• Publications
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Applications of random sampling in computational geometry, II
  • K. Clarkson
  • Computer Science, Mathematics
    SCG '88
  • 6 January 1988
TLDR
Asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets, are given.
Low-Rank Approximation and Regression in Input Sparsity Time
We design a new distribution over m × n matrices S so that, for any fixed n × d matrix A of rank r, with probability at least 9/10, ∥SAx∥2 = (1 ± ε)∥Ax∥2 simultaneously for all x ∈ Rd. Here, m is
Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm
  • K. Clarkson
  • Mathematics, Computer Science
    SODA '08
  • 20 January 2008
TLDR
These results are tied together, stronger convergence results are reviewed, and several coreset bounds are generalized or strengthened.
Optimal core-sets for balls
Given a set of points [email protected]?R^d and value @e>0, an @[email protected]?P has the property that the smallest ball containing S has radius within [email protected] of the radius of the
Numerical linear algebra in the streaming model
TLDR
Near-optimal space bounds are given in the streaming model for linear algebra problems that include estimation of matrix products, linear regression, low-rank approximation, and approximation of matrix rank; results for turnstile updates are proved.
Low rank approximation and regression in input sparsity time
TLDR
The fastest known algorithms for overconstrained least-squares regression, low-rank approximation, approximating all leverage scores, and l<sub>p</sub>-regression are obtained.
Smaller core-sets for balls
TLDR
It is shown that any point-set has an ∊-core-set of size [2/∊], and a fast algorithm is given that finds this core-set and implies the existence of small core-sets for solving approximate approximate <i>k</i>-center clustering and related problems.
New applications of random sampling in computational geometry
  • K. Clarkson
  • Mathematics, Computer Science
    Discret. Comput. Geom.
  • 1 June 1987
This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by
Approximation algorithms for shortest path motion planning
  • K. Clarkson
  • Computer Science, Mathematics
    STOC
  • 1 January 1987
This paper gives approximation algorithms of 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
Combinatorial complexity bounds for arrangements of curves and spheres
TLDR
Upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike are presented and it is proved that the maximum number of edges boundingm cells in an arrangement ofn lines is Θ(m2/3n 2/3 +n), and that it isO(m3/2β(m) forn unit-circles.
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