## On the Number of Congruent Simplices in a Point Set (2002)

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Venue: | DISCRETE & COMPUTATIONAL GEOMETRY |

Citations: | 12 - 2 self |

### BibTeX

@ARTICLE{Agarwal02onthe,

author = {Pankaj K. Agarwal and Micha Sharir},

title = {On the Number of Congruent Simplices in a Point Set},

journal = {DISCRETE & COMPUTATIONAL GEOMETRY},

year = {2002},

volume = {28},

pages = {123--150}

}

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### Abstract

We derive improved bounds on the number of k-dimensional simplices spanned by a set of n points in R(sup d) that are congruent to a given k-simplex, for k < or = d - 1. Let f(sup d)(sub K)(n) be the maximum number of k-simplices spanned by a set of n points in R(sup d) that are congruent to a given k-simplex. We prove that f(sup 3)(sub 2) (n) = O(n sup 5/3) times 2 sup O(alpha (sup 2)(n)), f(sup 4)(sub 2)(n) = O(n (sup 2) + epsilon), f(sup 5)(sub 2) (n)) = theta (n (sup 7/3), and f(sup 4)(sub 3)(n) = O(n (sup 9/4) plus epsilon). We also derive a recurrence to bound f(sup d)(sub k) (n) for arbitrary values of k and d, and use it to derive the bound f(sup d)(sub k(n) = O(n(sup d/2)) for d < or = 7 and k < or = d - 2. Following Erdo's and Purdy, we conjecture that this bound holds for larger values of d as well, and for k < or = d - 2.

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Citation Context ...garwal et al. [4], we compute a (1=r)-cutting of of size O(r d log r) as follows. Lift to a collection H of b + c hyperplanes in R d+1 , using the well-known lifting transformation, e.g. given in [16], which maps a sphere x 2 1 + + x 2 d = 1 x 1 + + d x d +sto the hyperplane x d+1 = 1 x 1 + + d x d +. The points of R d are lifted to the standard paraboloid : x d+1 = P d i... |

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Citation Context ...te the sphere (p) (resp. (q)). Set B = f (p) j p 2 Bg, C = f (q) j q 2 Cg, and = B [ C . A subdivision of R d into constant-description-complexity cells, in the sense defined in [25], is called a (1=r)-cutting of if each cell in is crossed by at most b=r (resp. c=r) spheres of B (resp. C ). A similar cutting is used in the algorithm sketched at the end of the previous secti... |

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Citation Context ...ce between any two points p 2 C 1 and q 2 C 2 is p 2, thereby obtaining a set P of n points with n 2 ) pairs of points at distance p 2. The known upper bounds for d = 2; 3 are f (2) 1 (n) = O(n 4=3 ) =-=[15, 26, -=-27] and f (3) 1 (n) = O(n 3=2s(n)) [15], wheres(n) = 2 ( 2 (n)) is a slowly growing function of n, defined in terms of the inverse Ackermann's function (n). However, neither of these bounds is known t... |

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Citation Context ...s of n points in R d and over all k-simplices in R d . We wish to obtain sharp bounds for f (d) k (n). The case k = 1 is the well-studied problem of repeated distances, originally considered by Erdos =-=[-=-17] in 1946: How many pairs of points of P lie at a prescribed distance from each other. This special case is interesting only for d = 2; 3 because f (d) 1 (n) = (n 2 ) for d 4. Indeed, Work on this... |

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Citation Context ...r (resp. c=r) spheres of B (resp. C ). A similar cutting is used in the algorithm sketched at the end of the previous section. By following the approach originally proposed by Chazelle and Friedman [1=-=-=-4] and refined by Agarwal et al. [4], we compute a (1=r)-cutting of of size O(r d log r) as follows. Lift to a collection H of b + c hyperplanes in R d+1 , using the well-known lifting transformatio... |

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Citation Context ...le . For any such pair�, we report the triangle� � �� � , as it is congruent to . Since�is a set of congruent circles, all lying on the sphere�, we can compute, by adapting the algorithm described in =-=[13, 21]-=- for computing incidences between points and lines, all incidences between and�in time time. Following the above analysis, we can conclude that the total running time of the algorithm is ���, for any�... |

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Citation Context ...y such pair (v; w), we report the triangle 4uvw, as it is congruent to . Since C u is a set of congruent circles, all lying on the sphere u , we can compute, by adapting the algorithm described in [1=-=3, 21]-=- for computing incidences between points and lines, all incidences between P u and C u in time O(m 2=3 u c 2=3 u log n+(m u + c u ) log n) time. Following the above analysis, we can conclude that the ... |

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Citation Context ...s a constant, we can compute the (1=r)-cutting described above by a randomized algorithm in O(b+ c) expected time. In fact, it can be computed by a detreministic algorithm in O(b + c) worst-case time [12]. For each cell 2 , A ; ^ B ; ^ C can be computed in an additional O(a + b + c) time. Following the proof of Lemma 3.1, the setss(A ; ^ B ; C; ) ands(A ; B; ^ C ; ) can be computed ... |

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Citation Context ...ONS 7 in that contain u. Since an arrangement of r spheres can be decomposed into O(r 3s(r)) cells of constant description complexity [15], one can use the divide-and-conquer algorithm described in [=-=13], to-=- compute the incidences between P and , and thus the sets P u , for all u 2 P , in O(n 3=2+" ) time. The sets P 0 u can be computed in exactly the same way. (ii) Put C u = fsuv j v 2 P 0 u g. For... |

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Citation Context ...acing the points of P on k + 1 mutually orthogonal unit-radius circles centered at the origin. Erdos and Purdy [19] proved that f (3) 2 (n) = O(n 19=9 ). The bound was later improved by Akutsu et al. =-=[5] to O(n 9=5 ) a-=-nd then by Brass [10] to O(n 7=4 ). Akutsu et al. [5] also proved that f (4) 2 (n) = O(n 65=23+" ) and f (4) 3 (n) = O(n 66=23+" ), for any " > 0. 1 By generalizing Lenz' construction, ... |

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Citation Context ...C ). A similar cutting is used in the algorithm sketched at the end of the previous section. By following the approach originally proposed by Chazelle and Friedman [14] and refined by Agarwal et al. [=-=-=-4], we compute a (1=r)-cutting of of size O(r d log r) as follows. Lift to a collection H of b + c hyperplanes in R d+1 , using the well-known lifting transformation, e.g. given in [16], which maps ... |

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Citation Context ...ce between any two points p 2 C 1 and q 2 C 2 is p 2, thereby obtaining a set P of n points with n 2 ) pairs of points at distance p 2. The known upper bounds for d = 2; 3 are f (2) 1 (n) = O(n 4=3 ) =-=[15, 26, -=-27] and f (3) 1 (n) = O(n 3=2s(n)) [15], wheres(n) = 2 ( 2 (n)) is a slowly growing function of n, defined in terms of the inverse Ackermann's function (n). However, neither of these bounds is known t... |

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Citation Context ...er bound can be proved by generalizing the construction for the case k = 1, namely, by placing the points of P on k + 1 mutually orthogonal unit-radius circles centered at the origin. Erdos and Purdy =-=[19] prov-=-ed that f (3) 2 (n) = O(n 19=9 ). The bound was later improved by Akutsu et al. [5] to O(n 9=5 ) and then by Brass [10] to O(n 7=4 ). Akutsu et al. [5] also proved that f (4) 2 (n) = O(n 65=23+" ... |

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Citation Context ...ll of the arrangement into simplices, using, e.g., bottom-vertex triangulation [14]. Let T be the set of simplices in the decomposition that intersect . The generalized zone theorem of Aronov et al. [=-=-=-7] implies that the number of simplices in T is O(r d log r). Let H4 H be the set of hyperplanes that cross a simplex 4 in T . Next, we construct a set 0 of pairwise-disjoint, constant-size cells, w... |

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Citation Context ... issn 4=3 ). This is obtained by placing one point at the origin and n 1 additional points on the unit sphere, so that there are n 4=3 ) pairs of those n 1 points at distance p 2 from each other (see =-=[1-=-8] for such a construction). The bound on f (4) 2 (n) is almost tight because as mentioned above, f (4) 2 (n) =sn 2 ). We conjecture that f (d) k (n) = (minfn k+1 ; n d=2 ) for even values of d 4 and... |

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Citation Context ... is O(n 3 (n)=j 11=2 + n 2 =j 3 + n=j), where (n) = (log n) O( 2 (n)) . Proof: The number of incidences between these t j circles and the points of P is at least jt j . A result by Aronov et al. [6] implies that the maximum number of incidences between m circles and n points is O(n 6=11 m 9=11 (n) +m 2=3 n 2=3 +n+m), where (n) = (log n) O( 2 (n)) . We thus have jt j = O(n 6=11 t 9=11 j (n)... |

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Citation Context ...an O(mn d )-time algorithm to determine whether E contains a congruent copy of P . For d = 3, Brass recently developed an O(mn 7=4s(n) log n)-time algorithm, which improves an earlier result by Boxer =-=[9]-=-. Our improved bounds can be applied to derive more efficient algorithms for the corresponding variants of this problem (see, e.g., a note to that effect at the end of Section 2). 2 Congruent Triangle... |

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6 |
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Citation Context ... [10] to O(n 7=4 ). Akutsu et al. [5] also proved that f (4) 2 (n) = O(n 65=23+" ) and f (4) 3 (n) = O(n 66=23+" ), for any " > 0. 1 By generalizing Lenz' construction, Abrego and Fern'=-=andez-Merchant [2-=-] proved that f (4) 2 (n) =sn 2 ) and f (5) 2 (n) =sn 7=3 ). Erdos and Purdy [20] conjectured that f (d) k (n) = O(n d=2 ) for even values of d 4. There has also been work on bounding the number of s... |

4 |
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Citation Context ...ally orthogonal unit-radius circles centered at the origin. Erdos and Purdy [19] proved that f (3) 2 (n) = O(n 19=9 ). The bound was later improved by Akutsu et al. [5] to O(n 9=5 ) and then by Brass =-=[10] to O(n 7=4 ). -=-Akutsu et al. [5] also proved that f (4) 2 (n) = O(n 65=23+" ) and f (4) 3 (n) = O(n 66=23+" ), for any " > 0. 1 By generalizing Lenz' construction, Abrego and Fern'andez-Merchant [2] p... |

3 | Zur Zerlegung von Punktmengen in solche kleineren - Lenz - 1955 |

2 |
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Citation Context ...Purdy [20] conjectured that f (d) k (n) = O(n d=2 ) for even values of d 4. There has also been work on bounding the number of simplices spanned by a point set that are similar to a given a simplex [=-=1, 2, 3]. 1 We follow the co-=-nvention that an upper bound that involves the parameter " holds for any " > 0 and the constant of proportionality depends on ", and generally tends to infinity as " tends to 0. CO... |

1 |
Point set pattern matching in d-dimensons, Algorithmica 13
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Citation Context ...hether all the other points of P map to points of E under that motion. The efficiency of such an algorithm depends on the number of congruent copies of in E. Using this approach, de Rezende and Lee [=-=24]-=- developed an O(mn d )-time algorithm to determine whether E contains a congruent copy of P . For d = 3, Brass recently developed an O(mn 7=4s(n) log n)-time algorithm, which improves an earlier resul... |

1 | Testing the congruence of�-dimensional point sets - Brass, Knauer |

1 | On a set of distances ofÒpoints - Erdős - 1946 |

1 |
Point set pattern matching in�-dimensons, Algorithmica 13
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(Show Context)
Citation Context ...check whether all the other points of map to points of�under that motion. The efficiency of such an algorithm depends on the number of congruent copies of in�. Using this approach, de Rezende and Lee =-=[24]-=- developed an �-time algorithm to determine whether�contains a congruent copy of . For ��, Brass recently developed an �-time algorithm, which improves an earlier result by Boxer [9]. Our improved bou... |