•interatomic force constants in real space;•third-order anharmonic phonon lifetimes;•Infrared and Raman(nonresonant)cross section.PHonon can be used whenever PWscf can be used,with the exceptions of DFT+U and hybrid functionals.PAW is not implemented for higher-order response calculations.Calculations,in the Quasi-Harmonic approximations,of the vibrational free energy can be performed using the QHA package.PostProc can perform the following types of calculations:•Scanning Tunneling Microscopy(STM)images;•plots of Electron Localization Functions(ELF);•Density of States(DOS)and Projected DOS(PDOS);•L¨o wdin charges;•planar and spherical averages;plus interfacing with a number of graphical utilities and with external codes.CP can perform Car-Parrinello molecular dynamics,including variable-cell dynamics.1.2PeopleIn the following,the cited affiliation is either the current one or the one where the last known contribution was done.The maintenance and further development of the Quantum ESPRESSO distribution is promoted by the DEMOCRITOS National Simulation Center of IOM-CNR under the coor-dination of Paolo Giannozzi(Univ.Udine,Italy)and Layla Martin-Samos(Democritos)with the strong support of the CINECA National Supercomputing Center in Bologna under the responsibility of Carlo Cavazzoni.The PWscf package(which included PHonon and PostProc in earlier releases)was origi-nally developed by Stefano Baroni,Stefano de Gironcoli,Andrea Dal Corso(SISSA),Paolo Giannozzi,and many others.We quote in particular:•Matteo Cococcioni(Univ.Minnesota)for DFT+U implementation;•David Vanderbilt’s group at Rutgers for Berry’s phase calculations;•Ralph Gebauer(ICTP,Trieste)and Adriano Mosca Conte(SISSA,Trieste)for noncolinear magnetism;•Andrea Dal Corso for spin-orbit interactions;•Carlo Sbraccia(Princeton)for NEB,Strings method,for improvements to structural optimization and to many other parts;•Paolo Umari(Democritos)forfinite electricfields;•Renata Wentzcovitch and collaborators(Univ.Minnesota)for variable-cell molecular dynamics;•Lorenzo Paulatto(Univ.Paris VI)for PAW implementation,built upon previous work by Guido Fratesi(ano Bicocca)and Riccardo Mazzarello(ETHZ-USI Lugano);•Ismaila Dabo(INRIA,Palaiseau)for electrostatics with free boundary conditions.For PHonon,we mention in particular:•Michele Lazzeri(Univ.Paris VI)for the2n+1code and Raman cross section calculation with2nd-order response;•Andrea Dal Corso for USPP,noncollinear,spin-orbit extensions to PHonon.For PostProc,we mention:•Andrea Benassi(SISSA)for the epsilon utility;•Norbert Nemec(U.Cambridge)for the pw2casino utility;•Dmitry Korotin(Inst.Met.Phys.Ekaterinburg)for the wannier ham utility.The CP package is based on the original code written by Roberto Car and Michele Parrinello. CP was developed by Alfredo Pasquarello(IRRMA,Lausanne),Kari Laasonen(Oulu),Andrea Trave,Roberto Car(Princeton),Nicola Marzari(Univ.Oxford),Paolo Giannozzi,and others. FPMD,later merged with CP,was developed by Carlo Cavazzoni,Gerardo Ballabio(CINECA), Sandro Scandolo(ICTP),Guido Chiarotti(SISSA),Paolo Focher,and others.We quote in particular:•Carlo Sbraccia(Princeton)for NEB;•Manu Sharma(Princeton)and Yudong Wu(Princeton)for maximally localized Wannier functions and dynamics with Wannier functions;•Paolo Umari(Democritos)forfinite electricfields and conjugate gradients;•Paolo Umari and Ismaila Dabo for ensemble-DFT;•Xiaofei Wang(Princeton)for META-GGA;•The Autopilot feature was implemented by Targacept,Inc.Other packages in Quantum ESPRESSO:•PWcond was written by Alexander Smogunov(SISSA)and Andrea Dal Corso.For an introduction,see http://people.sissa.it/~smogunov/PWCOND/pwcond.html•GIPAW()was written by Davide Ceresoli(MIT),Ari Seitsonen (Univ.Zurich),Uwe Gerstmann,Francesco Mauri(Univ.Paris VI).•PWgui was written by Anton Kokalj(IJS Ljubljana)and is based on his GUIB concept (http://www-k3.ijs.si/kokalj/guib/).•atomic was written by Andrea Dal Corso and it is the result of many additions to the original code by Paolo Giannozzi and others.Lorenzo Paulatto wrote the PAW extension.•iotk(http://www.s3.infm.it/iotk)was written by Giovanni Bussi(SISSA).•XSPECTRA was written by Matteo Calandra(Univ.Paris VI)and collaborators.•VdW was contributed by Huy-Viet Nguyen(SISSA).•GWW was written by Paolo Umari and Geoffrey Stenuit(Democritos).•QHA amd PlotPhon were contributed by Eyvaz Isaev(Moscow Steel and Alloy Inst.and Linkoping and Uppsala Univ.).Other relevant contributions to Quantum ESPRESSO:•Andrea Ferretti(MIT)contributed the qexml and sumpdos utility,helped withfile formats and with various problems;•Hannu-Pekka Komsa(CSEA/Lausanne)contributed the HSE functional;•Dispersions interaction in the framework of DFT-D were contributed by Daniel Forrer (Padua Univ.)and Michele Pavone(Naples Univ.Federico II);•Filippo Spiga(ano Bicocca)contributed the mixed MPI-OpenMP paralleliza-tion;•The initial BlueGene porting was done by Costas Bekas and Alessandro Curioni(IBM Zurich);•Gerardo Ballabio wrote thefirst configure for Quantum ESPRESSO•Audrius Alkauskas(IRRMA),Uli Aschauer(Princeton),Simon Binnie(Univ.College London),Guido Fratesi,Axel Kohlmeyer(UPenn),Konstantin Kudin(Princeton),Sergey Lisenkov(Univ.Arkansas),Nicolas Mounet(MIT),William Parker(Ohio State Univ), Guido Roma(CEA),Gabriele Sclauzero(SISSA),Sylvie Stucki(IRRMA),Pascal Thibaudeau (CEA),Vittorio Zecca,Federico Zipoli(Princeton)answered questions on the mailing list, found bugs,helped in porting to new architectures,wrote some code.An alphabetical list of further contributors includes:Dario Alf`e,Alain Allouche,Francesco Antoniella,Francesca Baletto,Mauro Boero,Nicola Bonini,Claudia Bungaro,Paolo Cazzato, Gabriele Cipriani,Jiayu Dai,Cesar Da Silva,Alberto Debernardi,Gernot Deinzer,Yves Ferro, Martin Hilgeman,Yosuke Kanai,Nicolas Lacorne,Stephane Lefranc,Kurt Maeder,Andrea Marini,Pasquale Pavone,Mickael Profeta,Kurt Stokbro,Paul Tangney,Antonio Tilocca,Jaro Tobik,Malgorzata Wierzbowska,Silviu Zilberman,and let us apologize to everybody we have forgotten.This guide was mostly written by Paolo Giannozzi.Gerardo Ballabio and Carlo Cavazzoni wrote the section on CP.1.3ContactsThe web site for Quantum ESPRESSO is /.Releases and patches can be downloaded from this site or following the links contained in it.The main entry point for developers is the QE-forge web site:/.The recommended place where to ask questions about installation and usage of Quantum ESPRESSO,and to report bugs,is the pw forum mailing list:pw forum@.Here you can receive news about Quantum ESPRESSO and obtain help from the developers and from knowledgeable users.You have to be subscribed in order to post to the list.Please browse or search the archive–links are available in the”Contacts”page of the Quantum ESPRESSO web site,/contacts.php–before posting: many questions are asked over and over again.NOTA BENE:only messages that appear to come from the registered user’s e-mail address,in its exact form,will be accepted.Messages”waiting for moderator approval”are automatically deleted with no further processing(sorry,too much spam).In case of trouble,carefully check that your return e-mail is the correct one(i.e.the one you used to subscribe).Since pw forum averages∼10message a day,an alternative low-traffic mailing list,pw users@,is provided for those interested only in Quantum ESPRESSO-related news,such as e.g.announcements of new versions,tutorials,etc..You can subscribe(but not post)to this list from the Quantum ESPRESSO web site.If you need to contact the developers for specific questions about coding,proposals,offersof help,etc.,send a message to the developers’mailing list:user q-e-developers,address.1.4Terms of useQuantum ESPRESSO is free software,released under the GNU General Public License. See /licenses/old-licenses/gpl-2.0.txt,or thefile License in the distribution).We shall greatly appreciate if scientific work done using this code will contain an explicit acknowledgment and the following reference:P.Giannozzi,S.Baroni,N.Bonini,M.Calandra,R.Car,C.Cavazzoni,D.Ceresoli,G.L.Chiarotti,M.Cococcioni,I.Dabo,A.Dal Corso,S.Fabris,G.Fratesi,S.deGironcoli,R.Gebauer,U.Gerstmann,C.Gougoussis,A.Kokalj,zzeri,L.Martin-Samos,N.Marzari,F.Mauri,R.Mazzarello,S.Paolini,A.Pasquarello,L.Paulatto, C.Sbraccia,S.Scandolo,G.Sclauzero, A.P.Seitsonen, A.Smo-gunov,P.Umari,R.M.Wentzcovitch,J.Phys.:Condens.Matter21,395502(2009),/abs/0906.2569Note the form Quantum ESPRESSO for textual citations of the code.Pseudopotentials should be cited as(for instance)[]We used the pseudopotentials C.pbe-rrjkus.UPF and O.pbe-vbc.UPF from.2Installation2.1DownloadPresently,Quantum ESPRESSO is only distributed in source form;some precompiled exe-cutables(binaryfiles)are provided only for PWgui.Stable releases of the Quantum ESPRESSO source package(current version is4.2.0)can be downloaded from this URL:/download.php.Uncompress and unpack the core distribution using the command:tar zxvf espresso-X.Y.Z.tar.gz(a hyphen before”zxvf”is optional)where X.Y.Z stands for the version number.If your version of tar doesn’t recognize the”z”flag:gunzip-c espresso-X.Y.Z.tar.gz|tar xvf-A directory espresso-X.Y.Z/will be created.Given the size of the complete distribution,you may need to download more packages and to unpack them following the same procedure(they will unpack into the same directory).Plug-ins should instead be downloaded into subdirectory plugin/archive but not unpacked or uncompressed:command make will take care of this during installation.Occasionally,patches for the current version,fixing some errors and bugs,may be distributed as a”diff”file.In order to install a patch(for instance):cd espresso-X.Y.Z/patch-p1</path/to/the/diff/file/patch-file.diffIf more than one patch is present,they should be applied in the correct order.Daily snapshots of the development version can be downloaded from the developers’site :follow the link”Quantum ESPRESSO”,then”SCM”.Beware:the develop-ment version is,well,under development:use at your own risk!The bravest may access the development version via anonymous CVS(Concurrent Version System):see the Developer Manual(Doc/developer man.pdf),section”Using CVS”.The Quantum ESPRESSO distribution contains several directories.Some of them are common to all packages:Modules/sourcefiles for modules that are common to all programsinclude/files*.h included by fortran and C sourcefilesclib/external libraries written in Cflib/external libraries written in Fortraniotk/Input/Output Toolkitinstall/installation scripts and utilitiespseudo/pseudopotentialfiles used by examplesupftools/converters to unified pseudopotential format(UPF)examples/sample input and outputfilesDoc/general documentationwhile others are specific to a single package:PW/PWscf:sourcefiles for scf calculations(pw.x)pwtools/PWscf:sourcefiles for miscellaneous analysis programstests/PWscf:automated testsPP/PostProc:sourcefiles for post-processing of pw.x datafilePH/PHonon:sourcefiles for phonon calculations(ph.x)and analysisGamma/PHonon:sourcefiles for Gamma-only phonon calculation(phcg.x)D3/PHonon:sourcefiles for third-order derivative calculations(d3.x)PWCOND/PWcond:sourcefiles for conductance calculations(pwcond.x)vdW/VdW:sourcefiles for molecular polarizability calculation atfinite frequency CPV/CP:sourcefiles for Car-Parrinello code(cp.x)atomic/atomic:sourcefiles for the pseudopotential generation package(ld1.x) atomic doc/Documentation,tests and examples for atomicGUI/PWGui:Graphical User Interface2.2PrerequisitesTo install Quantum ESPRESSO from source,you needfirst of all a minimal Unix envi-ronment:basically,a command shell(e.g.,bash or tcsh)and the utilities make,awk,sed. MS-Windows users need to have Cygwin(a UNIX environment which runs under Windows) installed:see /.Note that the scripts contained in the distribution assume that the local language is set to the standard,i.e.”C”;other settings may break them. Use export LC ALL=C(sh/bash)or setenv LC ALL C(csh/tcsh)to prevent any problem when running scripts(including installation scripts).Second,you need C and Fortran-95compilers.For parallel execution,you will also need MPI libraries and a“parallel”(i.e.MPI-aware)compiler.For massively parallel machines,or for simple multicore parallelization,an OpenMP-aware compiler and libraries are also required.Big machines with specialized hardware(e.g.IBM SP,CRAY,etc)typically have a Fortran-95compiler with MPI and OpenMP libraries bundled with the software.Workstations or“commodity”machines,using PC hardware,may or may not have the needed software.If not,you need either to buy a commercial product(e.g Portland)or to install an open-source compiler like gfortran or g95.Note that several commercial compilers are available free of charge under some license for academic or personal usage(e.g.Intel,Sun).2.3configureTo install the Quantum ESPRESSO source package,run the configure script.This is ac-tually a wrapper to the true configure,located in the install/subdirectory.configure will(try to)detect compilers and libraries available on your machine,and set up things accordingly. Presently it is expected to work on most Linux32-and64-bit PCs(all Intel and AMD CPUs)and PC clusters,SGI Altix,IBM SP machines,NEC SX,Cray XT machines,Mac OS X,MS-Windows PCs.It may work with some assistance also on other architectures(see below).Instructions for the impatient:cd espresso-X.Y.Z/./configuremake allSymlinks to executable programs will be placed in the bin/subdirectory.Note that both Cand Fortran compilers must be in your execution path,as specified in the PATH environment variable.Additional instructions for CRAY XT,NEC SX,Linux PowerPC machines with xlf:./configure ARCH=crayxt4./configure ARCH=necsx./configure ARCH=ppc64-mnconfigure Generates the followingfiles:install/make.sys compilation rules andflags(used by Makefile)install/configure.msg a report of the configuration run(not needed for compilation)install/config.log detailed log of the configuration run(may be needed for debugging) include/fft defs.h defines fortran variable for C pointer(used only by FFTW)include/c defs.h defines C to fortran calling conventionand a few more definitions used by CfilesNOTA BENE:unlike previous versions,configure no longer runs the makedeps.sh shell scriptthat updates dependencies.If you modify the sources,run./install/makedeps.sh or type make depend to updatefiles make.depend in the various subdirectories.You should always be able to compile the Quantum ESPRESSO suite of programs without having to edit any of the generatedfiles.However you may have to tune configure by specifying appropriate environment variables and/or command-line ually the tricky part is toget external libraries recognized and used:see Sec.2.4for details and hints.Environment variables may be set in any of these ways:export VARIABLE=value;./configure#sh,bash,kshsetenv VARIABLE value;./configure#csh,tcsh./configure VARIABLE=value#any shellSome environment variables that are relevant to configure are:ARCH label identifying the machine type(see below)F90,F77,CC names of Fortran95,Fortran77,and C compilersMPIF90name of parallel Fortran95compiler(using MPI)CPP sourcefile preprocessor(defaults to$CC-E)LD linker(defaults to$MPIF90)(C,F,F90,CPP,LD)FLAGS compilation/preprocessor/loaderflagsLIBDIRS extra directories where to search for librariesFor example,the following command line:./configure MPIF90=mpf90FFLAGS="-O2-assume byterecl"\CC=gcc CFLAGS=-O3LDFLAGS=-staticinstructs configure to use mpf90as Fortran95compiler withflags-O2-assume byterecl, gcc as C compiler withflags-O3,and to link withflag-static.Note that the value of FFLAGS must be quoted,because it contains spaces.NOTA BENE:do not pass compiler names with the leading path included.F90=f90xyz is ok,F90=/path/to/f90xyz is not.Do not use environmental variables with configure unless they are needed!try configure with no options as afirst step.If your machine type is unknown to configure,you may use the ARCH variable to suggest an architecture among supported ones.Some large parallel machines using a front-end(e.g. Cray XT)will actually need it,or else configure will correctly recognize the front-end but not the specialized compilation environment of those machines.In some cases,cross-compilation requires to specify the target machine with the--host option.This feature has not been extensively tested,but we had at least one successful report(compilation for NEC SX6on a PC).Currently supported architectures are:ia32Intel32-bit machines(x86)running Linuxia64Intel64-bit(Itanium)running Linuxx8664Intel and AMD64-bit running Linux-see note belowaix IBM AIX machinessolaris PC’s running SUN-Solarissparc Sun SPARC machinescrayxt4Cray XT4/5machinesmacppc Apple PowerPC machines running Mac OS Xmac686Apple Intel machines running Mac OS Xcygwin MS-Windows PCs with Cygwinnecsx NEC SX-6and SX-8machinesppc64Linux PowerPC machines,64bitsppc64-mn as above,with IBM xlf compilerNote:x8664replaces amd64since v.4.1.Cray Unicos machines,SGI machines with MIPS architecture,HP-Compaq Alphas are no longer supported since v.4.2.0.Finally,configure recognizes the following command-line options:--enable-parallel compile for parallel execution if possible(default:yes)--enable-openmp compile for openmp execution if possible(default:no)--enable-shared use shared libraries if available(default:yes)--disable-wrappers disable C to fortran wrapper check(default:enabled)--enable-signals enable signal trapping(default:disabled)and the following optional packages:--with-internal-blas compile with internal BLAS(default:no)--with-internal-lapack compile with internal LAPACK(default:no)--with-scalapack use ScaLAPACK if available(default:yes)If you want to modify the configure script(advanced users only!),see the Developer Manual.2.3.1Manual configurationIf configure stops before the end,and you don’tfind a way tofix it,you have to write working make.sys,include/fft defs.h and include/c defs.hfiles.For the latter twofiles,follow the explanations in include/defs.h.README.If configure has run till the end,you should need only to edit make.sys.A few templates (each for a different machine type)are provided in the install/directory:they have names of the form Make.system,where system is a string identifying the architecture and compiler.The template used by configure is also found there as make.sys.in and contains explanations of the meaning of the various variables.The difficult part will be to locate libraries.Note that you will need to select appropriate preprocessingflags in conjunction with the desired or available libraries(e.g.you need to add-D FFTW)to DFLAGS if you want to link internal FFTW).For a correct choice of preprocessingflags,refer to the documentation in include/defs.h.README.NOTA BENE:If you change any settings(e.g.preprocessing,compilationflags)after a previous(successful or failed)compilation,you must run make clean before recompiling,unless you know exactly which routines are affected by the changed settings and how to force their recompilation.2.4LibrariesQuantum ESPRESSO makes use of the following external libraries:•BLAS(/blas/)and•LAPACK(/lapack/)for linear algebra•FFTW(/)for Fast Fourier TransformsA copy of the needed routines is provided with the distribution.However,when available, optimized vendor-specific libraries should be used:this often yields huge performance gains. BLAS and LAPACK Quantum ESPRESSO can use the following architecture-specific replacements for BLAS and LAPACK:MKL for Intel Linux PCsACML for AMD Linux PCsESSL for IBM machinesSCSL for SGI AltixSUNperf for SunIf none of these is available,we suggest that you use the optimized ATLAS library:see /.Note that ATLAS is not a complete replacement for LAPACK:it contains all of the BLAS,plus the LU code,plus the full storage Cholesky code. Follow the instructions in the ATLAS distributions to produce a full LAPACK replacement.Sergei Lisenkov reported success and good performances with optimized BLAS by Kazushige Goto.They can be freely downloaded,but not redistributed.See the”GotoBLAS2”item at /tacc-projects/.FFT Quantum ESPRESSO has an internal copy of an old FFTW version,and it can use the following vendor-specific FFT libraries:IBM ESSLSGI SCSLSUN sunperfNEC ASLAMD ACMLconfigure willfirst search for vendor-specific FFT libraries;if none is found,it will search for an external FFTW v.3library;if none is found,it will fall back to the internal copy of FFTW.If you have recent versions of MKL installed,you may try the FFTW interface provided with MKL.You will have to compile them(only sources are distributed with the MKL library) and to modifyfile make.sys accordingly(MKL must be linked after the FFTW-MKL interface)MPI libraries MPI libraries are usually needed for parallel execution(unless you are happy with OpenMP multicore parallelization).In well-configured machines,configure shouldfind the appropriate parallel compiler for you,and this shouldfind the appropriate libraries.Since often this doesn’t happen,especially on PC clusters,see Sec.2.7.5.Other libraries Quantum ESPRESSO can use the MASS vector math library from IBM, if available(only on AIX).2.4.1If optimized libraries are not foundThe configure script attempts tofind optimized libraries,but may fail if they have been in-stalled in non-standard places.You should examine thefinal value of BLAS LIBS,LAPACK LIBS, FFT LIBS,MPI LIBS(if needed),MASS LIBS(IBM only),either in the output of configure or in the generated make.sys,to check whether it found all the libraries that you intend to use.If some library was not found,you can specify a list of directories to search in the envi-ronment variable LIBDIRS,and rerun configure;directories in the list must be separated by spaces.For example:./configure LIBDIRS="/opt/intel/mkl70/lib/32/usr/lib/math"If this still fails,you may set some or all of the*LIBS variables manually and retry.For example:./configure BLAS_LIBS="-L/usr/lib/math-lf77blas-latlas_sse"Beware that in this case,configure will blindly accept the specified value,and won’t do any extra search.。
安普乐克 Model G 端子压接机 使用说明书
Fast Facts•Quieter•Improved guarding and lighting •Toolless changeover of applicators •Precision manual- or auto-adjust for crimp height•Total and batch counter•Accepts all existing TE miniature applicators, with minor modifications •Reduced maintenance requirements •Split-cycle operation for both lead-making and bench-top use•Standardized 15/8 [41.3] stroke •Cannot be operated during normal production with guards open •Operates on either 120 or 220 VAC, 50 or 60 Hz•Available with CQM G-Adapter for crimp quality monitoring using miniature applicators•Produced under a Quality Management System certified to ISO 9001AMP-O-LECTRIC Model GTerminating MachineThe Model “G” is the most advanceddesign in the long-standing series ofAMP-O-LECTRIC machines for terminat-ing wire using reeled terminals and con-tacts. It features a reliable direct motordrive system. Its modular constructionalso enables us to provide versions foruse either as bench-top units, or in com-bination with fully-automatic lead-mak-ing equipment.For operator convenience, we’veimproved access to, and lighting in thetarget area, and moved the CrimpQuality Monitor (CQM II) to eye level forbench-top use. The Model “G” is alsoquieter, plus improved guarding meetsapplicable European and domestic safe-ty requirements.All Model “G” versions include precisionadjustment for crimp height — an impor-tant feature, which, in combination withthe optional CQM, will enable you tomaintain the tight tolerances requiredfor today’s demands for higher levels ofquality. By simply resetting a dial, anoperator can adjust crimp height in.0005 [0.013] increments over a .018[0.457] range.Another user-friendly feature is its con-trol system. At the push of a button, youcan power the motor on or off, jog for-ward or reverse, switch to split-cycleoperation, turn the work light off or on,power the air feed on or off, or reset thebatch counter. And, it’s all reported onan LCD display. The controls will alsodetect and report errors in the machine,and stop cycling when a bad crimp isdetected by the CQM. The operatormust then push the reset button to con-tinue.The Model “G” will accept all existing TEminiature applicators, with minor modifi-cations. See Applicator Instructions 408-8053, Conversion Guide for TE MiniatureQuick-Change Applicators, for details.Most of our existing applicators may beused for crimp quality monitoring onversions equipped with CQM and CQMG-Adapter. To use CQM for Tab-Lok,FASTON flag terminals, a CQM hinge-barapplicator is required. You can’t useCQM to monitor the application ofclosed-barrel pre-insulated terminals.Catalog 65828 / 04-12 / Application ToolingAMP, AMP-O-LECTRIC, FASTON, the TE Connectivity (logo) and TE Connectivity are Tyco Electronics Corporation, Harrisburg, PA 17105, Phone: 888-777-5917or717-810-2080;email:*******************AMP-O-LECTRIC Model G Terminating MachineSpecifications and DimensionsAMP-O-LECTRIC Model “G” Machine VersionsSpecificationsCapacity: 5000 lb [2268 kg]maximum crimp force Deflection: .003 [0.076]maximum per 1 000 lb [454 kg] crimp force Noise: 76 dB maximum at 5000 lb [2 268 kg] full capacityWeight: Approx.240 lb [110 kg]Height: 20 [510]Electrical: 120/220 VAC, 50/60 Hz• Avg 2.6 A at 120 VAC when used as a bench-top unit at 2000 cycles per hour operating rate•Avg 6.5 A at 120 VAC when used in combination with automatic lead-making equipment at 5 000 cycles per hour operating rateAir:90-100 psi [6.21-6.90 bar], 6 scfm [0.00282 m 3/s] (when required for use with air-feed applicators)CQM:• .0002 [0.0051] avg error, .0002 [0.0051] standard deviation for crimp height measurements using CQMapplicators• .0005 [0.0127] avg error, .0005 [0.0127] standard deviation for crimp height measurements using regular applicators in a CQM G-Adapter systemError is the difference between CQM measurements and measurements using a micrometer. These specifica-tions are based on properly following the applicable procedures for wire preparation and termination.32.7[831]31.4[798]28.1[714]25.6[650]24.0[610]38.0[965]25.3[643]21.5[546]19.0[483]17.4[442]Top View With Side-Feed Applicator Terminal GuideOptionalStripping ModuleAMP-O-LECTRIC Model G Terminator with Stripping Module1490501-1Manual Precision Adjust1-1490501-5Manual Precision Adjust and CQM Sensors 1-1490501-6Auto-Precision Adjust and CQM Sensors Field Retrofit Kits for Model G Terminator2119999-1G Terminator without CQM II*2161270-1G Terminator with CQM II** For machines delivered 10/1/11 or newer. Please contact your local TE representative or the Tooling Assistance Center (1-800-522-6752) to add CQM II onto an older Model G Terminator.The combination of the Stripping Module with the AMP-O-LECTRIC Model G Terminator provides an economical and proficient method of stripping wire and crimping terminals on the same machine. Wires are strippedmoments before crimping, mean-ing there is virtually no chance of damaging wire conductors dur-ing handling or storage. Once the wire is fed into the start sensor the Stripping Module does the rest, improving placement accu-racy. Order Catalog 1309085 for more information.In the interest of continuous improvement, TE reserves the right to modify, discontinue, or replace any products.。
算法设计与分析基础课后习题答案solu8
1. a. Both techniques are based on dividing a problem’s instance into smaller instances of the same problem. b. Typically, divide-and-conquer divides an instance into smaller instances with no intersection whereas dynamic programming deals with problems in which smaller instances overlap. Consequently, divide-and-conquer algorithms do not explicitly store solutions to smaller instances and dynamic programming algorithms do. 2. a. 0 1 2 3 4 5 6 0 1 1 1 1 1 1 1 1 1 2 3 4 5 6 2 3
This file contains the exercises, hints, and solutions for Chapter 8 of the book ”Introduction to the Design and Analysis of Algorithms,” 2nd edition, by A. Levitin. The problems that might be challenging for at least some students are marked by ; those that might be difficult for a majority of students are marked by .
A Robust Optimization Approach to Inventory theory
A Robust Oபைடு நூலகம்timization Approach to Inventory Theory
Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, E53-363, Cambridge, Massachusetts 02139, dbertsim@ Department of Industrial and Systems Engineering, Lehigh University, Mohler Building, Bethlehem, Pennsylvania 18015, aurelie.thiele@
1. Introduction
Optimal supply chain management has been extensively studied in the past with much theoretical success. Dynamic programming has long emerged as the standard tool for this purpose, and has led to significant breakthroughs as early as 1960, when Clark and Scarf (1960) proved the optimality of base-stock policies for series systems in their landmark paper. Although dynamic programming is a powerful technique as to the theoretical characterization of the optimal policy for simple systems, the complexity of the underlying recursive equations over a growing number of state variables makes it ill suited for the computation of the actual policy parameters, which is crucial for real-life applications. Approximation algorithms have been developed to address those issues. These include stochastic approximation (see Koshner and Clark 1978) and infinitesimal perturbation analysis (IPA) (see Glasserman 1991, Ho and Cao 1991), where a class of policies, e.g., base-stock, characterized by a set of parameters, is optimized using simulation-based methods (see Fu 1994, Glasserman and Tayur 1995, Kapuscinski and Tayur 1999). IPA-based methods assume knowledge of the underlying probability distributions and restrict their attention to cer150
PP is closed under intersection
al. 27] have shown that a depth-(d + 1) circuit of threshold gates which com-
putes the parity on n inputs requires (dn1=d=log2n) threshold gates. Recently,
simulated by a circuit with a single threshold gate (at the root) with depth
greater by a constant and only a limited increase in size. If the original circuit
under polynomial-time conjunctive and disjunctive reductions. Consequently,
PP is closed under polynomial-time truth-table reductions in which the truth
Beigel 6], extending our techniques, has shown that that no constant depth cir-
cuit with no(1) threshold gates, 2no(1) AND, OR, and NOT gates, and 2no(1) wires
threshold gates, 2no(1) AND, OR, and NOT gates (in arbitrary positions), and
2no(1) wires can compute parity. This is the rst natural example of a function
稀疏恢复和傅里叶采样
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leslie A. Kolodziejski Chair, Department Committee on Graduate Students
2
Sparse Recovery and Fourier Sampling by Eric Price
Submitted to the Department of Electrical Engineering and Computer Science on August 26, 2013, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Department of Electrical Engineering and Computer Science August 26, 2013
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piotr Indyk Professor Thesis Supervisor
D-ary Bounded-Length Huffman Coding
arXiv:cs/0701012v2 [cs.IT] 17 Apr 2007
Abstract— Efficient optimal prefix coding has long been accomplished via the Huffman algorithm. However, there is still room for improvement and exploration regarding variants of the Huffman problem. Length-limited Huffman coding, useful for many practical applications, is one such variant, in which codes are restricted to the set of codes in which none of the n codewords is longer than a given length, lmax . Binary lengthlimited coding can be done in O(nlmax ) time and O(n) space via the widely used Package-Merge algorithm. In this paper the Package-Merge approach is generalized without increasing complexity in order to introduce a minimum codeword length, lmin , to allow for objective functions other than the minimization of expected codeword length, and to be applicable to both binary and nonbinary codes; nonbinary codes were previously addressed using a slower dynamic programming approach. These extensions have various applications — including faster decompression — and can be used to solve the problem of finding an optimal code with limited fringe, that is, finding the best code among codes with a maximum difference between the longest and shortest codewords. The previously proposed method for solving this problem was nonpolynomial time, whereas solving this using the novel algorithm requires only O(n(lmax − lmin )2 ) time and O(n) space.
瓶颈 和 约束条件
瓶颈和约束条件英文回答：Bottleneck refers to a point in a process or system where the flow or progress is limited or slowed down due to a constraint or limitation. It can occur in various contexts, such as manufacturing, project management, or even in personal situations. A bottleneck can arise from various factors, including limited resources, inefficient processes, or a lack of coordination among different components.For example, in a manufacturing plant, if there is a machine that can only produce a certain number of units per hour, it becomes a bottleneck for the entire production line. Even if other machines or workers can work at afaster pace, the overall output will still be limited by the capacity of that particular machine. This creates a situation where the flow of production is constrained, and it becomes a challenge to increase the overall efficiencyof the process.Another example can be seen in project management. If there is a team member who is responsible for a critical task but is unable to complete it within the required timeframe, it becomes a bottleneck for the project. The progress of the entire project will be delayed until the task is completed, impacting the overall timeline and potentially causing delays in other dependent tasks.Constraints, on the other hand, refer to limitations or restrictions that affect the design, implementation, or operation of a system or process. Constraints can be external, such as regulatory requirements or market conditions, or internal, such as budget limitations or resource availability. They set boundaries and define what is possible or permissible within a given context.For instance, in software development, there may be constraints on the available programming languages or platforms that can be used for a particular project. These constraints can impact the design decisions and the overallfunctionality of the software. Similarly, in construction projects, there may be constraints on the maximum load-bearing capacity of a structure or the height limitations imposed by local regulations.Constraints can also be seen in personal situations.For example, if someone has a limited budget for a vacation, it becomes a constraint that influences their choices and options. They may have to prioritize certain expenses or make compromises in order to stay within their budget.In summary, bottlenecks and constraints both refer to limitations or restrictions that affect the flow, progress, or design of a process or system. Bottlenecks are specific points where the flow is limited, while constraints aremore general limitations or boundaries. Understanding and managing bottlenecks and constraints are crucial for optimizing processes, improving efficiency, and achieving desired outcomes.中文回答：瓶颈是指在一个过程或系统中，由于约束或限制，流程或进展受到限制或减慢的点。