Analyzing the worst-case complexity of the k-level in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O(nk
|
1541
|
Computational Geometry: An Introduction
– Preparata, Shamos
- 1985
|
|
611
|
Algorithms in combinatorial geometry
– Edelsbrunner
- 1987
|
|
345
|
Applications of random sampling in computational geometry
– Clarkson
- 1989
|
|
322
|
Davenport-Schinzel Sequences and Their Geometric Applications
– Sharir, Agarwal
- 1995
|
|
274
|
Computational Geometry. An introduction through randomized algorithms
– Mulmuley
- 1994
|
|
194
|
Geometric range searching and its relatives
– Agarwal, Erickson
- 1999
|
|
116
|
Applications of random sampling
– Clarkson, Shor
- 1989
|
|
69
|
Kinetic data structures — a state of the art report
– Guibas
- 1998
|
|
67
|
Modern Graph Theory, Graduate Texts
– Bollobás
- 1998
|
|
52
|
On k-sets in arrangements of curves and surfaces
– Sharir
- 1991
|
|
50
|
Some dynamic computational geometry problems
– Atallah
- 1985
|
|
47
|
Arrangements and their applications
– Agarwal, Sharir
- 2000
|
|
46
|
The colored Tverberg's problem and complexes of injective functions
– Zivaljevi'c, Vre'cica
- 1992
|
|
44
|
Proximity problems on moving points
– Basch, Guibas, et al.
- 1997
|
|
43
|
Allowable sequences and order types in discrete and computational geometry
– Goodman, Pollack
- 1993
|
|
41
|
On the number of halving lines
– Lov'asz
- 1971
|
|
39
|
On levels in arrangements of lines, segments, planes, and triangles
– Agarwal, Aronov, et al.
- 1997
|
|
35
|
On geometric optimization with few violated constraints
– Matoušek
- 1995
|
|
32
|
Halfspace range search: an algorithmic application of k-sets
– Chazelle, Preparata
- 1986
|
|
30
|
Dissection graphs of planar point sets
– Erdos, Lov'asz, et al.
- 1973
|
|
29
|
Points and triangles in the plane and halving planes in space, Discrete Comput
– Aronov, Chazelle, et al.
- 1991
|
|
28
|
Point sets with many k-sets
– Toth
- 2000
|
|
27
|
Algorithms for ham-sandwich cuts
– Lo, Matouˇsek, et al.
- 1994
|
|
25
|
The number of small semispaces of a finite set of points in the plane
– Alon, Gyori
- 1986
|
|
24
|
An upper bound on the number of planar k-sets
– Pach, Steiger, et al.
- 1992
|
|
20
|
Cutting circles into pseudo-segments and improved bounds for incidences and complexity of many faces. Discrete Comput. Geom
– Aronov, Sharir
- 2002
|
|
19
|
Parametric and Kinetic Minimum Spanning Trees
– Agarwal, Eppstein, et al.
- 1998
|
|
19
|
Counting triangle crossings and halving planes, Discrete Comput
– Dey, Edelsbrunner
- 1994
|
|
16
|
Lenses in arrangements of pseudo-circles and their applications
– Agarwal, Nevo, et al.
- 2004
|
|
16
|
Improved bounds on planar k-sets and k-levels
– Dey
- 1998
|
|
16
|
Taking a walk in a planar arrangement
– Har-Peled
- 1999
|
|
15
|
Generalizing ham sandwich cuts to equitable subdivisions
– Bespamyatnikh, Kirkpatrick, et al.
|
|
15
|
Arrangements of curves in the plane–topology, combinatorics, and algorithms
– Edelsbrunner, Guibas, et al.
- 1992
|
|
15
|
An optimal algorithm for the (≤ k)-levels, with applications to separation and transversal problems
– Everett, Robert, et al.
- 1996
|
|
15
|
A characterization of planar graphs by pseudo-line arrangements
– Tamaki, Tokuyama
- 1997
|
|
15
|
Counting triangle crossings and halving planes
– Dey, Edelsbrunner
- 1994
|
|
14
|
Geometric lower bounds for parametric matroid optimization
– Eppstein
- 1995
|
|
14
|
An improved bound for k-sets in three dimensions
– Sharir, Smorodinsky, et al.
|
|
13
|
On the number of halving planes
– B'ar'any, Furedi, et al.
- 1990
|
|
13
|
How to cut pseudo-parabolas into segments
– Tamaki, Tokuyama
- 1998
|
|
12
|
2D arrangements
– Wein, Fogel, et al.
- 2007
|
|
10
|
k-sets and j-facets: a tour of discrete geometry
– Andrzejak, Welzl
- 1997
|
|
10
|
Remarks on k-level algorithms in the plane
– Chan
- 1999
|
|
10
|
Improved bounds for intersecting triangles and halving planes
– Eppstein
- 1993
|
|
10
|
On the number of k-subsets of a set of n points in the plane
– Goodman, Pollack
- 1984
|
|
10
|
On minimum and maximum spanning trees of linearly moving points
– Katoh, Tokuyama, et al.
- 1995
|
|
9
|
An upper bound on the number of planar k-sets. Discrete Comput. Geom
– Pach, Steiger, et al.
- 1992
|
|
8
|
Algorithms for ham-sandwich cuts. Discrete Comput. Geom
– Lo, Matousek, et al.
- 1994
|
|
5
|
Bounds for the parametric minimum spanning tree problem
– Gusfield
- 1979
|
|
4
|
Lovasz's lemma for the three-dimensional k-level of concave surfaces and its applications
– Katoh, Tokuyama
- 1999
|