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21
The Exact Computation Paradigm
, 1994
"... We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next ..."
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Cited by 95 (10 self)
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We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next we survey some recent applications of this paradigm. Finally, we outline some basic theory and techniques in this paradigm. 1 This paper will appear as a chapter in the 2nd edition of Computing in Euclidean Geometry, edited by D.Z. Du and F.K. Hwang, published by World Scientific Press, 1994. 1 1 Two Numerical Computing Paradigms Computation has always been intimately associated with numbers: computability theory was early on formulated as a theory of computable numbers, the first computers have been number crunchers and the original massproduced computers were pocket calculators. Although one's first exposure to computers today is likely to be some nonnumerical application, numeri...
Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
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Cited by 94 (12 self)
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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
Exact Computational Geometry and Tolerancing Metrology
, 1994
"... We describe the relevance of Computational Geometry to tolerancing metrology. We outline the basic issues and define the class of zone problems that is central in this area. In the context of the exact computation paradigm, these problems are prime candidates for "exact solution" since we show that ..."
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Cited by 26 (6 self)
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We describe the relevance of Computational Geometry to tolerancing metrology. We outline the basic issues and define the class of zone problems that is central in this area. In the context of the exact computation paradigm, these problems are prime candidates for "exact solution" since we show that they have boundeddepth. Metrologists in this field have mounted a quest for a reference software which will impose some certainty in a confusing market of metrology software. The use of exact computation in the reference software will solve many intractable difficulties associated with current approaches. In short, here is a practical area in which CG and exact computation can have a real impact. 1 Introduction Researchers in Computational Geometry (CG) have always been convinced that their subject is relevant to a variety of application areas. But CG'ers have often assumed that the application areas would come to CG to find answers to their questions. To what extent is this valid? I will d...
Approximation and Exact Algorithms for MinimumWidth Annuli and Shells
 Discrete Comput. Geom
, 1999
"... Let S be a set of n points in R d . The "roundness" of S can be measured by computing the width ! = ! (S) of the thinnest spherical shell (or annulus in R 2 ) that contains S. This paper contains three main results related to computing ! : (i) For d = 2, we can compute in O(n log n) tim ..."
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Cited by 22 (14 self)
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Let S be a set of n points in R d . The "roundness" of S can be measured by computing the width ! = ! (S) of the thinnest spherical shell (or annulus in R 2 ) that contains S. This paper contains three main results related to computing ! : (i) For d = 2, we can compute in O(n log n) time an annulus containing S whose width is at most 2! (S). We extend this algorithm, so that for any given parameter " ? 0, an annulus containing S whose width is at most (1 + ")! , is computed in time O(n log n + n=" 2 ). (ii) For d 3, given a parameter " ? 0, we can compute a shell containing S of width at most (1+ ")! either in time O \Gamma n " d log( \Delta ! " ) \Delta or in time O \Gamma n " d\Gamma2 \Gamma log n + 1 " \Delta log \Gamma \Delta ! " \Delta\Delta . Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by NSF grants EIA9870724, and CCR9732787, by an NYI award, and by a grant from ...
A Complete Roundness Classification Procedure
 IN PROC. 13TH ANNU. ACM SYMPOS. COMPUT. GEOM
, 1997
"... We describe a roundness classification procedure, that is, a procedure to determine if the roundness of a planar object I is within some ffl 0 from an ideal circle. The procedure consists of a probing strategy and an evaluation algorithm working in a feedback loop. This approach of combining probin ..."
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Cited by 18 (0 self)
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We describe a roundness classification procedure, that is, a procedure to determine if the roundness of a planar object I is within some ffl 0 from an ideal circle. The procedure consists of a probing strategy and an evaluation algorithm working in a feedback loop. This approach of combining probing with evaluation is new in computational metrology. For several definitions of roundness, our procedure uses O(1=qual(I)) probes and runs in time O(1=qual(I) 2 ). Here, the quality qual(I) of I measures how far the roundness of I is from the acceptreject criterion. Hence our algorithms are "quality sensitive".
Computing Constrained MinimumWidth Annuli of Point Sets
, 2000
"... We study the problem of determining whether a manufactured disc of certain radius r is within tolerance. More precisely, we present algorithms that, given a set of n probe points on the surface of the manufactured object, compute the thinnest annulus whose outer (or inner, or median) radius is r and ..."
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Cited by 15 (4 self)
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We study the problem of determining whether a manufactured disc of certain radius r is within tolerance. More precisely, we present algorithms that, given a set of n probe points on the surface of the manufactured object, compute the thinnest annulus whose outer (or inner, or median) radius is r and that contains all the probe points. Our algorithms run in O(n log n) time.
No coreset, no cry
 In Proceedings 24th Conference on Foundations of Software Technology and Theoretical Computer Science
, 2004
"... We show that coresets do not exist for the problem of 2slabs in IR 3, thus demonstrating that the natural approach for solving approximately this problem efficiently is infeasible. On the positive side, for a point set P in IR 3, we describe a near linear time algorithm for computing a (1 + ε)appr ..."
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Cited by 15 (3 self)
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We show that coresets do not exist for the problem of 2slabs in IR 3, thus demonstrating that the natural approach for solving approximately this problem efficiently is infeasible. On the positive side, for a point set P in IR 3, we describe a near linear time algorithm for computing a (1 + ε)approximation to the minimum width 2slab cover of P. This is a first step in providing an efficient approximation algorithm for the problem of covering a point set with kslabs. 1
Approximate Shape Fitting via Linearization
 In Proc. 42nd Annu. IEEE Sympos. Found. Comput. Sci
, 2001
"... Shape fitting is a fundamental optimization problem in computer science. In this paper, we present a general and unified technique for solving a certain family of such problems. Given a point set P in R d, this technique can be used to εapproximate: (i) the minwidth annulus and shell that contains ..."
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Cited by 14 (7 self)
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Shape fitting is a fundamental optimization problem in computer science. In this paper, we present a general and unified technique for solving a certain family of such problems. Given a point set P in R d, this technique can be used to εapproximate: (i) the minwidth annulus and shell that contains P, (ii) minimum width cylindrical shell containing P, (iii) diameter, width, minimum volume bounding box of P, and (iv) all the previous measures for the case the points are moving. The running time of the resulting algorithms is O(n + 1/ε c), where c is a constant that depends on the problem at hand. Our new general technique enable us to solve those problems without resorting to a careful and painful case by case analysis, as was previously done for those problems. Furthermore, for several of those problems our results are considerably simpler and faster than what was previously known. In particular, for the minimum width cylindrical shell problem, our solution is the first algorithm whose running time is subquadratic in n. (In fact we get running time linear in n.) 1
Issues in the Metrology of Geometric Tolerancing
 Robotics Motion and Manipulation
, 1996
"... Introduction Geometric tolerancing is concerned with the specification of geometric shapes for use in the manufacture of mechanical parts. Since manufacturing processes are inherently imprecise, it is imperative that such geometric designs be accompanied by tolerance specifications. Figure 1 is a s ..."
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Cited by 14 (1 self)
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Introduction Geometric tolerancing is concerned with the specification of geometric shapes for use in the manufacture of mechanical parts. Since manufacturing processes are inherently imprecise, it is imperative that such geometric designs be accompanied by tolerance specifications. Figure 1 is a simple example of such a specification. Evidently, the ideal object indicated there is a 3 \Theta 5 5 \Sigma 0:1 3 \Sigma 0:05 Figure 1: Tolerancing a rectangle. rectangle, with some amount of deviation to be tolerated in its manufacture. There is a naive interpretation of this symbology, namely, "any manufactured rectangle of dimension L \Theta H where jL \Gamma 5j 0:1 and jH \Gamma 3j 0:05 is considered w