Fast searching in a real algebraic manifold with applications to geometric complexity

  • Bernard Chazelle
Colloquium On Trees In Algebra And Programming Algorithms And Combinatorics
Part of the Lecture Notes in Computer Science book series (LNCS, volume 185)


This paper generalizes the multidimensional searching scheme of Dobkin and Lipton [SIAM J. Comput. 5(2), pp. 181–186, 1976] for the case of arbitrary (as opposed to linear) real algebraic varieties. Let d,r be two positive constants and let P 1,...,P n be n rational r-variate polynomials of degree ≤d. Our main result is an \(O(n^{2^{r + 6} } )\) data structure for computing the predicate [∃i (1≤in)|P i (x)=0] in O(log n) time, for any xE r . The method is intimately based on a decomposition technique due to Collins [Proc. 2nd GI Conf. on Automata Theory and Formal Languages, pp. 134–183, 1975]. The algorithm can be used to solve problems in computational geometry via a locus approach. We illustrate this point by deriving an o(n 2) algorithm for computing the time at which the convex hull of n (algebraically) moving points in E 2 reaches a steady state.


Convex Hull Real Root Rational Coefficient Computational Geometry Binary Search 
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.


  1. [A]
    Atallah, M.J. Dynamic computational geometry, Proc. 24th Annual FOCS Symp., pp. 92–99, Nov. 1983.Google Scholar
  2. [BT]
    Brown, W., Traub, J.F. On Euclid's algorithm and the theory of subresultants, J. ACM 18, pp. 505–514, 1971.CrossRefGoogle Scholar
  3. [Co]
    Cole, R. Searching and storing similar lists, Tech. Rep. No. 88, New York University, Oct. 1983.Google Scholar
  4. [C]
    Collins, G.E., Quantifier elimination for real closed fields by cylindric algebraic decomposition, Proc. 2nd GI Conf. on Automata Theory and Formal Languages, Springer-Verlag, JNCS 35, Berlin, pp. 134–183, 1975Google Scholar
  5. [DL]
    Dobkin, D.P., Lipton, R.J. Multidimensional searching problems, SIAM J. Comput. 5(2), pp. 181–186, 1976.Google Scholar
  6. [EGS]
    Edelsbrunner, H., Guibas, L., Stolfi, J. Optimal point location in a monotone subdivision, to appear.Google Scholar
  7. [O]
    Overmars, M.H. The locus approach, Tech. Rept. RUU-CS-83-12, Univ. Utrecht, July 1983.Google Scholar
  8. [SS]
    Schwartz, J.T., Sharir, M. On the “piano movers” problem. II: General techniques for computing topological properties of real algebraic manifolds, Adv. in Appl. Math 4, pp. 298–351, 1983.Google Scholar
  9. [T]
    Tarski, A. A decision method for elementary algebra and geometry, Univ. of Calif. Press, 1948, 2nd edition, 1951.Google Scholar
  10. [W]
    van der Waerden, B.L. Modern Algebra, Ungar Co., New York, 1953.Google Scholar
  11. [Y]
    Yao, A.C. On constructing minimum spanning tree in k-dimensional space and related problems, SIAM J. Comput. 11(4), pp. 721–736, 1982.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Bernard Chazelle
    • 1
  1. 1.Department of Computer ScienceBrown UniveristyProvidenceUSA

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