The number of small semispaces of a finite set of points in the plane
Journal of Combinatorial Theory Series A
Constructing arrangements of lines and hyperplanes with applications
SIAM Journal on Computing
On k-hulls and related problems
SIAM Journal on Computing
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Topologically sweeping an arrangement
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
Sharp upper and lower bounds on the length of general Davenport-Schinzel Sequences
Journal of Combinatorial Theory Series A
A fast planar partition algorithm, II
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
On levels in arrangements and Voronoi diagrams
Discrete & Computational Geometry
An optimal algorithm for intersecting line segments in the plane
Journal of the ACM (JACM)
An on-line algorithm for fitting straight lines between data ranges
Communications of the ACM
Constructing levels in arrangements and higher order Voronoi diagrams
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
On geometric optimization with few violated constraints
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
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This paper gives an optimal Onlogn+nk time algorithm for constructing the levels1,…,k in an arrangement ofn lines in the plane. This algorithmis extended to compute these levels in an arrangement ofn unboundedx-monotone polygonal convex chains,of which each pair intersects at most a constant number of times.These algorithms can be used to solve the following separation andtransversal problems. For a set nblue points and an set of n redpoints, find a line that separates the two sets in such a way that thesum, m, of the number of red pointsabove the line and the number of blue points below the line isminimized. Such an optimal line can be found in Onmlogm+nlogn time. For a set ofnline segments in the plane, find aline that intersects the maximum number of the line segments. Such anoptimal line can be found in Onmlogm+nlogn time for vertical segments and in Onmlogm+nlog2nan expected time for arbitrary line segments, wherem denotes the number ofline segments not intersected by the optimal line.