A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
On approximate range counting and depth
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
An optimal randomized algorithm for d-variate zonoid depth
Computational Geometry: Theory and Applications
On the bichromatic k-set problem
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Output-sensitive algorithms for Tukey depth and related problems
Statistics and Computing
An Efficient Algorithm for 2D Euclidean 2-Center with Outliers
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Algorithms for optimal outlier removal
Journal of Discrete Algorithms
A general approach for cache-oblivious range reporting and approximate range counting
Proceedings of the twenty-fifth annual symposium on Computational geometry
Enclosing weighted points with an almost-unit ball
Information Processing Letters
A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries
Journal of the ACM (JACM)
Coloring geometric range spaces
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
A general approach for cache-oblivious range reporting and approximate range counting
Computational Geometry: Theory and Applications
On the bichromatic k-set problem
ACM Transactions on Algorithms (TALG)
Thin partitions: isoperimetric inequalities and a sampling algorithm for star shaped bodies
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Covering points by disjoint boxes with outliers
Computational Geometry: Theory and Applications
Minimizing the error of linear separators on linearly inseparable data
Discrete Applied Mathematics
On the coarseness of bicolored point sets
Computational Geometry: Theory and Applications
Covering a bichromatic point set with two disjoint monochromatic disks
Computational Geometry: Theory and Applications
Absolute approximation of Tukey depth: Theory and experiments
Computational Geometry: Theory and Applications
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Two decades ago, Megiddo and Dyer showed that linear programming (LP) in two and three dimensions (and subsequently any constant number of dimensions) can be solved in linear time. In this paper, we consider the LP problem with at most k violations, i.e., finding a point inside all but at most k halfspaces, given a set of n halfspaces. We present a simple algorithm in two dimensions that runs in O((n+k2)log n) expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matousek (1994) for many values of k and is probably near-optimal. An extension of our algorithm in three dimensions runs in near O(n+k11/4n1/4) expected time. Interestingly, the idea is based on concave-chain decompositions (or covers) of the $(\le k)$-level, previously used in proving combinatorial k-level bounds.Applications in the plane include improved algorithms for finding a line that misclassifies the fewest among a set of bichromatic points, and finding the smallest circle enclosing all but k points. We also discuss related problems of finding local minima in levels.