## On Levels in Arrangements of Curves (2002)

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Venue: | Proc. 41st IEEE |

Citations: | 21 - 3 self |

### BibTeX

@INPROCEEDINGS{Chan02onlevels,

author = {Timothy M. Chan},

title = {On Levels in Arrangements of Curves},

booktitle = {Proc. 41st IEEE},

year = {2002},

pages = {219--227}

}

### OpenURL

### Abstract

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 9 2 s 3 )) for curves that are graphs of polynomial functions of an arbitrary fixed degree s. Previously, nontrivial results were known only for the case s = 1 and s = 2. We also improve the earlier bound for pseudo-parabolas (curves that pairwise intersect at most twice) to O(nk k). The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudo-parabolas into pseudo-segments, as well as a new observation for cutting pseudo-segments into pieces that can be extended to pseudo-lines. We mention applications to parametric and kinetic minimum spanning trees.

### Citations

1828 |
Computational Geometry — An Introduction
- Preparata, Shamos
- 1988
(Show Context)
Citation Context ...ies that (s 1 ; s 3 ) 2 G(S), so hs 1 ; s 3 ; : : : ; s ` ; s 1 i would be a shorter cycle: contradiction. 2 Our key result regarding pseudo-segments can now be proved by a standard segment-tree idea =-=[43]-=-. Theorem 3.3 Any arrangement of n pseudo-segments can be cut into an arrangement of O(n log n) extendible pseudo-segments. Proof: Given n intervals, the standard tree construction gives us a collecti... |

702 |
Algorithms in Combinatorial Geometry
- Edelsbrunner
- 1987
(Show Context)
Citation Context ... 1 Figure 1: The 3-level in an arrangement ofsve 3-intersecting curves. (see below). Generally, arrangements of curves and surfaces have for a long time been a central topic in computational geometry =-=[6, 23, 24, 33, 45]-=-, and levels in arrangements have been used in the design of algorithms for range searching [4, 19], geometric optimization with k violations [28, 39], and partitioning of point sets [15, 38] (see the... |

420 | Davenport-Schinzel sequences and their geometric applications
- Agarwal, Sharir
- 1998
(Show Context)
Citation Context ... 1 Figure 1: The 3-level in an arrangement ofsve 3-intersecting curves. (see below). Generally, arrangements of curves and surfaces have for a long time been a central topic in computational geometry =-=[6, 23, 24, 33, 45]-=-, and levels in arrangements have been used in the design of algorithms for range searching [4, 19], geometric optimization with k violations [28, 39], and partitioning of point sets [15, 38] (see the... |

393 | Applications of random sampling in computational geometry
- Clarkson, Shor
- 1989
(Show Context)
Citation Context ...s of nonlinear surfaces. In contrast, tight worst-case bounds on the 1-level, 2level, . . . , k-level combined are easier to obtain. An O(nk) bound for lines was well-known [5, 24]. Clarkson and Shor =-=[1-=-5] gave a proof by random sampling that extends to higher dimensions. As shown by Sharir [39], this proof also yields an O(nk(n=k)) bound for pseudo-segments, and an O( s (n=k)k 2 ) bound for general ... |

286 |
Computational Geometry: An Introduction Through Randomized Algorithms
- Mulmuley
- 1994
(Show Context)
Citation Context ...been used in the design of algorithms for range searching [4, 19], geometric optimization with k violations [28, 39], and partitioning of point sets [15, 38] (see the surveys [6, 8, 33] and the books =-=[23, 40, 41, 45]-=- for more details). History. The k-level has a reputation of being one of the more dicult substructures of arrangements to analyze, even in the simplest case of lines (s = 1) in the plane, where in th... |

257 | Geometric range searching and its relatives
- Agarwal, Erickson
- 1998
(Show Context)
Citation Context ... curves and surfaces have for a long time been a central topic in computational geometry [6, 23, 24, 33, 45], and levels in arrangements have been used in the design of algorithms for range searching =-=[4, 19]-=-, geometric optimization with k violations [28, 39], and partitioning of point sets [15, 38] (see the surveys [6, 8, 33] and the books [23, 40, 41, 45] for more details). History. The k-level has a re... |

164 |
Modern Graph Theory, Graduate Texts
- Bollobás
- 1998
(Show Context)
Citation Context ... segments) as vertices, and pairs that form lenses in L as edges. As it turns out, for pseudo-parabolas, G(L) cannot contain K c;3 as a subgraph for some constant c. It then follows easily (e.g., see =-=[16]-=-) that the number of edges in G(L) is O(n 5=3 ). Agarwal et al.'s improvement requires some more involved techniques to bound the maximal number of nonoverlapping lenses. Here, we will follow Tamaki a... |

152 |
Applications of random sampling
- Clarkson, Shor
- 1989
(Show Context)
Citation Context ... of nonlinear surfaces. In contrast, tight worst-case bounds on the 1-level, 2-level, . . . , k-level combined are easier to obtain. An O(nk) bound for lines was well-known [7, 29]. Clarkson and Shor =-=[2-=-0] gave a proof by random sampling that extends to higher dimensions. As shown by Sharir [44], this proof also yields an O(nk(n=k)) bound for pseudo-segments, and an O( s (n=k)k 2 ) bound for general ... |

92 | Kinetic data structures — a state of the art report
- Guibas
- 1998
(Show Context)
Citation Context ...ong the simplest in the combinatorial analysis of so-called \kinetic data structures," an area that has received much recent attention, dealing with the maintenance of structures of objects in mo=-=tion [11, 31-=-]; in particular, there is a direct relevance to the study of kinetic and parametric minimum spanning trees [26, 32, 36] A preliminary version of this paper appeared in Proc. 41st IEEE Sympos. Found.... |

79 | Arrangements and their applications
- Agarwal, Sharir
- 2000
(Show Context)
Citation Context ... 1 Figure 1: The 3-level in an arrangement ofsve 3-intersecting curves. (see below). Generally, arrangements of curves and surfaces have for a long time been a central topic in computational geometry =-=[6, 23, 24, 33, 45]-=-, and levels in arrangements have been used in the design of algorithms for range searching [4, 19], geometric optimization with k violations [28, 39], and partitioning of point sets [15, 38] (see the... |

60 |
On k-sets in arrangements of curves and surfaces
- Sharir
- 1991
(Show Context)
Citation Context ... , k-level combined are easier to obtain. An O(nk) bound for lines was well-known [7, 29]. Clarkson and Shor [20] gave a proof by random sampling that extends to higher dimensions. As shown by Sharir =-=[4-=-4], this proof also yields an O(nk(n=k)) bound for pseudo-segments, and an O( s (n=k)k 2 ) bound for general s-intersecting curves. 2 Extendible pseudo-segments Agarwal et al. [1] observed that the cl... |

59 |
The colored Tverberg’s problem and complexes of injective functions
- ˘Zivaljević, Vrećica
- 1992
(Show Context)
Citation Context ...ived considerable attention in the linear case (arrangements of planes or 3 triangles); see [1, 9, 12, 22, 25, 35, 46]. A nontrivial combinatorial bound for hyperplanes in higher dimensions was known =-=[50]-=-. However, there has so far been no successful generalization to families of nonlinear surfaces. In contrast, tight worst-case bounds on the 1-level, 2-level, . . . , k-level combined are easier to ob... |

50 | Proximity problems on moving points
- Basch, Guibas, et al.
- 1997
(Show Context)
Citation Context ...r parabolas [5], the bound can be further reduced to O(n 8=3 (log n) O( 2 (n)) ). For algorithms on the parametric/kinetic minimum spanning tree problem in both its graph and geometric settings, see [=-=3, 13]-=-. Very recently, our Theorem 3.3 was used by Agarwal, Aronov, and Sharir [2] to bound the combinatorial complexity of multiple faces in arrangements of pseudo-segments and of circles. Our Theorem 6.2/... |

49 | On the number of halving lines - Lovasz - 1971 |

48 |
Some dynamic computational geometry problems
- Atallah
- 1985
(Show Context)
Citation Context ...ong the simplest in the combinatorial analysis of so-called \kinetic data structures," an area that has received much recent attention, dealing with the maintenance of structures of objects in mo=-=tion [11, 31-=-]; in particular, there is a direct relevance to the study of kinetic and parametric minimum spanning trees [26, 32, 36] A preliminary version of this paper appeared in Proc. 41st IEEE Sympos. Found.... |

46 |
On geometric optimization with few violated constraints
- Matousek
- 1995
(Show Context)
Citation Context ...entral topic in computational geometry [6, 23, 24, 33, 45], and levels in arrangements have been used in the design of algorithms for range searching [4, 19], geometric optimization with k violations =-=[28, 39]-=-, and partitioning of point sets [15, 38] (see the surveys [6, 8, 33] and the books [23, 40, 41, 45] for more details). History. The k-level has a reputation of being one of the more dicult substructu... |

46 | Point sets with many k-sets - Tóth |

45 |
Allowable sequences and order types in discrete and computational geometry
- Goodman, Pollack
- 1993
(Show Context)
Citation Context ...y [21] with a short elegant proof; Dey's result remains the current record. Like the classical proof, Dey's proof generalizes to any arrangement of 1-intersecting curves, commonly called pseudo-lines =-=[23, -=-30], as shown by Tamaki and Tokuyama [47]. Both proofs can also be adapted for an arrangement of line segments, the latter yielding an O(nk 1=3 (n=k)) bound, as shown by Agarwal et al. [1], where () d... |

42 | On levels in arrangements of lines, segments, planes, and triangles
- Agarwal, Aronov, et al.
- 1997
(Show Context)
Citation Context ...do-lines [23, 30], as shown by Tamaki and Tokuyama [47]. Both proofs can also be adapted for an arrangement of line segments, the latter yielding an O(nk 1=3 (n=k)) bound, as shown by Agarwal et al. [1], where () denotes the inverse Ackermann function. (Note that the worst-case complexity of the 1-level, i.e., lower envelope, of n line segments is (n(n)) [45].) Whether the same bound holds for a... |

38 |
Dissection graphs of planar point sets
- Erdős, Lovász, et al.
- 1973
(Show Context)
Citation Context ...der the name of the k-set problem (given an n-point set P in the plane, bound the number of subsets of size k that can be formed by intersecting P with a halfplane). In the early 1970s, Erd}os et al. =-=[2-=-7] and Lovasz [37] started the investigation by establishing a nontrivial O(n p k) upper bound and ansn log k) lower bound, but an improvement did not come until 1989, when Pach, Steiger, and Szemered... |

38 |
The number of small semispaces of a finite set of points in the plane
- Alon, Győri
- 1986
(Show Context)
Citation Context ...l generalization to families of nonlinear surfaces. In contrast, tight worst-case bounds on the 1-level, 2level, . . . , k-level combined are easier to obtain. An O(nk) bound for lines was well-known =-=[5, 2-=-4]. Clarkson and Shor [15] gave a proof by random sampling that extends to higher dimensions. As shown by Sharir [39], this proof also yields an O(nk(n=k)) bound for pseudo-segments, and an O( s (n=k)... |

37 |
Algorithms for ham-sandwich cuts
- Lo, Matousek, et al.
- 1994
(Show Context)
Citation Context ..., 18, 19, 28, 40], and levels in arrangements have been used in the design of algorithms for range searching [3, 14], geometric optimization with k violations [23, 34], and partitioning of point sets =-=[12, 33]-=- (see the surveys [4, 6, 28] and the books [18, 35, 36, 40] for more details). History. The k-level has a reputation of being one of the more difficult substructures of arrangements to analyze, even i... |

34 |
Halfspace range search: An algorithmic application of k-sets
- Chazelle, Preparata
- 1986
(Show Context)
Citation Context ... curves and surfaces have for a long time been a central topic in computational geometry [6, 23, 24, 33, 45], and levels in arrangements have been used in the design of algorithms for range searching =-=[4, 19]-=-, geometric optimization with k violations [28, 39], and partitioning of point sets [15, 38] (see the surveys [6, 8, 33] and the books [23, 40, 41, 45] for more details). History. The k-level has a re... |

31 |
Points and triangles in the plane and halving planes in space
- Aronov, Chazelle, et al.
- 1991
(Show Context)
Citation Context ...ed by many researchers; for recent work, see [17, 34]. The k-level problem in three dimensions has also received considerable attention in the linear case (arrangements of planes or 3 triangles); see =-=[1, 9, 12, 22, 25, 35, 46]-=-. A nontrivial combinatorial bound for hyperplanes in higher dimensions was known [50]. However, there has so far been no successful generalization to families of nonlinear surfaces. In contrast, tigh... |

31 | 2D Arrangements
- Wein, Fogel, et al.
- 2006
(Show Context)
Citation Context |

30 | Parametric and kinetic minimum spanning trees
- Agarwal, Eppstein, et al.
- 1998
(Show Context)
Citation Context ...r parabolas [5], the bound can be further reduced to O(n 8=3 (log n) O( 2 (n)) ). For algorithms on the parametric/kinetic minimum spanning tree problem in both its graph and geometric settings, see [=-=3, 13]-=-. Very recently, our Theorem 3.3 was used by Agarwal, Aronov, and Sharir [2] to bound the combinatorial complexity of multiple faces in arrangements of pseudo-segments and of circles. Our Theorem 6.2/... |

26 | Taking a walk in a planar arrangement
- HAR-PELED
- 1999
(Show Context)
Citation Context ...lexity of many faces in curve arrangements [2]. Other related work. The algorithmic problem of constructing the k-level of lines and curves has been examined by many researchers; for recent work, see =-=[17, 34]-=-. The k-level problem in three dimensions has also received considerable attention in the linear case (arrangements of planes or 3 triangles); see [1, 9, 12, 22, 25, 35, 46]. A nontrivial combinatoria... |

25 |
An upper bound on the number of planar k-sets
- Pach, Steiger, et al.
- 1992
(Show Context)
Citation Context ... and Lovasz [32] started the investigation by establishing a nontrivial O(n p k) upper bound and ansn log k) lower bound, but an improvement did not come until 1989, when Pach, Steiger, and Szemeredi =-=[37-=-] managed to reduce the upper bound by a small log k factor. In 1997, a breakthroughsO(nk 1=3 ) upper bound was obtained by Dey [16] with a short elegant proof; Dey's result remains the current recor... |

24 |
Kreveld, An optimal algorithm for the (< k)-levels, with applications to separation and transversal problems
- Everett, Robert, et al.
- 1993
(Show Context)
Citation Context ...entral topic in computational geometry [6, 23, 24, 33, 45], and levels in arrangements have been used in the design of algorithms for range searching [4, 19], geometric optimization with k violations =-=[28, 39]-=-, and partitioning of point sets [15, 38] (see the surveys [6, 8, 33] and the books [23, 40, 41, 45] for more details). History. The k-level has a reputation of being one of the more dicult substructu... |

24 | G.: An improved bound for k-sets in three dimensions
- Sharir, Smorodinsky, et al.
- 2001
(Show Context)
Citation Context ...ed by many researchers; for recent work, see [17, 34]. The k-level problem in three dimensions has also received considerable attention in the linear case (arrangements of planes or 3 triangles); see =-=[1, 9, 12, 22, 25, 35, 46]-=-. A nontrivial combinatorial bound for hyperplanes in higher dimensions was known [50]. However, there has so far been no successful generalization to families of nonlinear surfaces. In contrast, tigh... |

23 | Lenses in arrangements of pseudo-circles and their applications
- Agarwal, Nevo, et al.
- 2004
(Show Context)
Citation Context ...gh polynomials are the most natural instances, the above proof clearly works for any curve families whose (s 1)-th derivatives obey the pseudo-line property. 7 An improvement Recently, Agarwal et al. =-=[5-=-] have improved Tamaki and Tokuyama's theorem [48] in various special cases; in particular, they have proved that every family of n parabolas can be cut into O(n 3=2 (log n) O( 2 (n)) ) pseudo-segment... |

23 | Generalizing ham sandwich cuts to equitable subdivisions
- Bespamyatnikh, Kirkpatrick, et al.
(Show Context)
Citation Context ..., 23, 24, 33, 45], and levels in arrangements have been used in the design of algorithms for range searching [4, 19], geometric optimization with k violations [28, 39], and partitioning of point sets =-=[15, 38]-=- (see the surveys [6, 8, 33] and the books [23, 40, 41, 45] for more details). History. The k-level has a reputation of being one of the more dicult substructures of arrangements to analyze, even in t... |

22 | Cutting circles into pseudo-segments and improved bounds for incidences
- Aronov, Sharir
- 2002
(Show Context)
Citation Context ...g points in anysxed dimension [36]. Very recently, the techniques of this paper have also found applications in other fundamental problems of combinatorial geometry, concerning point-curve incidences =-=[10]-=- and the complexity of many faces in curve arrangements [2]. Other related work. The algorithmic problem of constructing the k-level of lines and curves has been examined by many researchers; for rece... |

22 | On the number of halving planes
- Barany, Furedi, et al.
- 1990
(Show Context)
Citation Context ...ed by many researchers; for recent work, see [17, 34]. The k-level problem in three dimensions has also received considerable attention in the linear case (arrangements of planes or 3 triangles); see =-=[1, 9, 12, 22, 25, 35, 46]-=-. A nontrivial combinatorial bound for hyperplanes in higher dimensions was known [50]. However, there has so far been no successful generalization to families of nonlinear surfaces. In contrast, tigh... |

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

19 |
A characterization of planar graphs by pseudo-line arrangements
- Tamaki, Tokuyama
(Show Context)
Citation Context ...esult remains the current record. Like the classical proof, Dey's proof generalizes to any arrangement of 1-intersecting curves, commonly called pseudo-lines [23, 30], as shown by Tamaki and Tokuyama =-=[-=-47]. Both proofs can also be adapted for an arrangement of line segments, the latter yielding an O(nk 1=3 (n=k)) bound, as shown by Agarwal et al. [1], where () denotes the inverse Ackermann function.... |

19 |
Counting triangle crossings and halving planes
- Dey, Edelsbrunner
- 1994
(Show Context)
Citation Context ...ned by many researchers; for recent work, see [13, 29]. The klevel problem in three dimensions has also received consid2 erable attention in the linear case (arrangements of planes or triangles); see =-=[1, 7, 9, 17, 20, 30, 41]-=-. A nontrivial combinatorial bound for hyperplanes in higher dimensions was known [45]. However, there has so far been no successful generalization to families of nonlinear surfaces. In contrast, tigh... |

16 | Improved bounds on planar k-sets and k-levels
- Dey
- 1998
(Show Context)
Citation Context ...vement did not come until 1989, when Pach, Steiger, and Szemeredi [42] managed to reduce the upper bound by a small log k factor. In 1997, a breakthrough O(nk 1=3 ) upper bound was obtained by Dey [2=-=1]-=- with a short elegant proof; Dey's result remains the current record. Like the classical proof, Dey's proof generalizes to any arrangement of 1-intersecting curves, commonly called pseudo-lines [23, 3... |

16 |
Arrangements of curves in the plane: Topology, combinatorics, and algorithms
- Edelsbrunner, Guibas, et al.
- 1992
(Show Context)
Citation Context |

16 | Improved bounds for intersecting triangles and halving planes
- Eppstein
- 1993
(Show Context)
Citation Context |

16 | Geometric lower bounds for parametric matroid optimization
- Eppstein
- 1995
(Show Context)
Citation Context ...ch recent attention, dealing with the maintenance of structures of objects in motion [11, 31]; in particular, there is a direct relevance to the study of kinetic and parametric minimum spanning trees =-=[26, 32, 36-=-] A preliminary version of this paper appeared in Proc. 41st IEEE Sympos. Found. Comput. Sci. [18]. y Supported by an NSERC Research Grant. 1 Figure 1: The 3-level in an arrangement ofsve 3-intersect... |

16 |
On the number of k-subsets of a set of n points in the plane
- Goodman, Pollack
- 1984
(Show Context)
Citation Context ... generalization to families of nonlinear surfaces. In contrast, tight worst-case bounds on the 1-level, 2-level, . . . , k-level combined are easier to obtain. An O(nk) bound for lines was well-known =-=[7, 2-=-9]. Clarkson and Shor [20] gave a proof by random sampling that extends to higher dimensions. As shown by Sharir [44], this proof also yields an O(nk(n=k)) bound for pseudo-segments, and an O( s (n=k)... |

12 | Remarks on k-level algorithms in the plane
- Chan
(Show Context)
Citation Context ...lexity of many faces in curve arrangements [2]. Other related work. The algorithmic problem of constructing the k-level of lines and curves has been examined by many researchers; for recent work, see =-=[17, 34]-=-. The k-level problem in three dimensions has also received considerable attention in the linear case (arrangements of planes or 3 triangles); see [1, 9, 12, 22, 25, 35, 46]. A nontrivial combinatoria... |

11 |
On minimum and maximum spanning trees of linearly moving points, Discrete Comput
- Katoh, Tokuyama, et al.
- 1995
(Show Context)
Citation Context ...ch recent attention, dealing with the maintenance of structures of objects in motion [11, 31]; in particular, there is a direct relevance to the study of kinetic and parametric minimum spanning trees =-=[26, 32, 36-=-] A preliminary version of this paper appeared in Proc. 41st IEEE Sympos. Found. Comput. Sci. [18]. y Supported by an NSERC Research Grant. 1 Figure 1: The 3-level in an arrangement ofsve 3-intersect... |

11 |
How to cut pseudo-parabolas into segments
- Tamaki, Tokuyama
- 1995
(Show Context)
Citation Context ...l families of curves appears even more challenging, as the preceding techniques simply fail. Thesrst nontrivial result for quadratic functions (s = 2) was obtained only in 1995 by Tamaki and Tokuyama =-=[48]-=-. They proved a theorem on how to cut an arrangement of pseudo-parabolas (2-intersecting curves) into an arrangement of pseudo-segments. When combined with the classical result for pseudo-lines, this ... |

9 |
Algorithms for ham-sandwichcuts. Discrete Comput. Geom
- Lo, Matousek, et al.
- 1994
(Show Context)
Citation Context ..., 23, 24, 33, 45], and levels in arrangements have been used in the design of algorithms for range searching [4, 19], geometric optimization with k violations [28, 39], and partitioning of point sets =-=[15, 38]-=- (see the surveys [6, 8, 33] and the books [23, 40, 41, 45] for more details). History. The k-level has a reputation of being one of the more dicult substructures of arrangements to analyze, even in t... |

9 |
An upper bound on the number of planar k-sets, Discrete Comput
- Pach, Steiger, et al.
- 1992
(Show Context)
Citation Context ...nd Lovasz [37] started the investigation by establishing a nontrivial O(n p k) upper bound and ansn log k) lower bound, but an improvement did not come until 1989, when Pach, Steiger, and Szemeredi [4=-=2-=-] managed to reduce the upper bound by a small log k factor. In 1997, a breakthrough O(nk 1=3 ) upper bound was obtained by Dey [21] with a short elegant proof; Dey's result remains the current recor... |

8 | k-sets and j-facets: A tour of discrete geometry
- ANDRZEJAK, WELZL
- 1997
(Show Context)
Citation Context ...els in arrangements have been used in the design of algorithms for range searching [4, 19], geometric optimization with k violations [28, 39], and partitioning of point sets [15, 38] (see the surveys =-=[6, 8, 33]-=- and the books [23, 40, 41, 45] for more details). History. The k-level has a reputation of being one of the more dicult substructures of arrangements to analyze, even in the simplest case of lines (s... |

7 |
Bounds for the parametric minimum spanning tree problem
- Gusfield
(Show Context)
Citation Context ...uch recent attention, dealing with the maintenance of structures of objects in motion [8, 26]; in particular, there is a direct relevance to the study of kinetic and parametric minimum spanning trees =-=[21, 27, 31]-=- (see below) . Generally, arrangements of curves and surfaces have for a long time been a central topic in computational geometry [4, 18, 19, 28, 40], and levels in arrangements have been used in the ... |

6 | Lovász’s lemma for the three-dimensional k-level of concave surfaces and its applications. Discrete Comput
- Katoh, Tokuyama
(Show Context)
Citation Context |