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- Kenneth L. Clarkson, Peter W. Shor
- Discrete & Computational Geometry
- 1988

Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires <italic>&Ogr;</italic>(<italic>A</italic> + <italic>n</italic> log… (More)

- Kenneth L. Clarkson
- Discrete & Computational Geometry
- 1987

- Kenneth L. Clarkson, David P. Woodruff
- STOC
- 2013

We design a new distribution over <i>m</i> × <i>n</i> matrices <i>S</i> so that, for any fixed <i>n</i> × <i>d</i> matrix <i>A</i> of rank <i>r</i>, with probability at least 9/10, ∥<i>SAx</i>∥<sub>2</sub> = (1 ± ϵ)∥<i>Ax</i>∥<sub>2</sub> simultaneously for all <i>x</i> ∈ R<i><sup>d</sup></i>. Here, <i>m</i>… (More)

- Kenneth L. Clarkson
- J. ACM
- 1995

This paper gives an algorithm for solving linear programming problems. For a problem with n constraints and d variables, the algorithm requires an expected<inline-equation><f>O<fen lp="par">d<sup>2</sup>n<rp post="par"></fen>+<fen lp="par">logn<rp post="par"></fen>O<fen lp="par">d<rp post="par"></fen><sup>d/2+O<fen lp="par">1<rp… (More)

- Kenneth L. Clarkson
- SODA
- 2008

The problem of maximizing a concave function <i>f(x)</i> in the unit simplex Δ can be solved approximately by a simple greedy algorithm. For given <i>k</i>, the algorithm can find a point <i>x</i><sub>(<i>k</i>)</sub> on a <i>k</i>-dimensional face of Δ, such that <i>f</i>(<i>x</i><sub>(<i>k</i>)</sub> ≥ <i>f(x</i><sub>*</sub>) −… (More)

This paper gives approximation algorithms for solving the following motion planning problem: Given a set of polyhedral obstacles and points s and t, find a shortest path from s to t that avoids the obstacles. The paths found by the algorithms are piecewise linear, and the length of a path is the sum of the lengths of the line segments making up the path.… (More)

- Mihai Badoiu, Kenneth L. Clarkson
- SODA
- 2003

Given a set of points <i>P</i> ⊂ <i>R</i><sup><i>d</i></sup> and value ∊ > 0, an ∊-core-set <i>S</i> ⊂ <i>P</i> has the property that the smallest ball containing <i>S</i> is an ∊-approximation of the smallest ball containing <i>P</i>. This paper shows that any point-set has an ∊-core-set of size [2/∊]. We also give… (More)

- Mihai Badoiu, Kenneth L. Clarkson
- Comput. Geom.
- 2008

Given a set of points P ⊂ R and value > 0, an core-set S ⊂ P has the property that the smallest ball containing S is within of the smallest ball containing P . This paper shows that any point set has an -core-set of size d1/ e, and this bound is tight in the worst case. A faster algorithm given here finds an core-set of size at most 2/ . These results imply… (More)

- Kenneth L. Clarkson
- SIAM J. Comput.
- 1988

- Kenneth L. Clarkson, David P. Woodruff
- STOC
- 2009

We give near-optimal space bounds in the streaming model for linear algebra problems that include estimation of matrix products, linear regression, low-rank approximation, and approximation of matrix rank. In the streaming model, sketches of input matrices are maintained under updates of matrix entries; we prove results for turnstile updates, given in an… (More)