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

Mohab Safey El Din 1 Éric Schost 2
1 SALSA - Solvers for Algebraic Systems and Applications
LIP6 - Laboratoire d'Informatique de Paris 6, Inria Paris-Rocquencourt
Abstract : 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 algorithm of deterministic cost $s^{d+1} D^{O(n^2)}$ for the more general problem of computing roadmaps of semi-algebraic sets ($d \le n$ is the dimension of an associated object). We give a Monte Carlo algorithm of complexity $(nD)^{O(n^{1.5})}$ for the problem of computing a roadmap of a compact hypersurface $V$ of degree $D$ in $n$ variables; we also have to assume that $V$ has a finite number of singular points. Even under these extra assumptions, no previous algorithm featured a cost better than $D^{O(n^2)}$.
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Mohab Safey El Din, Éric Schost. A baby steps/giant steps Monte Carlo algorithm for computing roadmaps in smooth compact real hypersurfaces. Discrete and Computational Geometry, Springer Verlag, 2011, 45 (1), pp.181-220. ⟨10.1007/s00454-009-9239-2⟩. ⟨inria-00359748⟩

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