# Computing roadmaps of general semi-algebraic sets

## Abstract

In this paper we study the problem of determining whether two points lie in the same connected component of a semi-algebraic set *S*. Although we are mostly concerned with sets *S* ⊑ **R**^{ n }, our algorithm can also decide if points in an arbitrary set *S* ⊑ *R*^{ n } can be joined by a semi-algebraic path, for any real closed field *R*. Our algorithm computes a one-dimensional semi-algebraic subset ℜ(*S*) of *S* (actually of an embedding of *S* in a space \(\hat R^n \) for a certain real extension field \(\hat R\) of the given field *R*. ℜ(*S*) is called the roadmap of *S*. Our construction uses the original roadmap algorithm described in [Can88a], [Can88b] which worked only for compact, regularly stratified sets.

We measure the complexity of the formula describing the set *S* by the number of polynomials *k*, their maximum degree *d*, the maximum length of their coefficients in bits *c*, and the number of variables *n*. With respect to the above measures, the complexity of our new algorithm is \((k^n \log ^2 k)d^{O(n^2 )} c^2 \) randomized, or \((k^n \log k)d^{O(n^4 )} c^2 \) deterministic. Note that the combinatorial complexity (complexity in terms of *k*) in both cases is within a polylog factor of the worst-case lower bound for the number of connected components Ω(*k*^{ n }).

## Keywords

Boolean Function Homotopy Type Atomic Formula Boolean Formula Deformation Retract## Preview

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## References

- [BCR87]J. Bochnak, M. Coste, and M-F. Roy.
*Géométrie algébrique réelle*. Number 12 in Ergebnisse der mathematik 3. Springer-Verlag, Berlin, 1987.Google Scholar - [BOKR86]M. Ben-Or, D. Kozen, and J. Reif. The complexity of elementary algebra and geometry.
*J. Comp. and Sys. Sci.*, 32:251–264, 1986.Google Scholar - [Can88a]J.F. Canny.
*The Complexity of Robot Motion Planning*. M.I.T. Press, Cambridge, 1988.Google Scholar - [Can88b]J.F. Canny. Constructing roadmaps of semi-algebraic sets I: Completeness.
*Artificial Intelligence*, 37:203–222, 1988.Google Scholar - [Can88c]J.F. Canny. Some algebraic and geometric computations in PSPACE. In
*ACM Symposium on Theory of Computing*, pages 460–467, 1988.Google Scholar - [Can90]J.F. Canny. Computing roadmaps of semi-algebraic sets. Technical report, University of California, Berkeley, 1990.Google Scholar
- [Can91]J.F. Canny. An improved sign determination algorithm. In
*AAECC-91 (this volume)*, 1991.Google Scholar - [CGV91]J.F. Canny, D.Y. Grigor'ev, and N.N. Vorobjov. Finding connected components of a semialgebraic set in subexponential time.
*App. Algebra in Eng. Comm. and Comp. (submitted)*, 1991.Google Scholar - [FGM87]N. Fitchas, A. Galligo, and J. Morgenstern. Algorithmes rapides en séquential et en parallèle pour l'élimination des quantificateur en géométrie élémentaire.
*Sém. Structures Algébriques Ordonnées*, 1987.Google Scholar - [Gou91]L. Gournay. Construction of roadmaps in semialgebraic sets. manuscript, 1991.Google Scholar
- [GV89]D.Y. Grigor'ev and N.N. Vorobjov. Counting connected components of a semialgebraic set in subexponential time.
*Computational Complexity (submitted)*, 1989.Google Scholar - [GWDL76]C.G. Gibson, K. Wirthmüller, A.A. Du Plessis, and E.J.N. Looijenga.
*Topological Stability of Smooth Mappings*. Number 552 in Lecture Notes in Mathematics. Springer-Verlag, New York, 1976.Google Scholar - [HRS90a]J. Heintz, M.F. Roy, and P. Solerno. Single-exponential path finding in semialgebraic sets I: The case of a smooth bounded hypersurface. In
*AAECC-90, Springer LNCS (to appear)*, 1990.Google Scholar - [HRS90b]J. Heintz, M.F. Roy, and P. Solerno. Single-exponential path finding in semialgebraic sets II: The general case. In
*To appear Proc. 60th Birthday Conf. for S. Abhyankar*, 1990.Google Scholar - [KY85]D. Kozen and C. Yap. Algebraic cell decomposition in NC. In
*IEEE Conference on Foundations of Computer Science*, pages 515–521, 1985.Google Scholar - [SS83]J.T. Schwartz and M. Sharir. On the piano movers' problem, II: General techniques for computing topological properties of real algebraic manifolds.
*Advances in Applied Mathematics*, 4:298–351, 1983.Google Scholar - [Tro89]D. Trotman. On Canny's roadmap algorithm: Orienteering in semialgebraic sets. Technical report, Univ. Aix-Marseille, 1989.Google Scholar