Sample Spaces with Small Bias on Neighborhoods and Error-Correcting Communication Protocols

We give a deterministic algorithm which, on input an integer n , collection \cal F of subsets of {1,2,\ldots,n} , and ɛ∈ (0,1) , runs in time polynomial in n| \cal F |/ɛ and produces a \pm 1 -matrix M with n columns and m=O(log | \cal F |/ɛ ² ) rows with the following property: for any subset F ∈ \cal F , the fraction of 1's in the n -vector obtained by coordinatewise multiplication of the column vectors indexed by F is between (1-ɛ)/2 and (1+ɛ)/2 . In the case that \cal F is the set of all subsets of size at most k , k constant, this gives a polynomial time construction for a k -wise ɛ -biased sample space of size O(log n/ɛ ² ) , compared with the best previous constructions in [NN] and [AGHP] which were, respectively, O(log n/ɛ ⁴ ) and O(log ² n/ɛ ² ) . The number of rows in the construction matches the upper bound given by the probabilistic existence argument. Such constructions are of interest for derandomizing algorithms. As an application, we present a family of essentially optimal deterministic communication protocols for the problem of checking the consistency of two files. Received October 30, 1997; revised September 17, 1999, and April 17, 2000.     

Using a biarc filter to compute curvature extremes of NURBS curves

A method to compute curvature minima and maxima of parametric curves (represented in NURBS format) is presented in this paper. Since the curvature changes vary rapidly along the path of (even smooth) curves, a biarc filter is employed to approximate the curvature function with a piecewise constant function. This allows the isolation of curvature extreme values that are found within-engineering tolerances via repeated biarc approximation followed by golden section search. Because the derivative of the curvature is numerically very unstable, only optimization without derivatives is feasible. However, given the excellent isolation property of biarc filters, curvature extremes are found within 10–20 steps even for high accuracy requirements ranging from 10⁻⁴ to 10⁻⁶.     

Universal routing schemes

Summary.  In this paper, we deal with the compact routing problem, that is implementing routing schemes that use a minimum memory size on each router. A universal routing scheme is a scheme that applies to all n-node networks. In [31], Peleg and Upfal showed that one cannot implement a universal routing scheme with less than a total of Ω(n ^1+1/(2s+4)) memory bits for any given stretch factor s≧1. We improve this bound for stretch factors s, 1≦s<2, by proving that any near-shortest path universal routing scheme uses a total of Ω(n ²) memory bits in the worst-case. This result is obtained by counting the minimum number of routing functions necessary to route on all n-node networks. Moreover, and more fundamentally, we give a tight bound of Θ(n log n) bits for the local minimum memory requirement of universal routing scheme of stretch factors s, 1≦s<2. More precisely, for any fixed constant ɛ, 0<ɛ<1, there exists a n-node network G on which at least Ω(n ^ɛ) routers require Θ(n log n) bits each to code any routing function on G of stretch factor <2. This means that, whatever you choose the routing scheme, there exists a network on which one cannot compress locally the routing information better than routing tables do. Received: August 1995 / Accepted: August 1996     

A 3-Party Simultaneous Protocol for SUM-INDEX

Abstract. We show that the SUM-INDEX function can be computed by a 3-party simultaneous protocol in which one player sends only O(n ^ɛ ) bits and the other sends O(n ^1-C(ɛ) ) bits (0<C(ɛ)<1 ). This implies that, in the Valiant—Nisan—Wigderson approach for proving circuit lower bounds, the SUM-INDEX function is not suitable as a target function.     

Uniform Pointwise Convergence of Difference Schemes for Convection-Diffusion Problems on Layer-Adapted Meshes

We consider two convection-diffusion boundary value problems in conservative form: for an ordinary differential equation and for a parabolic equation. Both the problems are discretized using a four-point second-order upwind space difference operator on arbitrary and layer-adapted space meshes. We give ɛ-uniform maximum norm error estimates O(N ⁻²ln² N(+τ)) and O(N ⁻²(+τ)), respectively, for the Shishkin and Bakhvalov space meshes, where N is the space meshnodes number, τ is the time meshinterval. The smoothness condition for the Bakhvalov mesh is replaced by a weaker condition. Received December 14, 1999; revised September 13, 2000     

Correlation between the gradients of the quadratic functional and its clipped prototype

Applicability of clipping of quadratic functional E = −0.5x ⁺ Tx + Bx in the minimization problem is considered (here x is the configurational vector and B ∈ R ^N is real valued vector). The probability that the gradient of this functional and the gradient of clipped functional ɛ = −0.5x ⁺ τx + bx are collinear is shown to be very high (the matrix τ is obtained by clipping of original matrix T: τ_ij = sgnT _ij ). It allows the conclusion that minimization of functional ɛ implies minimization of functional E. We can therefore replace the laborious process of minimizing functional E by the minimization of its clipped prototype ɛ. Use of the clipped functional allows sixteen-times reduction of the computation time and computer memory usage.     

On Parabolic Boundary Layers for Convection–Diffusion Equations in a Channel: Analysis and Numerical Applications

In this article we discuss singularly perturbed convection–diffusion equations in a channel in cases producing parabolic boundary layers. It has been shown that one can improve the numerical resolution of singularly perturbed problems involving boundary layers, by incorporating the structure of the boundary layers into the finite element spaces, when this structure is available; see e.g. [Cheng, W. and Temam, R. (2002). Comput. Fluid. V.31, 453–466; Jung, C. (2005). Numer. Meth. Partial Differ. Eq. V.21, 623–648]. This approach is developed in this article for a convection–diffusion equation. Using an analytical approach, we first derive an approximate (simplified) form of the parabolic boundary layers (elements) for our problem; we then develop new numerical schemes using these boundary layer elements. The results are performed for the perturbation parameter ε in the range 10⁻¹–10⁻¹⁵ whereas the discretization mesh is in the range of order 1/10–1/100 in the x-direction and of order 1/10–1/30 in the y-direction. Indications on various extensions of this work are briefly described at the end of the Introduction.Dedicated to David Gottlieb on his 60th birthday.     

A Near-Linear Area Bound for Drawing Binary Trees

Abstract. We present several simple methods to construct planar, strictly upward, strongly order-preserving, straight-line drawings of any n -node binary tree. In particular, it is shown that O(n ^1+ɛ ) area is always sufficient for an arbitrary constant ɛ>0 .     

Curve interpolation with directional constraints for engineering design

An algorithm to interpolate data points with directional constraints is given in this paper. The interpolating B-spline curve passes through the data points and assumes tangent directions at arbitrarily selected points. The advantages of the method is that the user is free to select any number of directional constraints and the method produces full parametric continuity up to C ^{p − 1} for any degree p interpolation. The method is a generalization of Piegl and Tiller (The NURBS book, 2nd edn. Springer, New York, 1997] in two directions: (1) there is no restriction on the degree, and (2) no need to specify derivatives at all data points.     

Parametric G n blending of curves and surfaces

We introduce a simple blending method for parametric curves and surfaces that produces families of parametrically defined, G ⁿ –continuous blending curves and surfaces. The method depends essentially on the parameterizations of the curves/surfaces to be blended. Hence, the flexibility of the method relies on the existence of suitable parameter transformations of the given curves/surfaces. The feasibility of the blending method is shown by several examples. The shape of the blend curve/surface can be changed in a predictable way with the aid of two design parameters (thumb weight and balance).     

A Randomized Linear-Work EREW PRAM Algorithm to Find a Minimum Spanning Forest

We present a randomized EREW PRAM algorithm to find a minimum spanning forest in a weighted undirected graph. On an n -vertex graph the algorithm runs in o(( log n) ^{1+ ɛ} ) expected time for any ɛ >0 and performs linear expected work. This is the first linear-work, polylog-time algorithm on the EREW PRAM for this problem. This also gives parallel algorithms that perform expected linear work on two general-purpose models of parallel computation—the QSM and the BSP.     

Filling n-sided regions with G 1 triangular Coons B-spline patches

Filling n-sided regions is an essential operation in shape and surface modeling. Positional and tangential continuities are highly required in designing and manufacturing. We propose a method for filling n-sided regions with untrimmed triangular Coons B-spline patches, preserving G ¹ continuity exactly. The algorithm first computes a central point, a central normal, the central, and the corner derivative vectors. Then the region is split into n triangular areas by connecting the central point to each corner of the boundary. These inner curves and all cross-boundary derivatives are computed fulfilling G ¹ compatibility conditions. And finally, the triangular patches are generated in the Coons B-spline form, one boundary of which is regressed to the central vertex. Neither positional nor tangential error is introduced by this method. And only one degree elevation is needed.     

Self-intersection elimination in metamorphosis of two-dimensional curves

H :[0, 1]× ³→ ³, where H(t, r) for t=0 and t=1 are two given planar curves C ₁(r) and C ₂(r). The first t parameter defines the time of fixing the intermediate metamorphosis curve. The locus of H(t, r) coincides with the ruled surface between C ₁(r) and C ₂(r), but each isoparametric curve of H(t, r) is self-intersection free. The second algorithm suits morphing operations of planar curves. First, it constructs the best correspondence of the relative parameterizations of the initial and final curves. Then it eliminates the remaining self-intersections and flips back the domains that self-intersect.     

Existence and computation of spherical rational quartic curves for Hermite interpolation

n degrees of freedom for any given Hermite data on S ⁿ , n≥2. A method is presented for generating all spherical rational quartic curves on S ⁿ interpolating given Hermite data.     

Bounding the Inefficiency of Length-Restricted Prefix Codes

Abstract. We consider an alphabet Σ= {a ₁ ,\ldots,a _n } with corresponding symbol probabilities p ₁ ,\ldots,p _n . For each prefix code associated to Σ , let l _i be the length of the codeword associated to a _i . The average code length c is defined by c=\sum _i=1 ⁿ p _i l _i . An optimal prefix code for Σ is one that minimizes c . An optimal L -restricted prefix code is a prefix code that minimizes c constrained to l _i ≤ L for i=1,\ldots,n . The value of the length restriction L is an integer no smaller than \lceil log n \rceil . Let A be the average length of an optimal prefix code for Σ . Also let A _L be the average length of an optimal L -restricted prefix code for Σ . The average code length difference ɛ is defined by ɛ=A _L -A . Let ψ be the golden ratio 1.618. In this paper we show that ɛ < 1/ψ ^{L-\lceil\log (n+\lceil\log n\rceil-L)\rceil-1} when L > \lceil log n \rceil . We also prove the sharp bound ɛ < \lceil log n \rceil -1 , when L = \lceil log n \rceil . By showing the lower bound 1/(ψ ^{L-\lceil\log n\rceil+2+\lceil\log (n/(n-L))\rceil} -1) on the maximum value of ɛ , we guarantee that our bound is asymptotically tight in the range \lceil log n \rceil < L ≤ n/2 . When L\geq \lceil log n \rceil +11 , the bound guarantees that ɛ < 0.01 . From a practical point of view, this is a negligible loss of compression efficiency. Furthermore, we present an O(n) time and space 1/ψ ^{L-\lceil\log (n+\lceil\log n\rceil-L)\rceil-1} -approximative algorithm to construct L -restricted prefix codes, assuming that the given probabilities are already sorted. The results presented in this paper suggest that one can efficiently implement length restricted prefix codes, obtaining also very effective codes.     

Solving Some Discrepancy Problems in NC

We show that several discrepancy-like problems can be solved in NC nearly achieving the discrepancies guaranteed by a probabilistic analysis and achievable sequentially. For example, we describe an NC algorithm that given a set system (X, S) , where X is a ground set and S⊆2 ^X , computes a set R⊆X so that for each S∈ S the discrepancy ||R S|-|R^-∩S|| is . Whereas previous NC algorithms could only achieve discrepancies with ɛ>0 , ours matches the probabilistic bound within a multiplicative factor 1+o(1) . Other problems whose NC solution we improve are lattice approximation, ɛ -approximations of range spaces with constant VC-exponent, sampling in geometric configuration spaces, approximation of integer linear programs, and edge coloring of graphs. Received June 26, 1998; revised February 18, 1999.     

利用距离与内能极小平滑链接Bézier曲线

目的 对于满足低阶连续的链接Bézier曲线,提高曲线之间的连续性以达到平滑的目的,需要对曲线的控制顶点进行相应调整。因此,可根据具体的目标对需要调整的控制顶点进行优化选取,使得平滑后的链接曲线满足相应的要求。针对这一问题,给出了3种目标下优化调整控制顶点的方法。方法 首先对讨论的问题进行描述,分别指出链接Bézier曲线从C⁰连续平滑为C¹连续和从C¹连续平滑为C²连续两种情形需调整的控制顶点;然后分别给出两种情形下,以新旧控制顶点距离极小为目标、曲线内能极小为目标、新旧控制顶点距离与曲线内能同时极小为目标,对链接Bézier曲线进行平滑的方法,最后对3种极小化方法进行对比,并指出了不同方法的适用场合。结果 数值算例表明,距离极小化方法调整后的控制顶点偏离原控制顶点的距离相对较小,适合于控制顶点取自于实物时的应用场合;内能极小化方法获得的链接曲线内能相对较小,适合于要求曲线能量尽可能小的应用场合;距离与内能同时极小化方法兼顾了新旧控制顶点的距离和链接曲线的内能,适合于对两个目标都有要求的应用场合。结论 提出的方法为链接Bézier曲线的平滑提供了3种有效手段,且易于实现,对其他类型链接曲线的平滑具有参考价值。     

Approximation Algorithm Based on Chain Implication for Constrained Minimum Vertex Covers in Bipartite Graphs

The constrained minimum vertex cover problem on bipartite graphs (the Min-CVCB problem) is an important NP-complete problem. This paper presents a polynomial time approximation algorithm for the problem based on the technique of chain implication. For any given constant ε > 0, if an instance of the Min-CVCB problem has a minimum vertex cover of size (ku, kl), our algorithm constructs a vertex cover of size (ku*, kl* ), satisfying max{ku*/ku, kl* /kl} 1 ε.     

A Nonoverlapping Domain Decomposition Method for Legendre Spectral Collocation Problems

We consider the Dirichlet boundary value problem for Poisson’s equation in an L-shaped region or a rectangle with a cross-point. In both cases, we approximate the Dirichlet problem using Legendre spectral collocation, that is, polynomial collocation at the Legendre–Gauss nodes. The L-shaped region is partitioned into three nonoverlapping rectangular subregions with two interfaces and the rectangle with the cross-point is partitioned into four rectangular subregions with four interfaces. In each rectangular subregion, the approximate solution is a polynomial tensor product that satisfies Poisson’s equation at the collocation points. The approximate solution is continuous on the entire domain and its normal derivatives are continuous at the collocation points on the interfaces, but continuity of the normal derivatives across the interfaces is not guaranteed. At the cross point, we require continuity of the normal derivative in the vertical direction. The solution of the collocation problem is first reduced to finding the approximate solution on the interfaces. The discrete Steklov–Poincaré operator corresponding to the interfaces is self-adjoint and positive definite with respect to the discrete inner product associated with the collocation points on the interfaces. The approximate solution on the interfaces is computed using the preconditioned conjugate gradient method. A preconditioner is obtained from the discrete Steklov–Poincaré operators corresponding to pairs of the adjacent rectangular subregions. Once the solution of the discrete Steklov–Poincaré equation is obtained, the collocation solution in each rectangular subregion is computed using a matrix decomposition method. The total cost of the algorithm is O(N ³), where the number of unknowns is proportional to N ².      

Hardness and approximation results for packing steiner trees

We study approximation algorithms and hardness of approximation for several versions of the problem of packing Steiner trees. For packing edge-disjoint Steiner trees of undirected graphs, we show APX-hardness for four terminals. For packing Steiner-node-disjoint Steiner trees of undirected graphs, we show a logarithmic hardness result, and give an approximation guarantee ofO (√n logn), wheren denotes the number of nodes. For the directed setting (packing edge-disjoint Steiner trees of directed graphs), we show a hardness result of Θ(m ^1/3/−ɛ) and give an approximation guarantee ofO(m ^1/2/+ɛ), wherem denotes the number of edges. We have similar results for packing Steiner-node-disjoint priority Steiner trees of undirected graphs. Supported by NSERC Grant No. OGP0138432. Supported by an NSERC postdoctoral fellowship, Department of Combinatorics and Optimization at University of Waterloo, and a University start-up fund at University of Alberta.