# A baby steps/giant steps Monte Carlo algorithm for computing roadmaps in smooth compact real hypersurfaces

@article{Din2009ABS, title={A baby steps/giant steps Monte Carlo algorithm for computing roadmaps in smooth compact real hypersurfaces}, author={Mohab Safey El Din and {\'E}ric Schost}, journal={ArXiv}, year={2009}, volume={abs/0902.1612} }

We consider the problem of constructing roadmaps of real algebraic sets. The problem was introduced by Canny to answer connectivity questions and solve motion planning problems. Given $s$ polynomial equations with rational coefficients, of degree $D$ in $n$ variables, Canny's algorithm has a Monte Carlo cost of $s^n\log(s) D^{O(n^2)}$ operations in $\mathbb{Q}$; a deterministic version runs in time $s^n \log(s) D^{O(n^4)}$. The next improvement was due to Basu, Pollack and Roy, with an…

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A natural construction is introduced that allows to tackle the case of a non-smooth real hypersurface by means of a reduction to a smooth complete intersection.

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