## Improved Bounds on Planar k-sets and k-levels (1997)

### Cached

### Download Links

- [www.cis.ohio-state.edu]
- [www.cse.ohio-state.edu]
- DBLP

### Other Repositories/Bibliography

Venue: | Discrete Comput. Geom |

Citations: | 16 - 0 self |

### BibTeX

@ARTICLE{Dey97improvedbounds,

author = {Tamal K. Dey},

title = {Improved Bounds on Planar k-sets and k-levels},

journal = {Discrete Comput. Geom},

year = {1997},

volume = {19},

pages = {156--161}

}

### Years of Citing Articles

### OpenURL

### Abstract

We prove an O(nk 1=3 ) upper bound for planar k-sets. This is the first considerable improvement on this bound after its early solutions approximately twenty seven years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of k-levels in arrangements of line segments, k convex polygons in the union of n lines, parametric minimum spanning trees and parametric matroids in general. 1 Introduction The problem of determining the optimum asymptotic bound on the number of k-sets is one of the most tantalizing open problems in combinatorial geometry. Due to its importance in analyzing geometric algorithms [8, 9, 18], the problem has caught the attention of the computational geometers as well [5, 6, 7, 13, 17, 26, 28]. Given a set P of n points in ! d , and a positive integer k n, a k-set is a subset P 0 ` P such that P 0 = P " H for a halfspace H and jP 0 j = k. A close to optimal solution for the problem remains elusive even in ! ...

### Citations

688 |
Algorithms in Combinatorial Geometry
- Edelsbrunner
- 1987
(Show Context)
Citation Context ...lusive even in ! 2 . In spite of several attempts, no considerable improvement could be made from its early bound of O(nk 1=2 ) given by [19, 24]. Several proofs exist for this well known upper bound =-=[3, 5, 16, 28]-=- which is quite far away from the best known lower bound of \Omega\Gamma n log k) [16]. Pach et al. made the first dent on this upper bound improving it to O(nk 1=2 = log k). Even such a small improve... |

151 |
Applications of random sampling
- Clarkson, Shor
- 1989
(Show Context)
Citation Context ... this miserable state of the problem, exact asymptotic bound is known for the number of i-sets summed over all isk. Alon and Gyori [2] showed that this number is \Theta(nk) in ! 2 . Clarkson and Shor =-=[11]-=- generalized the bound to \Theta(n bd=2c k dd=2e ) for ! d . Other than k-sets, our proof technique also applies to establish a new O(nk 1=3 + n 2=3 k 2=3 ) complexity bound for k convex polygons whos... |

143 |
Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput
- Clarkson, Edelsbrunner, et al.
- 1990
(Show Context)
Citation Context ... are drawn from n lines. The complexity of these polygons is the total number of vertices 3 they have altogether. If they are interior-wise disjoint, an optimal \Theta(n 2=3 k 2=3 + n) bound is known =-=[10, 22]-=-. However, these analysis techniques fail if the polygons overlap in their interiors. Our proof technique can be used to establish an optimal \Theta(nk 1=3 +n 2=3 k 2=3 ) bound for this case. First, w... |

107 | Crossing numbers and hard Erdős problems in discrete geometry
- Székely
- 1997
(Show Context)
Citation Context ... was first used by us to prove an O(n 8=3 ) bound on 3dimensionalsk-sets [13]. Crossings in geometric graphs have been successfully used for many problems in combinatorial geometry. See, for example, =-=[12, 15, 27]-=-. It is expected that our approach would open up new avenues to solve the d-dimensional k-set problem, which remains largely unsolved for d ? 3 till date. The only nontrivial bound known for d ? 3 is ... |

82 |
New lower bound techniques for VLSI
- Leighton
- 1984
(Show Context)
Citation Context ...ph G we have exactly jS k\Gamma1 j edges. We can assume jS k\Gamma1 j ? 4n. Otherwise, the claim follows trivially. It follows from the result of Ajtai, Chv'atal, Newborn, Szemer'edi [1] and Leighton =-=[23]-=- that there are at least c \Delta t 3 =n 2 crossings among t ? 4n edges connecting pairs of points in a set of n points in plane, where c is some constant. Plugging t = jS k\Gamma1 j and using the res... |

71 |
Crossing-free subgraphs
- Ajtai, Chvátal, et al.
- 1982
(Show Context)
Citation Context ...e author acknowledges the support of NSF grant CCR9321799, USA and a DST young scientist grant, India. Our proof technique is surprisingly simple. It uses the concept of crossings in geometric graphs =-=[1]-=- which was first used by us to prove an O(n 8=3 ) bound on 3dimensionalsk-sets [13]. Crossings in geometric graphs have been successfully used for many problems in combinatorial geometry. See, for exa... |

62 |
On k-hulls and related problems
- Cole, Sharir, et al.
- 1987
(Show Context)
Citation Context ...lem of determining the optimum asymptotic bound on the number of k-sets is one of the most tantalizing open problems in combinatorial geometry. Due to its importance in analyzing geometric algorithms =-=[8, 9, 18]-=-, the problem has caught the attention of the computational geometers as well [5, 6, 7, 13, 17, 26, 28]. Given a set P of n points in ! d , and a positive integer ksn, a k-set is a subset P 0 ` P such... |

60 |
On k-sets in arrangements of curves and surfaces
- Sharir
- 1991
(Show Context)
Citation Context ...st tantalizing open problems in combinatorial geometry. Due to its importance in analyzing geometric algorithms [8, 9, 18], the problem has caught the attention of the computational geometers as well =-=[5, 6, 7, 13, 17, 26, 28]. Giv-=-en a set P of n points in ! d , and a positive integer ksn, a k-set is a subset P 0 ` P such that P 0 = P " H for a halfspace H and jP 0 j = k. A close to optimal solution for the problem remains... |

57 |
The colored Tverberg’s problem and complexes of injective functions
- Zivaljević, Vrećica
- 1992
(Show Context)
Citation Context ...up new avenues to solve the d-dimensional k-set problem, which remains largely unsolved for d ? 3 till date. The only nontrivial bound known for d ? 3 is insignificantly better than the trivial bound =-=[4, 29]-=-. Inspite of this miserable state of the problem, exact asymptotic bound is known for the number of i-sets summed over all isk. Alon and Gyori [2] showed that this number is \Theta(nk) in ! 2 . Clarks... |

48 |
On the number of halving lines
- Lovász
- 1971
(Show Context)
Citation Context ... 0 j = k. A close to optimal solution for the problem remains elusive even in ! 2 . In spite of several attempts, no considerable improvement could be made from its early bound of O(nk 1=2 ) given by =-=[19, 24]-=-. Several proofs exist for this well known upper bound [3, 5, 16, 28] which is quite far away from the best known lower bound of \Omega\Gamma n log k) [16]. Pach et al. made the first dent on this upp... |

42 | On levels in arrangements of lines, segments, planes, and triangles
- Agarwal, Aronov, et al.
- 1997
(Show Context)
Citation Context ...lusive even in ! 2 . In spite of several attempts, no considerable improvement could be made from its early bound of O(nk 1=2 ) given by [19, 24]. Several proofs exist for this well known upper bound =-=[3, 5, 16, 28]-=- which is quite far away from the best known lower bound of \Omega\Gamma n log k) [16]. Pach et al. made the first dent on this upper bound improving it to O(nk 1=2 = log k). Even such a small improve... |

37 | The number of small semispaces of a finite set of points in the plane - Alon, Győri - 1986 |

37 |
Dissection graphs of planar point sets
- Erdős, Lovász, et al.
- 1973
(Show Context)
Citation Context ... 0 j = k. A close to optimal solution for the problem remains elusive even in ! 2 . In spite of several attempts, no considerable improvement could be made from its early bound of O(nk 1=2 ) given by =-=[19, 24]-=-. Several proofs exist for this well known upper bound [3, 5, 16, 28] which is quite far away from the best known lower bound of \Omega\Gamma n log k) [16]. Pach et al. made the first dent on this upp... |

34 |
Halfspace range search: An algorithmic application of k-sets
- Chazelle, Preparata
- 1986
(Show Context)
Citation Context ...lem of determining the optimum asymptotic bound on the number of k-sets is one of the most tantalizing open problems in combinatorial geometry. Due to its importance in analyzing geometric algorithms =-=[8, 9, 18]-=-, the problem has caught the attention of the computational geometers as well [5, 6, 7, 13, 17, 26, 28]. Given a set P of n points in ! d , and a positive integer ksn, a k-set is a subset P 0 ` P such... |

30 |
and triangles in the plane and halving planes in space, Discrete Comput
- Aronov, Chazelle, et al.
- 1991
(Show Context)
Citation Context ...st tantalizing open problems in combinatorial geometry. Due to its importance in analyzing geometric algorithms [8, 9, 18], the problem has caught the attention of the computational geometers as well =-=[5, 6, 7, 13, 17, 26, 28]. Giv-=-en a set P of n points in ! d , and a positive integer ksn, a k-set is a subset P 0 ` P such that P 0 = P " H for a halfspace H and jP 0 j = k. A close to optimal solution for the problem remains... |

25 |
An upper bound on the number of planar k-sets
- Pach, Steiger, et al.
- 1992
(Show Context)
Citation Context ...known lower bound of \Omega\Gamma n log k) [16]. Pach et al. made the first dent on this upper bound improving it to O(nk 1=2 = log k). Even such a small improvement in ! 2 was a distinguished result =-=[25]-=-. Recently Agarwal, Aronov and Sharir [3] attacked the problem with a fresh look. Although they could not improve the worstcase upper bound, several approaches were put forward. One of these approache... |

23 | Extremal problems for geometric hypergraphs
- Dey, Pach
- 1998
(Show Context)
Citation Context ... bounds of several other related problems. It remains to be seen if the technique can be used in higher dimensions, albeit with necessary modifications. The generalization of the result of [1] exists =-=[12, 14]-=-. However, the concept of convex chains do not generalize in higher dimensions in a straightforward manner. The author believes that the technique developed in this paper would make further inroads in... |

23 |
More on k-sets of finite sets in the plane
- Welzl
- 1986
(Show Context)
Citation Context ...st tantalizing open problems in combinatorial geometry. Due to its importance in analyzing geometric algorithms [8, 9, 18], the problem has caught the attention of the computational geometers as well =-=[5, 6, 7, 13, 17, 26, 28]. Giv-=-en a set P of n points in ! d , and a positive integer ksn, a k-set is a subset P 0 ` P such that P 0 = P " H for a halfspace H and jP 0 j = k. A close to optimal solution for the problem remains... |

20 | On the number of halving planes
- Bárány, Füredi, et al.
- 1990
(Show Context)
Citation Context |

19 |
Counting triangle crossings and halving planes
- Dey, Edelsbrunner
- 1994
(Show Context)
Citation Context |

19 |
On the number of line separations of a finite set in the plane
- Edelsbrunner, Welzl
- 1985
(Show Context)
Citation Context ...lem of determining the optimum asymptotic bound on the number of k-sets is one of the most tantalizing open problems in combinatorial geometry. Due to its importance in analyzing geometric algorithms =-=[8, 9, 18]-=-, the problem has caught the attention of the computational geometers as well [5, 6, 7, 13, 17, 26, 28]. Given a set P of n points in ! d , and a positive integer ksn, a k-set is a subset P 0 ` P such... |

18 |
Cutting dense point sets in half
- Edelsbrunner, Valtr, et al.
- 1997
(Show Context)
Citation Context |

16 | Geometric lower bounds for parametric matroid optimization
- Eppstein
- 1995
(Show Context)
Citation Context ...union of n lines. A number of other results follow from this bound. An optimal \Theta(nk 1=3 ) bound on the complexity of n-element parametric matroids with rank k follows due to a result of Eppstein =-=[20]-=-. As an immediate consequence, we obtain an O(EV 1=3 ) bound on the number of parametric minimum spanning trees of a graph with E edges and V vertices whose edge weights vary linearly with time. A new... |

11 | Point selections and weak "-nets for convex hulls - Alon, B'ar'any, et al. - 1991 |

8 |
for the Parametric Spanning Tree Problem
- Gusfield, “Bound
- 1979
(Show Context)
Citation Context ...cation of this result is the case of parametric minimum spanning trees of a graph with V vertices and E edges where the edge weights vary linearly with time. The previous O(EV 1=2 ) bound of Gusfield =-=[21]-=- is improved to O(EV 1=3 ) by our result. 4.2 Complexity of j consecutive levels Let L k ; L k\Gamma1 ; :::; L k\Gammaj+1 be j ? 0 consecutive levels in an arrangement of n lines. We are interested in... |

6 |
On counting triangulations in d dimensions
- Dey
- 1993
(Show Context)
Citation Context ... was first used by us to prove an O(n 8=3 ) bound on 3dimensionalsk-sets [13]. Crossings in geometric graphs have been successfully used for many problems in combinatorial geometry. See, for example, =-=[12, 15, 27]-=-. It is expected that our approach would open up new avenues to solve the d-dimensional k-set problem, which remains largely unsolved for d ? 3 till date. The only nontrivial bound known for d ? 3 is ... |

5 |
On the expected number of k-sets
- Bárány, Steiger
- 1994
(Show Context)
Citation Context |

3 |
Convex polygons made from few lines and convex decompositions of polyhedra
- Hershberger, Snoeyink
- 1992
(Show Context)
Citation Context ... are drawn from n lines. The complexity of these polygons is the total number of vertices 3 they have altogether. If they are interior-wise disjoint, an optimal \Theta(n 2=3 k 2=3 + n) bound is known =-=[10, 22]-=-. However, these analysis techniques fail if the polygons overlap in their interiors. Our proof technique can be used to establish an optimal \Theta(nk 1=3 +n 2=3 k 2=3 ) bound for this case. First, w... |

2 |
On the number of simplicial complexes in R d
- Dey, Shah
- 1997
(Show Context)
Citation Context ... was first used by us to prove an O(n 8=3 ) bound on 3dimensionalsk-sets [13]. Crossings in geometric graphs have been successfully used for many problems in combinatorial geometry. See, for example, =-=[12, 15, 27]-=-. It is expected that our approach would open up new avenues to solve the d-dimensional k-set problem, which remains largely unsolved for d ? 3 till date. The only nontrivial bound known for d ? 3 is ... |