Applicable Algebra in Engineering, Communication and Computing

, Volume 2, Issue 4, pp 217–238

Finding connected components of a semialgebraic set in subexponential time

  • J. Canny
  • D. Yu. Grigor'ev
  • N. N. VorobjovJr.
Article

DOI: 10.1007/BF01614146

Cite this article as:
Canny, J., Grigor'ev, D.Y. & Vorobjov, N.N. AAECC (1992) 2: 217. doi:10.1007/BF01614146

Abstract

Let a semialgebraic set be given by a quantifier-free formula of the first-order theory of real closed fields with atomic subformulae of type (fi ≥ 0), 1 ≤i ≤k where the polynomialsfi ε ℤ[X1,..., Xn] have degrees deg(fi <d and the absolute value of each (integer) coefficient offi is at most 2M. An algorithm is designed which finds the connected components of the semialgebraic set in time\(M^{O(1)} (kd)^{n^{O(1)} } \). The best previously known bound\(M^{O(1)} (kd)^{n^{O(n)} } \) for this problem follows from Collins' method of Cylindrical Algebraic Decomposition.

Key words

Semialgebraic setConnected componentSubexponential-time algorithmInfinitesimalsTarski algebra

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • J. Canny
    • 1
  • D. Yu. Grigor'ev
    • 2
  • N. N. VorobjovJr.
    • 2
  1. 1.Computer Science DivisionUniversity of BerkeleyCaliforniaUSA
  2. 2.V.A. Steklov Institute of Academy of SciencesSt. PetersburgRussia