## Cutting Circles into Pseudo-segments and Improved Bounds for Incidences (2000)

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Venue: | Geom |

Citations: | 22 - 11 self |

### BibTeX

@ARTICLE{Aronov00cuttingcircles,

author = {Boris Aronov and Micha Sharir},

title = {Cutting Circles into Pseudo-segments and Improved Bounds for Incidences},

journal = {Geom},

year = {2000},

volume = {28},

pages = {475--490}

}

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### Abstract

We show that n arbitrary circles in the plane can be cut into O(n 3/2+# ) arcs, for any # > 0, such that any pair of arcs intersect at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant maximum degree. 1 Introduction Let P be a finite set of points in the plane and C a finite set of circles. Let I = I(P, C) denote the number of incidences between the points and the circles. Let I(m, n) denote the maximum value of I(P, C), taken over all sets P of m points and sets C of n circles, and let I # (m, n, X) denote the maximum value of I(P, C), taken over all sets P of m points and sets C of n circles with at most X intersecting pairs. In this paper we derive improved upper bounds for I(m, n) and I # (m, n, X). The previous best upper bounds were I(m, n) = O(m 3/5 n 4/5 ...

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Citation Context ...y depend on #. The bound on I(m, n) is worst-case tight when m is larger than roughly n 5/4 . This follows from the construction of #(m 2/3 n 2/3 ) incidences between m points and n lines (see, e.g., =-=[10]-=-) which, after applying an inversion to the plane, becomes a configuration with #(m 2/3 n 2/3 ) incidences between m points and n circles. For sets C of circles with the additional property that every... |

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Citation Context ...most k, for an appropriate threshold parameter k whose value will be fixed momentarily. See Figure 2 for an illustration. By a straightforward application of the Clarkson-Shor probabilistic technique =-=[7]-=-, the number L k of bichromatic lenses at level at most k is O(k 2 ) times the number of empty bichromatic lenses in a random sample of n/k circles of A # # B # . Using Lemma 2.1, L k = O(k 2 (N/k)) =... |

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Citation Context ...d sets C of n circles with at most X intersecting pairs. In this paper we derive improved upper bounds for I(m, n) and I # (m, n, X). The previous best upper bounds were I(m, n) = O(m 3/5 n 4/5 +m+n) =-=[8, 15]-=-, and I # (m, n, X) = O(m 3/5 X 2/5 + m+ n) [4]. The bounds that we obtain are: I(m, n) = # O(m 2/3 n 2/3 +m) for m # n (5-3#)/(4-9#) O(m (6+3#)/11 n (9-#)/11 + n) for m # n (5-3#)/(4-9#) and I # (m, ... |

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Citation Context ...2 s+1 ( 1- 1 3 s-1 ) 41 3 s-2 - 2 s+1 + n # # m # n 2-1/(23 s-2 ) O(m 2/3 n 2/3 +m) m # n 2-1/(23 s-2 ) . (7) 5 Conclusion The main observation in this paper is that two recent techniques, of Szekely =-=[19]-=- and of Tamaki and Tokuyama [20], can be combined in a straightforward manner to yield improved incidence bounds for points and circles and for other families of curves. The improvement of Tamaki and ... |

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Citation Context ...een two curves is a point at which their relative interiors intersect transversally. An edge-crossing in (the drawing of) the graph is a pair of crossing edges. Lemma 2.2 (Leighton [12]; Ajtai et al. =-=[3]-=-; see also [14]). Any plane drawing of a simple graph G with e edges and n vertices must have## e 3 /n 2 ) edge-crossings, provided that e # 4n. Equivalently, if G can be drawn in the plane with X edg... |

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Citation Context ...ot contain) more than O(r 1-1/d ) of these simplices. In addition, the following refinement of Theorem 2.3, due to Agarwal and Matousek, will be useful in our analysis: Theorem 2.4 (Agarwal--Matousek =-=[1]-=-). Let A be a set of n points in R d that lie on an algebraic (d-1)-dimensional surface of constant degree, and let 1 # r # n be a given parameter. Then A can be partitioned into q # 2r subsets, A 1 ,... |

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Citation Context ...d sets C of n circles with at most X intersecting pairs. In this paper we derive improved upper bounds for I(m, n) and I # (m, n, X). The previous best upper bounds were I(m, n) = O(m 3/5 n 4/5 +m+n) =-=[8, 15]-=-, and I # (m, n, X) = O(m 3/5 X 2/5 + m+ n) [4]. The bounds that we obtain are: I(m, n) = # O(m 2/3 n 2/3 +m) for m # n (5-3#)/(4-9#) O(m (6+3#)/11 n (9-#)/11 + n) for m # n (5-3#)/(4-9#) and I # (m, ... |

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Citation Context ...d the curve that represents it in the drawing. 2.3 Simplicial partitioning of point sets in higher dimensions We also need the following well-known result of Matouˇsek: 3 a3 b3sTheorem 2.3 (Matouˇsek =-=[13]-=-). Let A be a set of n points in R d and 1 ≤ r ≤ n a given parameter. Then A can be partitioned into q ≤ 2r subsets, A1,... ,Aq, so that, for each i, |Ai| ≤ n/r, and Ai is contained in a (possibly low... |

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Citation Context ...mbine this new bound on the number of cuts with several other tools to derive the aforementioned improved bounds for I(m, n) (and I # (m, n, X)). These tools are reviewed in Section 2. Recently, Chan =-=[5]-=- has extended Tamaki and Tokuyama's result to the case of graphs of polynomials of any constant maximum degree. Using Chan's bound and extending, in a straightforward manner, the proof technique we us... |

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Citation Context ...es. A crossing between two curves is a point at which their relative interiors intersect transversally. An edge-crossing in (the drawing of) the graph is a pair of crossing edges. Lemma 2.2 (Leighton =-=[12]-=-; Ajtai et al. [3]; see also [14]). Any plane drawing of a simple graph G with e edges and n vertices must have## e 3 /n 2 ) edge-crossings, provided that e # 4n. Equivalently, if G can be drawn in th... |

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Citation Context ... edge and the curve that represents it in the drawing. 2.3 Simplicial partitioning of point sets in higher dimensions We also need the following well-known result of Matousek: 3 Theorem 2.3 (Matousek =-=[13]-=-). Let A be a set of n points in R d and 1 # r # n a given parameter. Then A can be partitioned into q # 2r subsets, A 1 , . . . , A q , so that, for each i, |A i | # n/r, and A i is contained in a (p... |

14 |
How to cut pseudo-parabolas into segments, Discrete Comput
- Tamaki, Tokuyama
- 1998
(Show Context)
Citation Context ... We show that n arbitrary circles in the plane can be cut into O(n 3/2+# ) arcs, for any # > 0, such that any pair of arcs intersect at most once. This improves a recent result of Tamaki and Tokuyama =-=[20]-=-. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant ... |

13 |
Distinct distances in the plane, Discrete Comput. Geom
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Citation Context ...ackle the problem of the number of distinct distances in a set of n points in the plane? The setup in this problem involves a collection of many circles that have relatively few centers (see [19] and =-=[18]-=- for details). 13 . Find applications of the new incidence bounds obtained in this paper. Two problems to which the new bounds might be applicable are the unit distance problem in three dimensions [8]... |

7 | The complexity of many cells in the overlay of many arrangements - Har-Peled - 1995 |

2 |
On the number of congruent simplices in higher dimensions
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Citation Context ... applicable are the unit distance problem in three dimensions [8] and the problem of bounding the maximum number of simplices spanned by a set of n points in R d and congruent to a given simplex (see =-=[2]-=- for work in progress on this problem). . Can the technique of this paper be adapted to yield similar improved bounds for the complexity of many faces in an arrangement of circles (see [8] for the cur... |

2 |
On the complexity of arrangements of circles in the plane, submitted to Discrete Comput. Geom
- Alon, Last, et al.
(Show Context)
Citation Context ...airs. In this paper we derive improved upper bounds for I(m, n) and I # (m, n, X). The previous best upper bounds were I(m, n) = O(m 3/5 n 4/5 +m+n) [8, 15], and I # (m, n, X) = O(m 3/5 X 2/5 + m+ n) =-=[4]-=-. The bounds that we obtain are: I(m, n) = # O(m 2/3 n 2/3 +m) for m # n (5-3#)/(4-9#) O(m (6+3#)/11 n (9-#)/11 + n) for m # n (5-3#)/(4-9#) and I # (m, n, X) = # O(m 2/3 X 1/3 +m) for m # X (1+6#)/(4... |

2 |
Gallai-Sylvester theorem for pairwise-intersecting unit circles
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(Show Context)
Citation Context ...es (contained in the interiors of the two defining circles) or lune-faces (contained in the interior of one defining circle and in the exterior of the other); see Figure 1. In recent papers, Pinchasi =-=[16]-=- and Alon et al. [4] have obtained various upper bounds for the number of these faces: It was shown that if every pair of circles in C intersect then the number of lens-faces and lune-faces is O(n). I... |