A nearly optimal algorithm for deciding connectivity queries in smooth and bounded real algebraic sets
Résumé
A roadmap for a semi-algebraic set $S$ is a curve which has a
non-empty and connected intersection with all connected components
of $S$. Hence, this kind of object, introduced by Canny, can be used
to answer connectivity queries (with applications, for instance, to
motion planning) but has also become of central importance in
effective real algebraic geometry, since it is used in higher-level
algorithms.
In this paper, we provide a probabilistic algorithm which computes
roadmaps for smooth and bounded real algebraic sets. Its output size
and running time are polynomial in $(nD)^{n\log(d)}$, where $D$ is
the maximum of the degrees of the input polynomials, $d$ is the
dimension of the set under consideration and $n$ is the number of
variables. More precisely, the running time of the algorithm is
essentially subquadratic in the output size. Even under our
assumptions, it is the first roadmap algorithm with output size and
running time polynomial in $(nD)^{n\log(d)}$.
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