Proceedings of the nineteenth annual symposium on Computational geometry
An optimal randomized algorithm for maximum Tukey depth
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
On the least median square problem
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
On approximating the depth and related problems
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
Bounds for point recolouring in geometric graphs
Computational Geometry: Theory and Applications
Approximating Points by a Piecewise Linear Function: II. Dealing with Outliers
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Computing the least median of squares estimator in time O(nd)
ICCSA'05 Proceedings of the 2005 international conference on Computational Science and its Applications - Volume Part I
Delineating boundaries for imprecise regions
ESA'05 Proceedings of the 13th annual European conference on Algorithms
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Outlier respecting points approximation
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
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Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in lineartime. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given halfspaces. We give a simple algorithm in 2-d that runs in 0((n + k^2 )\log n) expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matou驴ek (1994) and is probably near-optimal for all k \ll {n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}. A (theoretical) extension of our algorithm in 3-d runs in near 0(n + k^{{{11} \mathord{\left/ {\vphantom {{11} 4}} \right. \kern-\nulldelimiterspace} 4}} n^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}}) expected time. Interestingly, the idea is based on concave-chain decompositions (or covers) of the( \leqslant 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.