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The Area of Overlap of two Unions of Convex Objects under Translation
, 2003
"... Let A and B be two sets of n resp. m disjoint unit discs in the plane, with m n. We consider the problem of nding a translation of A that maximizes the total area of its overlap with B. We rst show that the maximum number of combinatorially distinct translations of A with respect to B can be a ..."
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Let A and B be two sets of n resp. m disjoint unit discs in the plane, with m n. We consider the problem of nding a translation of A that maximizes the total area of its overlap with B. We rst show that the maximum number of combinatorially distinct translations of A with respect to B can be as high as (n m). Moreover, the function describing the area of overlap is quite complex, even for combinatorially equivalent translations. Hence, we turn our attention to approximation algorithms: we give a fairly simple algorithm with running time O((nm= ) log(nm= )) that, for any given > 0, computes a translation for which the area of overlap is within a factor (1 ) of the optimum. Several other results that give constantfactor approximation schemes are also discussed. Our results generalize to the case where A and B consist of either possibly intersecting homothets of a xed planar convex object or possibly intersecting planar fat objects provided that (i) the ratio of the areas of any two objects in A [ B is bounded, and (ii) within each set, the maximum number of objects with a nonempty intersection is bounded.
Determining critical points in . . .
, 2015
"... We consider the problem of computing critical points of plane curves represented in a finite orthogonal polynomial basis. This is motivated by an approach to the recognition of handwritten mathematical symbols in which the initial data is in such an orthogonal basis and it is desired to avoid ill ..."
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We consider the problem of computing critical points of plane curves represented in a finite orthogonal polynomial basis. This is motivated by an approach to the recognition of handwritten mathematical symbols in which the initial data is in such an orthogonal basis and it is desired to avoid illconditioned basis conversions. Our main contribution is to assemble the relevant mathematical tools to perform all the necessary operations in the orthogonal polynomial basis. These include implicitization, differentiation, root finding and resultant computation.
Root Refinement for Real Polynomials using Quadratic Interval Refinement
"... We consider the problem of approximating all real roots of a squarefree polynomial f with real coefficients. Given isolating intervals for the real roots and an arbitrary positive integer L, the task is to approximate each root to L bits after the binary point. Abbott has proposed the quadratic in ..."
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We consider the problem of approximating all real roots of a squarefree polynomial f with real coefficients. Given isolating intervals for the real roots and an arbitrary positive integer L, the task is to approximate each root to L bits after the binary point. Abbott has proposed the quadratic interval refinement method (QIR for short), which is a variant of Regula Falsi combining the secant method and bisection. We formulate a variant of QIR, denoted AQIR (”Approximate QIR”), that considers only approximations of the polynomial coefficients and chooses a suitable working precision adaptively. It returns a certified result for any given real polynomial, whose roots are all simple. In addition, we propose several techniques to improve the asymptotic complexity of QIR: We prove a bound on the bit complexity of our algorithm in terms of the degree of the polynomial, the size and the separation of the roots, that is, parameters exclusively related to the geometric location of the roots. For integer coefficients, our variant improves, in theory and practice, the variant with exact integer arithmetic. Combining our approach with multipoint evaluation, we obtain nearoptimal bounds that essentially match the best known theoretical bounds on root approximation as obtained by very sophisticated algorithms.