Ray shooting and other applications of spanning trees with low stabbing number
SIAM Journal on Computing
Approximating center points with iterated radon points
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Cutting hyperplanes for divide-and-conquer
Discrete & Computational Geometry
Ray shooting and lines in space
Handbook of discrete and computational geometry
The discrepancy method: randomness and complexity
The discrepancy method: randomness and complexity
Lectures on Discrete Geometry
A Combinatorial Bound for Linear Programming and Related Problems
STACS '92 Proceedings of the 9th Annual Symposium on Theoretical Aspects of Computer Science
An optimal randomized algorithm for maximum Tukey depth
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Algorithms for center and Tverberg points
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
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A center-transversal line for two finite point sets in ℝ3 is a line with the property that any closed halfspace that contains it also contains at least one third of each point set. It is known that a center-transversal line always exists [12],[24] but the best known algorithm for finding such a line takes roughly n12 time. We propose an algorithm that finds a center-transversal line in ${\it O}({\it n}^{\rm 1+{\it \epsilon}}{\it \kappa}^{\rm 2}({\it n}))$ worst-case time, for any ${\it \epsilon}$0, where ${\it \kappa}({\it n})$ is the maximum complexity of a single level in an arrangement of n planes in ℝ3. With the current best upper bound ${\it \kappa}$(n)=O(n5/2) of [21], the running time is ${\it O}({\it n}^{\rm 6+{\it \epsilon}})$, for any ${\it \epsilon} 0$. We also show that the problem of deciding whether there is a center-transversal line parallel to a given direction u can be solved in O(nlogn) expected time. Finally, we We also extend the concept of center-transversal line to that of bichromatic depth of lines in space, and give an algorithm that computes a deepest line exactly in time ${\it O}({\it n}^{\rm 1+{\it \epsilon}}{\it \kappa}^{\rm 2}({\it n}))$, and a linear-time approximation algorithm that computes, for any specified ${\it \delta}0$, a line whose depth is at least $1-{\it \delta}$ times the maximum depth.