Kenneth L. Clarkson

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Random sampling is used for several new geometric algorithms. The algorithms are &#8220;Las Vegas,&#8221; 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)
We design a new distribution over <i>m</i> &#215; <i>n</i> matrices <i>S</i> so that, for any fixed <i>n</i> &#215; <i>d</i> matrix <i>A</i> of rank <i>r</i>, with probability at least 9/10, &par;<i>SAx</i>&par;<sub>2</sub> &equals; (1 &#177; &epsiv;)&par;<i>Ax</i>&par;<sub>2</sub> simultaneously for all <i>x</i> &#8712; R<i><sup>d</sup></i>. Here, <i>m</i>(More)
Given a set of points <i>P</i> &sub; <i>R</i><sup><i>d</i></sup> and value &#8714; &gt; 0, an &#8714;-core-set <i>S</i> &sub; <i>P</i> has the property that the smallest ball containing <i>S</i> is an &#8714;-approximation of the smallest ball containing <i>P</i>. This paper shows that any point-set has an &#8714;-core-set of size [2/&#8714;]. We also give(More)
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)