TY - GEN
T1 - New lower bounds for Hopcroft's problem
AU - Erickson, Jeff
N1 - Funding Information:
*This research was partially supported by NSF grant CCR-9058440. An earlier version of this paper was published as Technical Report A/04/94, Fachbereich Inforrnatik, Universitat des Saarlandes, Saarbriicken, Germany, November 1994.
Publisher Copyright:
© 1995 ACM.
PY - 1995/9/1
Y1 - 1995/9/1
N2 - We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in IRd, is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time Ω(n log m + n2/3 m2/3 + m log n) in two dimensions, or Ω(n log m + n5/6 m1/2 + n1/2m5/6 + m log n) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2O(log∗(n+m)) of the best known upper bound, due to Matoušek. Previously, the best known lower bound, in any dimension, was Ω(n log m + m log n). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called a monochromatic cover, and derive lower bounds on the complexity of this representation. We then show that the running time of any partitioning algorithm is bounded below by the size of some monochromatic cover.
AB - We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in IRd, is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time Ω(n log m + n2/3 m2/3 + m log n) in two dimensions, or Ω(n log m + n5/6 m1/2 + n1/2m5/6 + m log n) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2O(log∗(n+m)) of the best known upper bound, due to Matoušek. Previously, the best known lower bound, in any dimension, was Ω(n log m + m log n). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called a monochromatic cover, and derive lower bounds on the complexity of this representation. We then show that the running time of any partitioning algorithm is bounded below by the size of some monochromatic cover.
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U2 - 10.1145/220279.220293
DO - 10.1145/220279.220293
M3 - Conference contribution
AN - SCOPUS:84968398176
T3 - Proceedings of the Annual Symposium on Computational Geometry
SP - 127
EP - 137
BT - Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995
PB - Association for Computing Machinery
T2 - 11th Annual Symposium on Computational Geometry, SCG 1995
Y2 - 5 June 1995 through 7 June 1995
ER -