## On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences (2003)

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Venue: | In Proc. 44th IEEE Sympos. Found. Comput. Sci |

Citations: | 9 - 2 self |

### BibTeX

@INPROCEEDINGS{Chan03onlevels,

author = {Timothy M. Chan},

title = {On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences},

booktitle = {In Proc. 44th IEEE Sympos. Found. Comput. Sci},

year = {2003},

pages = {544--550}

}

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### Abstract

We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the k-level has subquadratic (O(n 2s )) complexity. This answers one of the main open problems from the author's previous paper (FOCS'00), which provided a weaker bound for a restricted class of curves (graphs of degree-s polynomials) only. When combined with existing tools (cutting curves, sampling, etc.), the new idea generates a slew of improved k-level results for most of the curve families studied earlier, including a near-O(n ) bound for parabolas.

### Citations

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Citation Context ...e analysis of geometric algorithms for ham-sandwich cuts [20], range searching [11], geometric optimization with violations [10], and kinetic/parametric minimum spanning trees [3]. See the many books =-=[14, 22, 23, 27-=-] and surveys [5, 7, 18] for more background information. In the discussion below, we focus on the most important case, when k = (n), because as Agarwal et al. [1] showed, an O(f(n)) upper bound for t... |

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Citation Context ... 1) that the maximum size of a depth-O(i) collection is O(i 2 (s) (n=i)). For the sake of completeness, we quickly provide the proof, which employs random sampling in the style of Clarkson and Shor [12]. Pick a random subset R of the given curve segments with jRj = n=i. For each lens 2 (s) i defined by curve segmentss1 ands2 , put in a subcollection (R) iffs1 ands2 are in R but all other curv... |

161 |
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Citation Context ...e analysis of geometric algorithms for ham-sandwich cuts [20], range searching [11], geometric optimization with violations [10], and kinetic/parametric minimum spanning trees [3]. See the many books =-=[14, 22, 23, 27-=-] and surveys [5, 7, 18] for more background information. In the discussion below, we focus on the most important case, when k = (n), because as Agarwal et al. [1] showed, an O(f(n)) upper bound for t... |

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Citation Context ...2 = log n) result from their (remarkably involved) FOCS'89 paper [24]. No significant progress was made for yet another decade until Dey's O(n 4=3 ) result from his (remarkably short) FOCS'97 paper [=-=13]-=-. The lower bound meanwhile has been increased to n2sp logn) by Toth in 2000 [32]; this lower bound was known some time ago by Klawe, Paterson, and Pippenger [14] for the weaker case of pseudo-lines (... |

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Citation Context ...t, where each element moves according to a nonlinear function in time.) This generalized problem is equally natural and fundamental, and has been studied intensively by several authors over the years =-=[4, 9, 26, 31]-=-. For example, our O(n 3=2 ) proof works immediately for pseudo-lines (x-monotone curves going from x = 1 to x = 1, where each pair intersects at most once) and pseudo-segments (x-monotone curve segme... |

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Citation Context ...r parabolas. 1. Introduction The k-level of lines. We begin by re-examining an old result on a famous open problem in two-dimensional combinatorial geometry. The problem, first investigated by Lovasz =-=[21]-=- and Erdos et al. [16] in the early 1970s, is the following: for a set P of n points in the plane, how many different subsets of size k (called k-sets) can be formed by intersecting P with a halfplane... |

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Citation Context ... i = O n 3=2 + n 11=6 j 5=6 j 20=27 i 1=3 log 26=27 j : Setting j = bn 2=5 c yields t i = O(n 97=54 i 1=3 log 26=27 n) for i = O(n 2=5 ). 2 6. Additional Remarks Toth's n2sp log n) lower bound [32]=-= rema-=-ins the current record, even for s-intersecting curves. At this point, it is conceivable that Erdos et al.'s o(n 1+" ) conjecture [16] might hold for curves. We do not know whether the main idea ... |

42 | On levels in arrangements of lines, segments, planes, and triangles
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Citation Context ...[3]. See the many books [14, 22, 23, 27] and surveys [5, 7, 18] for more background information. In the discussion below, we focus on the most important case, when k = (n), because as Agarwal et al. [=-=1] showed, a-=-n O(f(n)) upper bound for this case automatically implies a "k-sensitive" O((n=k)f(k)) bound, for any well-behaved function f . The initial papers by Lovasz [21] and Erdos et al. [16] establ... |

37 |
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Citation Context ...1 + O(i); where t n = O(n 2 ). It is straightforward to verify that t i = O(n 3=2 i 1=2 ). 2 Remark: Like previous proofs, a modification can directly show that the first k levels have O(nk) vertices =-=[6, 17]-=- and the k-level has O(n p k) vertices. For line segments, we can obtain another, albeit suboptimal, proof that the 1-level (the lower envelope [27]) has O(n log n) vertices; contrast this with Tagans... |

37 |
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Citation Context ...uction The k-level of lines. We begin by re-examining an old result on a famous open problem in two-dimensional combinatorial geometry. The problem, first investigated by Lovasz [21] and Erdos et al. =-=[16]-=- in the early 1970s, is the following: for a set P of n points in the plane, how many different subsets of size k (called k-sets) can be formed by intersecting P with a halfplane, asymptotically in th... |

36 |
Algorithms for ham-sandwich cuts
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Citation Context ...nt views of the problem help explain its central place in combinatorial and computational geometry. In particular, the problem is related to the analysis of geometric algorithms for ham-sandwich cuts =-=[20]-=-, range searching [11], geometric optimization with violations [10], and kinetic/parametric minimum spanning trees [3]. See the many books [14, 22, 23, 27] and surveys [5, 7, 18] for more background i... |

34 |
Halfspace range search: An algorithmic application of k-sets
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- 1986
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Citation Context ...m help explain its central place in combinatorial and computational geometry. In particular, the problem is related to the analysis of geometric algorithms for ham-sandwich cuts [20], range searching =-=[11]-=-, geometric optimization with violations [10], and kinetic/parametric minimum spanning trees [3]. See the many books [14, 22, 23, 27] and surveys [5, 7, 18] for more background information. In the dis... |

29 | Parametric and kinetic minimum spanning trees
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- 1998
(Show Context)
Citation Context ...problem is related to the analysis of geometric algorithms for ham-sandwich cuts [20], range searching [11], geometric optimization with violations [10], and kinetic/parametric minimum spanning trees =-=[3-=-]. See the many books [14, 22, 23, 27] and surveys [5, 7, 18] for more background information. In the discussion below, we focus on the most important case, when k = (n), because as Agarwal et al. [1]... |

28 | 2D arrangements
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(Show Context)
Citation Context ...ithms for ham-sandwich cuts [20], range searching [11], geometric optimization with violations [10], and kinetic/parametric minimum spanning trees [3]. See the many books [14, 22, 23, 27] and surveys =-=[5, 7, 18-=-] for more background information. In the discussion below, we focus on the most important case, when k = (n), because as Agarwal et al. [1] showed, an O(f(n)) upper bound for this case automatically ... |

25 |
An upper bound on the number of planar k-sets
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- 1992
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Citation Context ...1+" ) for any fixed " > 0. No progress on the upper bound was made for almost two decades until Pach, Steiger, and Szemeredi's O(n 3=2 = log n) result from their (remarkably involved) FOCS'=-=89 paper [24]-=-. No significant progress was made for yet another decade until Dey's O(n 4=3 ) result from his (remarkably short) FOCS'97 paper [13]. The lower bound meanwhile has been increased to n2sp logn) by Tot... |

25 | A new technique for analyzing substructures in arrangements of piecewise linear surfaces. Discrete Comput. Geom
- Tagansky
- 1996
(Show Context)
Citation Context ...p k) vertices. For line segments, we can obtain another, albeit suboptimal, proof that the 1-level (the lower envelope [27]) has O(n log n) vertices; contrast this with Tagansky's probabilistic proof =-=[29]-=-. 3. The Generalization to Curves The proof extends easily to pseudo-lines and pseudosegments. In this section, we consider how the bound is affected for general curve segments. The generalization of ... |

24 | G.: An improved bound for k-sets in three dimensions - Sharir, Smorodinsky, et al. - 2001 |

23 |
More on k-sets of finite sets in the plane
- Welzl
- 1986
(Show Context)
Citation Context ...bound follows by solving a difference equation. In contrast, previous proofs focused on a single k-level, and relationships between different levels were explored "after the fact" (for examp=-=le, as in [1, 8, 19, 33]-=-). The k-level of curves. Unfortunately, we see no obvious ways to obtain an o(n 3=2 ) bound with the new idea, let alone an improvement to Dey's result. However, as it turns out, the idea adapts beau... |

22 | Lenses in arrangements of pseudo-circles and their applications, manuscript, 2001. [8
- Sharir, Smorodinsky
- 2000
(Show Context)
Citation Context ...t, where each element moves according to a nonlinear function in time.) This generalized problem is equally natural and fundamental, and has been studied intensively by several authors over the years =-=[4, 9, 26, 31]-=-. For example, our O(n 3=2 ) proof works immediately for pseudo-lines (x-monotone curves going from x = 1 to x = 1, where each pair intersects at most once) and pseudo-segments (x-monotone curve segme... |

20 | On levels in arrangements of curves
- Chan
- 2000
(Show Context)
Citation Context ...int, where each point moves according to a nonlinear function in time.) This generalized problem is equally natural and fundamental, and has been studied intensively by several authors over the years =-=[4, 10, 29, 33]-=-. For example, our O(n 3/2 ) proof works immediately for pseudo-lines (x-monotone curves going from x = −∞ to x = ∞, where each pair intersects at most once) and pseudo-segments (x-monotone curve segm... |

18 |
A characterization of planar graphs by pseudo-line arrangements, manuscript
- Tamaki, Tokuyama
- 1997
(Show Context)
Citation Context ...n with the extendibility result from our previous paper [9]. Next, apply Lemma 5.1 to these segments, with Dey's level bound i (j) = O(j 4=3 i 2=3 ) [13], which holds for extendible pseudo-segments [30]: t i = O n 4=3 log n + n 3=2 j 1=2 j 1=3 i 2=3 : Setting j = bn 1=3 = log 2 nc yields t i = O(n 13=9 i 2=3 log 1=3 n) for i = O(n 1=3 = log 2 n). 2 Remark: Curiously, Theorem 5.2 is the only r... |

16 | Geometric lower bounds for parametric matroid optimization
- Eppstein
- 1995
(Show Context)
Citation Context ...s designed for this more general result. Many natural problems in combinatorial plane geometry have complexity (n 4=3 ), and indeed Dey's upper bound is tight for this multiple concave chain problem [=-=15]-=-. It would thus appear that the limit has been reached, unless somehow a substantially different approach for the k-set/klevel problem is discovered. To renew hope for Erdos et al.'s original conjectu... |

16 |
On the number of k-subsets of a set of n points in the plane
- Goodman, Pollack
- 1984
(Show Context)
Citation Context ...1 + O(i); where t n = O(n 2 ). It is straightforward to verify that t i = O(n 3=2 i 1=2 ). 2 Remark: Like previous proofs, a modification can directly show that the first k levels have O(nk) vertices =-=[6, 17]-=- and the k-level has O(n p k) vertices. For line segments, we can obtain another, albeit suboptimal, proof that the 1-level (the lower envelope [27]) has O(n log n) vertices; contrast this with Tagans... |

12 | Remarks on k-level algorithms in the plane
- Chan
(Show Context)
Citation Context ...bound follows by solving a difference equation. In contrast, previous proofs focused on a single k-level, and relationships between different levels were explored "after the fact" (for examp=-=le, as in [1, 8, 19, 33]-=-). The k-level of curves. Unfortunately, we see no obvious ways to obtain an o(n 3=2 ) bound with the new idea, let alone an improvement to Dey's result. However, as it turns out, the idea adapts beau... |

11 |
How to cut pseudo-parabolas into segments
- Tamaki, Tokuyama
- 1995
(Show Context)
Citation Context ...t, where each element moves according to a nonlinear function in time.) This generalized problem is equally natural and fundamental, and has been studied intensively by several authors over the years =-=[4, 9, 26, 31]-=-. For example, our O(n 3=2 ) proof works immediately for pseudo-lines (x-monotone curves going from x = 1 to x = 1, where each pair intersects at most once) and pseudo-segments (x-monotone curve segme... |

11 |
Intersection reverse sequences and geometric applications
- Marcus, Tardos
(Show Context)
Citation Context ...eated differentiation leads to a 1- or 2-intersecting family). For degree-s poly2− 1 2− 1 nomials, the previous bound was much worse— � O(n 9·2s−3 ) [10] (or � O(n 6·2s−3 ) using a recent improvement =-=[24]-=-). The notation � O hides polylogarithmic factors in this paper. As Agarwal et al. [1] showed also for curves and curve segments, an O(f(n)) upper bound implies a k-sensitive O((n/k)f(k)β(n/k)) bound,... |

8 | k-sets and j-facets: a tour of discrete geometry
- Andrzejak, Welzl
- 1997
(Show Context)
Citation Context ...ithms for ham-sandwich cuts [20], range searching [11], geometric optimization with violations [10], and kinetic/parametric minimum spanning trees [3]. See the many books [14, 22, 23, 27] and surveys =-=[5, 7, 18-=-] for more background information. In the discussion below, we focus on the most important case, when k = (n), because as Agarwal et al. [1] showed, an O(f(n)) upper bound for this case automatically ... |

8 | On k-Sets and their Generalizations
- Andrzejak
- 2000
(Show Context)
Citation Context ...nd. Comput. Sci. [11]. This work was supported in part by an NSERC Research Grant. 1sviolations [9], and kinetic/parametric minimum spanning trees [3]. See the many books [16, 25, 26, 30] and surveys =-=[5, 7, 20]-=- for more background information. In the discussion below, we focus on bounds that are functions of n only, because as Agarwal et al. [1] showed, an O(f(n)) upper bound for this case automatically imp... |

7 | Topological graphs with no self-intersecting cycle of length 4
- Pinchasi, Radoičić
- 2003
(Show Context)
Citation Context ...mbined with Lemma 4.1, the main inequality becomes t i 2it i + O(n 5=3 i 1=3 ); which solves to t i = O(n 5=3 i 1=3 ). (b) Agarwal et al. [4] observed that a recent result by Pinchasi and Radoicic [25] implies the improved bound (1) (n) = O(n 8=5 ) for pseudo-parabolas. The main inequality now becomes t i 2it i + O(n 8=5 i 2=5 ); which solves to t i = O(n 8=5 i 2=5 ). (c) Agarwal et al. [4] al... |

6 |
More on k-sets of finite sets in the plane, Discrete Comput
- Welzl
- 1986
(Show Context)
Citation Context ...= O(n 3/2 i 1/2 ). The size of the k-level is upper-bounded by t2, so the theorem follows. ✷ Remarks. • The O(n 3/2 i 3/2 ) bound on the complexity of O(i) consecutive levels was first shown by Welzl =-=[35]-=-. Dey’s proof [15] improved this to O(n 4/3 i 2/3 ). • A few previous papers (e.g., [1, 8, 21]) have also examined relationships among sizes of different levels, but the above inequality has not been ... |

4 |
Parametric Polymatroid Optimization and Its Geometric Applications
- Katoh, Tamaki, et al.
- 1999
(Show Context)
Citation Context ...bound follows by solving a difference equation. In contrast, previous proofs focused on a single k-level, and relationships between different levels were explored "after the fact" (for examp=-=le, as in [1, 8, 19, 33]-=-). The k-level of curves. Unfortunately, we see no obvious ways to obtain an o(n 3=2 ) bound with the new idea, let alone an improvement to Dey's result. However, as it turns out, the idea adapts beau... |

3 | On the complexity of many faces in arrangements of circles - Agarwal, Aronov, et al. - 2003 |

3 | On levels in arrangements of surfaces in three dimensions
- Chan
- 2005
(Show Context)
Citation Context ... levels, could help in resolving the original problem for lines in the plane. (For example, is it possible to reduce the coefficient 2 in Lemma 2.1 by increasing the overhead term?) An upcoming paper =-=[12]-=- demonstrates that the idea can be used for arrangements of surfaces in three dimensions, albeit with considerably more effort. Acknowledgements I would like to thank a referee for comments that impro... |

1 | On the complexity of many faces in arrangements of circles. To appear in Discrete and Computational Geometry: The Goodman-Pollack Festschrift - Agarwal, Aronov, et al. - 2001 |

1 |
On levels in arrangements of curves. Discrete Comput
- Chan
- 2000
(Show Context)
Citation Context |