TY - JOUR

T1 - Combinatorial complexity bounds for arrangements of curves and spheres

AU - Clarkson, Kenneth L.

AU - Edelsbrunner, Herbert

AU - Guibas, Leonidas J.

AU - Sharir, Micha

AU - Welzl, Emo

PY - 1990/12

Y1 - 1990/12

N2 - We present upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike. For example, we prove that the maximum number of edges bounding m cells in an arrangement of n lines is Θ(m2/3n2/3 +n), and that it is O(m2/3n2/3β(n) +n) for n unit-circles, where β(n) (and later β(m, n)) is a function that depends on the inverse of Ackermann's function and grows extremely slowly. If we replace unit-circles by circles of arbitrary radii the upper bound goes up to O(m3/5n4/5β(n) +n). The same bounds (without the β(n)-terms) hold for the maximum sum of degrees of m vertices. In the case of vertex degrees in arrangements of lines and of unit-circles our bounds match previous results, but our proofs are considerably simpler than the previous ones. The maximum sum of degrees of m vertices in an arrangement of n spheres in three dimensions is O(m4/7n9/7β(m, n) +n2), in general, and O(m3/4n3/4β(m, n) +n) if no three spheres intersect in a common circle. The latter bound implies that the maximum number of unit-distances among m points in three dimensions is O(m3/2β(m)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given.

AB - We present upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike. For example, we prove that the maximum number of edges bounding m cells in an arrangement of n lines is Θ(m2/3n2/3 +n), and that it is O(m2/3n2/3β(n) +n) for n unit-circles, where β(n) (and later β(m, n)) is a function that depends on the inverse of Ackermann's function and grows extremely slowly. If we replace unit-circles by circles of arbitrary radii the upper bound goes up to O(m3/5n4/5β(n) +n). The same bounds (without the β(n)-terms) hold for the maximum sum of degrees of m vertices. In the case of vertex degrees in arrangements of lines and of unit-circles our bounds match previous results, but our proofs are considerably simpler than the previous ones. The maximum sum of degrees of m vertices in an arrangement of n spheres in three dimensions is O(m4/7n9/7β(m, n) +n2), in general, and O(m3/4n3/4β(m, n) +n) if no three spheres intersect in a common circle. The latter bound implies that the maximum number of unit-distances among m points in three dimensions is O(m3/2β(m)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given.

UR - http://www.scopus.com/inward/record.url?scp=0002443160&partnerID=8YFLogxK

U2 - 10.1007/BF02187783

DO - 10.1007/BF02187783

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AN - SCOPUS:0002443160

SN - 0179-5376

VL - 5

SP - 99

EP - 160

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 1

ER -