Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
A randomized algorithm for closest-point queries
SIAM Journal on Computing
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Points and triangles in the plane and halving planes in space
Discrete & Computational Geometry
An upper bound on the number of planar K-sets
Discrete & Computational Geometry
Improved bounds for intersecting triangles and halving planes
Journal of Combinatorial Theory Series A
The colored Tverberg's problem and complexes of injective functions
Journal of Combinatorial Theory Series A
Dehn-Sommerville relations, upper bound theorem, and levels in arrangements
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Journal of Computer and System Sciences - Special issue: papers from the 22nd ACM symposium on the theory of computing, May 14–16, 1990
Lectures on Discrete Geometry
A Characterization of Planar Graphs by Pseudo-Line Arrangements
ISAAC '97 Proceedings of the 8th International Symposium on Algorithms and Computation
Lenses in arrangements of pseudo-circles and their applications
Journal of the ACM (JACM)
Low-Dimensional Linear Programming with Violations
SIAM Journal on Computing
On levels in arrangements of surfaces in three dimensions
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences
Discrete & Computational Geometry
Intersection reverse sequences and geometric applications
Journal of Combinatorial Theory Series A
On a Geometric Generalization of the Upper Bound Theorem
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
On levels in arrangements of curves, iii: further improvements
Proceedings of the twenty-fourth annual symposium on Computational geometry
An Efficient Algorithm for 2D Euclidean 2-Center with Outliers
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
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We study a bichromatic version of the well-known k-set problem: given two sets R and B of points of total size n and an integer k, how many subsets of the form (R∩h)∪(B\ h) can have size exactly k over all halfspaces h? In the dual, the problem is asymptotically equivalent to determining the worst-case combinatorial complexity of the k-level in an arrangement of n halfspaces. Disproving an earlier conjecture by Linhart (1993), we present the first nontrivial upper bound for all k ≪ n in two dimensions: O(nk1/3 + n5/6--∈k2/3+2∈ + k2) for any fixed ∈ 0. In three dimensions, we obtain the bound O(nk3/2 +n0.5034k2.4932 + k3). Incidentally, this also implies a new upper bound for the original k-set problem in four dimensions: O(n2k3/2+n1.5034k2.4932+nk3), which improves the best previous result for all k ≪ n0.923. Extensions to other cases, such as arrangements of disks, are also discussed.