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Probing a set of hyperplanes by lines and related problems

  • Yasukazu Aoki
  • Hiroshi Imai
  • Keiko Imai
  • David Rappaport
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 709)

Abstract

Suppose that for a set H of n unknown hyperplanes in the Euclidean d-dimensional space, a line probe is available which reports the set of intersection points of a query line with the hyperplanes. Under this model, this paper investigates the complexity to find a generic line for H and further to determine the hyperplanes in H. This problem arises in factoring the u-resultant to solve systems of polynomials (e.g., Renegar [13]). We prove that d+1 line probes are sufficient to determine H. Algorithmically, the time complexity to find a generic line and reconstruct H from O(dn) probed points of intersection is important. It is shown that a generic line can be computed in O(dn log n) time after d line probes, and by an additional d line probes, all the hyperplanes in H are reconstructed in O(dn log n) time. This result can be extended to the d-dimensional complex space. Also, concerning the factorization of the u-resultant using the partial derivatives on a generic line, we touch upon reducing the time complexity to compute the partial derivatives of the u-resultant represented as the determinant of a matrix.

Keywords

Grid Point Time Complexity Generic Direction Generic Point Dual Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Yasukazu Aoki
    • 1
  • Hiroshi Imai
    • 1
  • Keiko Imai
    • 2
  • David Rappaport
    • 3
  1. 1.Department of Information ScienceUniversity of TokyoTokyoJapan
  2. 2.Department of Information and System EngineeringChuo UniversityTokyoJapan
  3. 3.Department of Computing and Information ScienceQueen's UniversityKingstonCanada

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