Roughly, a roadmap $R$ of a semi-algebraic set $S$ is a subset $R\subset S$ of dimension $\le 1$ such that each connected component of $S$ contains exactly one connected component of $R$. The first algorithm for computing a roadmap in [SS] had double exponential complexity, and this was later reduced to single exponential in [C]. In [DS], a randomized baby step--giant step technique was introduced to further reduce the complexity. The goal of this paper is to give a deterministic version of the previous algorithm without increasing the complexity.

The giant step in the main algorithm is a subroutine originating in [DS]. The new baby step uses an infinitesimal deformation (developed in [BPR]) of a variety to a special variety. This procedure exploits properties of special varieties to achieve a deterministic algorithm without changing the asymptotic complexity. The authors make give concise construction and proofs whenever possible, and their most technical definitions and complicated proofs are necessary to avoid spoiling complexity.

[BPR] S. Basu, R. Pollack and M.-F. Roy: Algorithms in Real Algebraic Geometry, volume 10 of Algorithms Comput. Math., Second edition, Springer, Berlin, 2006.

[C] J. Canny: The Complexity of Robot Motion Planning, MIT Press, Cambridge, 1987.

[DS] M.S. el Din and E. Schost: A Baby Steps/Giant Steps Probabilistic Algorithm for Computing Roadmaps in Smooth Bounded Real Hypersurface, Discret. Comput. Geom. 45(1), pages 181--220, 2011.

[SS] J. Schwartz and M. Sharir: On the Piano Movers' Problem. II. General Techniques for Computing Topological Properties of Real Algebraic Manifolds, Adv. Appl. Math. 4, pages 298--351, 1983.