Finding a closest visible vertex pair between two polygons

Given nonintersecting simple polygonsP andQ, two verticespP andq Q are said to be visible if does not properly intersectP orQ. We present a parallel algorithm for finding a closest pair among all visible pairs (p,q),pP andqQ. The algorithm runs in time O(logn) using O(n) processors on a CREW PRAM, wheren=¦P¦+¦Q¦. This algorithm can be implemented serially in (n) time, which gives a new optimal sequential solution for this problem.This paper appeared in preliminary form as [1]. This work was supported in part by an AT&T BellLaboratories Graduate Fellowship, the Joint Services Electronics Program (U.S. Army, U.S. Navy, U.S. Air Force) under Contract N00014-90-J-1270, and NSF Grant CCR-89-22008. This work was done while the author was with the Department of Computer Science at the University of Illinois at Urbana-Champaign.     

The expected additive weight of trees

Summary We consider a general additive weight of random trees which depends on the structure of the subtrees, on weight functions defined on the number of internal and external nodes and on the degrees of the nodes appearing in the tree and its subtrees. Choosing particular weight functions, the corresponding weight is an important parameter appearing in the analysis of sorting and searching algorithms. For a simply generated family of rooted planar trees , we shall derive a general approach to the computation of the average weight of a tree T with n nodes and m leaves for arbitrary weight functions. This general result implies exact and asymptotic expressions for many types of average weights of a tree T with n nodes if the weight functions are arbitrary polynomials in the number of nodes and leaves with coefficients depending on the node degrees.     

Comparative complexity of the representation of languages by probabilistic automata

We analyze the effect of the degree of isolation of a cut point on the number of states P(U, ) of a probabilistic automaton representing the language U. We give an example of a language Uⁿ consisting of words of length n such that there exist numbers < for which P(Uⁿ, )/P(Uⁿ, )0 as n.Translated from Kibernetika, No. 3, pp. 21–25, May–June, 1989.     

On the Epsilon-entropy of One Class of Ellipsoids in a Hamming Space

The asymptotic behavior of the -entropy of ellipsoids in an n-dimensional Hamming space whose coefficients take only two different values is investigated as n . Explicit expressions for the main terms of the asymptotic expansion of -entropy of such ellipsoids are obtained under various relations between and parameters that define these ellipsoids.     

A Delay-Averaged Logistic Model

For the equation x(t) = x(t) (1-(1/) _t-- ^t- x(u)du), > 0, > 0, > 0, conditions for the stability of a nonzero stationary solution under small perturbations are determined.     

On computing Boolean connectives of characteristic functions

This paper is a study of the existence of polynomial time Boolean connective functions for languages. A languageL has an AND function if there is a polynomial timef such thatf(x,y) L x L andy L. L has an OR function if there is a polynomial timeg such thatg(x,y) xL oryL. While all NP complete sets have these functions, Graph Isomorphism, which is probably not complete, is also shown to have both AND and OR functions. The results in this paper characterize the complete sets for the classes D^p and p^SAT[O(logn)] in terms of AND and OR and relate these functions to the structure of the Boolean hierarchy and the query hierarchies. Also, this paper shows that the complete sets for the levels of the Boolean hierarchy above the second level cannot have AND or OR unless the polynomial hierarchy collapses. Finally, most of the structural properties of the Boolean hierarchy and query hierarchies are shown to depend only on the existence of AND and OR functions for the NP complete sets.The first author was supported in part by NSF Research Grants DCR-8520597 and CCR-88-23053, and by an IBM Graduate Fellowship.     

ε-Entropy and Fractal Dimension of a State for a Gaussian Measure

Kolmogorov introduced the concept of -entropy to analyze information in classical continuous systems. The fractal dimension of a geometric set was introduced by Mandelbrot as a new criterion to analyze the geometric complexity of the set. The -entropy and the fractal dimension of a state in a general quantum system were introduced by one of the present authors (MO) in order to characterize chaotic properties of general states.In this paper, we show that -entropy of a state includes Kolmogorov's -entropy, and that the fractal dimension of a state describes fractal structure of Gaussian measures.     

On characterizations of recursively enumerable languages

Summary Geffert has shown that earch recursively enumerable languageL over can be expressed in the formL{h(x) ^–1 g(x)x in ⁺} ^* where is an alphabet andg, h is a pair of morphisms. Our purpose is to give a simple proof for Geffert's result and then sharpen it into the form where both of the morphisms are nonerasing. In our method we modify constructions used in a representation of recursively enumerable languages in terms of equality sets and in a characterization of simple transducers in terms of morphisms. As direct consequences, we get the undecidability of the Post correspondence problem and various representations ofL. For instance,L =(L ₀) ^* whereL ₀ is a minimal linear language and is the Dyck reductiona, A.     

Tracking Drifting Concepts By Minimizing Disagreements

In this paper we consider the problem of tracking a subset of a domain (called thetarget) which changes gradually over time. A single (unknown) probability distribution over the domain is used to generate random examples for the learning algorithm and measure the speed at which the target changes. Clearly, the more rapidly the target moves, the harder it is for the algorithm to maintain a good approximation of the target. Therefore we evaluate algorithms based on how much movement of the target can be tolerated between examples while predicting with accuracy . Furthermore, the complexity of the classH of possible targets, as measured byd, its VC-dimension, also effects the difficulty of tracking the target concept. We show that if the problem of minimizing the number of disagreements with a sample from among concepts in a classH can be approximated to within a factork, then there is a simple tracking algorithm forH which can achieve a probability of making a mistake if the target movement rate is at most a constant times ²/(k(d +k) ln 1/), whered is the Vapnik-Chervonenkis dimension ofH. Also, we show that ifH is properly PAC-learnable, then there is an efficient (randomized) algorithm that with high probability approximately minimizes disagreements to within a factor of 7d + 1, yielding an efficient tracking algorithm forH which tolerates drift rates up to a constant times ²/(d ² ln 1/). In addition, we prove complementary results for the classes of halfspaces and axisaligned hyperrectangles showing that the maximum rate of drift that any algorithm (even with unlimited computational power) can tolerate is a constant times ²/d.     

Decision procedures for elementary sublanguages of set theory: XIII. Model graphs,reflection and decidability

Positive solutions to the decision problem for a class of quantified formulae of the first order set theoretic language based on , , =, involving particular occurrences of restricted universal quantifiers and for the unquantified formulae of , , =, {...}, , where {...} is the tuple operator and is a general choice operator, are obtained. To that end a method is developed which also provides strong reflection principles over the hereditarily finite sets. As far as finite satisfiability is concerned such results apply also to the unquantified extention of , , =, {...}, , obtained by adding the operators of binary union, intersection and difference and the relation of inclusion, provided no nested term involving is allowed.     

On uniform convergence rates for Eulerian and Lagrangian finite element approximations of convection-dominated diffusion problems

Abstract Standard error estimates in the literature for finite element approximations of nonstationary convection-diffusion problems depend either reciprocally on the diffusion parameter or on higher order norms of the solution. Therefore, the estimates generally become worthless in the convection-dominated case with 0 < 1. In this work we develop a rigorous -uniform convergence theory for finite element discretizations of convection-dominated diffusion problems in Eulerian and Lagrangian coordinates. Here, the constants that arise in the error estimates depend on norms of the data and not of the solution and remain bounded in the hyperbolic limit 0. In particular in the Lagrangian case this requires modifications to standard finite element error analyses. In the Eulerian formulation -uniform convergence of order one half is proven whereas in the Lagrangian framework -uniform convergence of optimal order is established. The estimates are based on -uniform a priori estimates for the solution of the continuous problems which are derived first.     

Exhaustive testing of almost all devices with outputs depending on limited number of inputs

The problem is considered of efficient test design for combinational devices with large numbern of input variables, where each output depends on at mosts input variables. It is shown that the number of test patterns can be reduced drastically if we allow a small fraction of all possible outputs not to be tested exhaustively. The effect holds even if approaches zero with the increase ofn. Upper and lower bounds for the number of test patterns in such tests (called (, s)-exhaustive tests) are derived and constructions of the tests are suggested.     

Expansion in Terms of Power of a Small Parameter of the Maximum Rank Distribution of a Random Boolean Matrix

The paper considers an N × n matrix (N n) over a field GF(2) that consists of random values with a distribution depending on a small parameter . The expansion is found in terms of the power of the parameter of the probability that the matrix rank is equal to n. Exact values of the first three coefficients are indicated.     

Numerical stability of a convex hull algorithm for simple polygons

A numerically stable and optimalO(n)-time implementation of an algorithm for finding the convex hull of a simple polygon is presented. Stability is understood in the sense of a backward error analysis. A concept of the condition number of simple polygons and its impact on the performance of the algorithm is discussed. It is shown that if the condition number does not exceed (1+O())/(3), then, in floating-point arithmetic with the unit roundoff, the algorithm produces the vertices of a convex hull for slightly perturbed input points. The relative perturbation does not exceed 3(1+O()).J. W. Jaromczyk was partially supported by a grant from the Center for Robotics and Manufacturing Systems at the University of Kentucky and G. W. Wasilkowski was partially supported by the National Science Foundation under Grants CCR-89-05371 and CCR-91-14042.     

Convergence of MUSCL Relaxing Schemes to the Relaxed Schemes for Conservation Laws with Stiff Source Terms

We consider the convergence and stability property of MUSCL relaxing schemes applied to conservation laws with stiff source terms. The maximum principle for the numerical schemes will be established. It will be also shown that the MUSCL relaxing schemes are uniformly l ¹- and TV-stable in the sense that they are bounded by a constant independent of the relaxation parameter , the Lipschitz constant of the stiff source term and the time step t. The Lipschitz constant of the l ¹ continuity in time for the MUSCL relaxing schemes is shown to be independent of and t. The convergence of the relaxing schemes to the corresponding MUSCL relaxed schemes is then established.     

Lipschitz continuity of theε-subdifferential and nonmonotonic methods with averaging ofε-subgradients for nonsmooth optimization

Inertial methods with averaging of-subgradient are developed for convex optimization problems. Convergence is proved using Lipschitz continuity of the-subdifferential.Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 70–78, March–April, 1992.     

An Estimation Problem for Quasilinear Stochastic Partial Differential Equations

A problem of estimating a functional parameter (x) and functionals () based on observation of a solution u (t, x) of the stochastic partial differential equation is considered. The asymptotic problem setting, as the noise intensity 0, is investigated.     

Range Estimation Is NP-Hard for ε2 Accuracy and Feasible for ε2−δ

The basic problem of interval computations is: given a function f(x ₁,..., x _n) and n intervals [x _i, x _i], find the (interval) range yof the given function on the given intervals. It is known that even for quadratic polynomials f(x ₁,..., x _n), this problem is NP-hard. In this paper, following the advice of A. Neumaier, we analyze the complexity of asymptotic range estimation, when the bound on the width of the input intervals tends to 0. We show that for small c > 0, if we want to compute the range with an accuracy c ₂, then the problem is still NP-hard; on the other hand, for every > 0, there exists a feasible algorithm which asymptotically, estimates the range with an accuracy c ^2–.     

Improving known solutions is hard

In this paper, we study the complexity of computing better solutions to optimization problems given other solutions. We use a model of computation suitable for this purpose, the counterexample computation model. We first prove that, if PH _P ³ , polynomial time transducers cannot compute optimal solutions for many problems, even givenn ^1– non-trivial solutions, for any >0. These results are then used to establish sharp lower bounds for several problems in the counterexample model. We extend the model by defining probabilistic counterexample computations and show that our results hold even in the presence of randomness.     

Implementation of Generalized Finite Element Methods for Homogenization Problems

The implementation of a generalized Finite Element Method (FEM) for problems with coefficients or geometry that oscillate locally at a small length scale 1 is described. Two-scale FE-spaces are combined conformingly with standard FE. Numerical experiments show that the complexity of the algorithm is independent of the micro length scale .