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Workshop on Algorithms and Data Structures

WADS 1993: Algorithms and Data Structures pp 72–82Cite as

Probing a set of hyperplanes by lines and related problems

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Part of the book series: Lecture Notes in Computer Science ((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.

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Author information

Authors and Affiliations

  1. Department of Information Science, University of Tokyo, Hongo, Bunkyo-ku, 113, Tokyo, Japan

    Yasukazu Aoki & Hiroshi Imai

  2. Department of Information and System Engineering, Chuo University, Kasuga, Bunkyo-ku, 112, Tokyo, Japan

    Keiko Imai

  3. Department of Computing and Information Science, Queen's University, K7L 3N6, Kingston, Ontario, Canada

    David Rappaport

Authors
  1. Yasukazu Aoki
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  2. Hiroshi Imai
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  3. Keiko Imai
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  4. David Rappaport
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Editor information

Frank Dehne Jörg-Rüdiger Sack Nicola Santoro Sue Whitesides

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© 1993 Springer-Verlag Berlin Heidelberg

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Aoki, Y., Imai, H., Imai, K., Rappaport, D. (1993). Probing a set of hyperplanes by lines and related problems. In: Dehne, F., Sack, JR., Santoro, N., Whitesides, S. (eds) Algorithms and Data Structures. WADS 1993. Lecture Notes in Computer Science, vol 709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57155-8_237

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  • DOI: https://doi.org/10.1007/3-540-57155-8_237

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57155-1

  • Online ISBN: 978-3-540-47918-5

  • eBook Packages: Springer Book Archive

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