Computing roadmaps of semi-algebraic sets

on a variety

Authors:
Saugata Basu, Richard Pollack and Marie-Françoise Roy

Journal:
J. Amer. Math. Soc. **13** (2000), 55-82

MSC (1991):
Primary 14P10, 68Q25; Secondary 68Q40

Published electronically:
July 20, 1999

MathSciNet review:
1685780

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a semi-algebraic set defined by polynomials in variables which is contained in an algebraic variety . The variety is assumed to have real dimension the polynomial and the polynomials defining have degree at most . We present an algorithm which constructs a *roadmap* on . The complexity of this algorithm is . We also present an algorithm which, given a point of defined by polynomials of degree at most , constructs a path joining this point to the roadmap. The complexity of this algorithm is These algorithms easily yield an algorithm which, given two points of defined by polynomials of degree at most , decides whether or not these two points of lie in the same semi-algebraically connected component of and if they do computes a semi-algebraic path in connecting the two points.

**1.**Saugata Basu, Richard Pollack, and Marie-Françoise Roy,*On the combinatorial and algebraic complexity of quantifier elimination*, J. ACM**43**(1996), no. 6, 1002–1045. MR**1434910**, 10.1145/235809.235813**2.**S. BASU, R. POLLACK, M.-F. ROY*Computing Roadmaps of Semi-algebraic Sets*,*Proc. 28th Annual ACM Symposium on the Theory of Computing*, 168-173, (1996). CMP**97:06****3.**Saugata Basu, Richard Pollack, and Marie-Françoise Roy,*On computing a set of points meeting every cell defined by a family of polynomials on a variety*, J. Complexity**13**(1997), no. 1, 28–37. MR**1449758**, 10.1006/jcom.1997.0434**4.**S. BASU, R. POLLACK, M.-F. ROY*Computing Roadmaps of Semi-algebraic Sets on a Variety*,*Foundations of Computational Mathematics*, F. Cucker and M. Shub, Eds., 1-15, (1997). CMP**99:05****5.**J. BOCHNAK, M. COSTE, M.-F. ROY Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 36, Berlin: Springer-Verlag (1998). CMP**99:07****6.**John Canny,*The complexity of robot motion planning*, ACM Doctoral Dissertation Awards, vol. 1987, MIT Press, Cambridge, MA, 1988. MR**952555****7.**John Canny,*Computing roadmaps of general semi-algebraic sets*, Comput. J.**36**(1993), no. 5, 504–514. MR**1234122**, 10.1093/comjnl/36.5.504**8.**J. Canny, D. Yu. Grigor′ev, and N. N. Vorobjov Jr.,*Finding connected components of a semialgebraic set in subexponential time*, Appl. Algebra Engrg. Comm. Comput.**2**(1992), no. 4, 217–238. MR**1325530**, 10.1007/BF01614146**9.**George E. Collins,*Quantifier elimination for real closed fields by cylindrical algebraic decomposition*, Automata theory and formal languages (Second GI Conf., Kaiserslautern, 1975), Springer, Berlin, 1975, pp. 134–183. Lecture Notes in Comput. Sci., Vol. 33. MR**0403962****10.**M. Coste and M.-F. Roy,*Thom’s lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets*, J. Symbolic Comput.**5**(1988), no. 1-2, 121–129. MR**949115**, 10.1016/S0747-7171(88)80008-7**11.**Michel Coste and Masahiro Shiota,*Nash triviality in families of Nash manifolds*, Invent. Math.**108**(1992), no. 2, 349–368. MR**1161096**, 10.1007/BF02100609**12.**D. Yu. Grigor′ev and N. N. Vorobjov Jr.,*Counting connected components of a semialgebraic set in subexponential time*, Comput. Complexity**2**(1992), no. 2, 133–186. MR**1190827**, 10.1007/BF01202001**13.**L. Gournay and J.-J. Risler,*Construction of roadmaps in semi-algebraic sets*, Appl. Algebra Engrg. Comm. Comput.**4**(1993), no. 4, 239–252. MR**1235859**, 10.1007/BF01200148**14.**Robert M. Hardt,*Semi-algebraic local-triviality in semi-algebraic mappings*, Amer. J. Math.**102**(1980), no. 2, 291–302. MR**564475**, 10.2307/2374240**15.**Joos Heintz, Marie-Françoise Roy, and Pablo Solernó,*Single exponential path finding in semialgebraic sets. I. The case of a regular bounded hypersurface*, Applied algebra, algebraic algorithms and error-correcting codes (Tokyo, 1990) Lecture Notes in Comput. Sci., vol. 508, Springer, Berlin, 1991, pp. 180–196. MR**1123950**, 10.1007/3-540-54195-0_50**16.**Joos Heintz, Marie-Françoise Roy, and Pablo Solernó,*Single exponential path finding in semi-algebraic sets. II. The general case*, Algebraic geometry and its applications (West Lafayette, IN, 1990), Springer, New York, 1994, pp. 449–465. MR**1272047****17.**J.-C. LATOMBE*Robot Motion Planning*, The Kluwer International Series in Engineering and Computer Science. 124. Dordrecht: Kluwer Academic Publishers Group (1991).**18.**J. Milnor,*Morse theory*, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR**0163331****19.**James Renegar,*On the computational complexity and geometry of the first-order theory of the reals. I. Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals*, J. Symbolic Comput.**13**(1992), no. 3, 255–299. MR**1156882**, 10.1016/S0747-7171(10)80003-3

James Renegar,*On the computational complexity and geometry of the first-order theory of the reals. II. The general decision problem. Preliminaries for quantifier elimination*, J. Symbolic Comput.**13**(1992), no. 3, 301–327. MR**1156883**, 10.1016/S0747-7171(10)80004-5

James Renegar,*On the computational complexity and geometry of the first-order theory of the reals. III. Quantifier elimination*, J. Symbolic Comput.**13**(1992), no. 3, 329–352. MR**1156884**, 10.1016/S0747-7171(10)80005-7**20.**F. ROUILLIER, M.-F. ROY, M. SAFEY*Finding at least a point in each connected component of a real algebraic set defined by a single equation,*to appear in Journal of Complexity.**21.**Marie-Françoise Roy and Nicolai Vorobjov,*Finding irreducible components of some real transcendental varieties*, Comput. Complexity**4**(1994), no. 2, 107–132. MR**1285473**, 10.1007/BF01202285**22.**M.-F. ROY, N. VOROBJOV*Computing the Complexification of a Semi-algebraic Set*, Proc. of International Symposium on Symbolic and Algebraic Computations, 1996, 26-34 (complete version to appear in Math. Zeitschrift).**23.**Jacob T. Schwartz and Micha Sharir,*On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds*, Adv. in Appl. Math.**4**(1983), no. 3, 298–351. MR**712908**, 10.1016/0196-8858(83)90014-3

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (1991):
14P10,
68Q25,
68Q40

Retrieve articles in all journals with MSC (1991): 14P10, 68Q25, 68Q40

Additional Information

**Saugata Basu**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Email:
saugata@math.lsa.umich.edu

**Richard Pollack**

Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, New York 10012

Email:
pollack@cims.nyu.edu

**Marie-Françoise Roy**

Affiliation:
IRMAR (URA CNRS 305), Université de Rennes, Campus de Beaulieu 35042 Rennes cedex, France

Email:
mfroy@univ-rennes1.fr

DOI:
http://dx.doi.org/10.1090/S0894-0347-99-00311-2

Keywords:
Roadmaps,
semi-algebraic sets,
variety

Received by editor(s):
November 25, 1997

Received by editor(s) in revised form:
April 14, 1999

Published electronically:
July 20, 1999

Additional Notes:
The first author was supported in part by NSF Grants CCR-9402640 and CCR-9424398.

The second author was supported in part by NSF Grants CCR-9402640, CCR-9424398, DMS-9400293, CCR-9711240 and CCR-9732101.

The third author was supported in part by the project ESPRIT-LTR 21024 FRISCO and by European Community contract CHRX-CT94-0506.

Article copyright:
© Copyright 1999
American Mathematical Society