Algorithms for vertical and orthogonal L1 linear approximation of points
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Algorithms for a Minimum Volume Enclosing Simplex in Three Dimensions
SIAM Journal on Computing
SIAM Journal on Computing
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STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Approximating extent measures of points
Journal of the ACM (JACM)
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Journal of the ACM (JACM)
Smaller coresets for k-median and k-means clustering
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Subgradient and sampling algorithms for l1 regression
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Robust shape fitting via peeling and grating coresets
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Analysis of incomplete data and an intrinsic-dimension Helly theorem
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Matrix approximation and projective clustering via volume sampling
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Coresets and sketches for high dimensional subspace approximation problems
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Geometric optimization and sums of algebraic functions
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Geometric optimization and sums of algebraic functions
ACM Transactions on Algorithms (TALG)
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In this paper, we study the problem of L1-fitting a shape to a set of n points in Rd (where d is a fixed constant), where the target is to minimize the sum of distances of the points to the shape, or alternatively the sum of squared distances. We present a general technique for computing a (1 + ε)-approximation for such a problem, with running time O(n + poly(logn, 1/ε)), where poly(logn, 1/ε) is a polynomial of constant degree of logn and 1/ε (the power of the polynomial is a function of d). This is a linear time algorithm for a fixed ε 0, and is the first subquadratic algorithm for this problem. Applications of the algorithm include best fitting either a circle, a sphere or a cylinder to a set of points when minimizing the sum of distances (or squared distances) to the respective shape.