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 many higher-level algorithms. For a long time, the best known complexity result for computing roadmaps, due to Basu, Pollack and Roy, was $s^{d+1} D^{O(n^2)}$, where the input is given by $s$ polynomials of degree $D$ in $n$ variables, with $d \le n$ the dimension of an associated geometric object. In 2011, we introduced new proof techniques for establishing connectivity results in real algebraic sets. This gave us more freedom for the design of algorithms computing roadmaps and led us to a first probabilistic roadmap algorithm for smooth and bounded real hypersurfaces running in time $(nD)^{O(n^{1.5})}$. With Basu and Roy, we then obtained a deterministic algorithm for general real algebraic sets running in time $D^{O(n^{1.5})}$. Recently, Basu and Roy improved this result to obtain an algorithm computing a roadmap of degree polynomial in $n^{n\log^2(n)} D^{n\log(n)}$, in time polynomial in $n^{n\log^3(n)} D^{n\log^2(n)}$; this is close to the expected optimal $D^n$. In this paper, we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets such that the output size and the running time are polynomial in $(nD)^{n\log(n)}$. More precisely, the running time of the algorithm is essentially subquadratic in the output size. Even under these extra assumptions, it is the first roadmap algorithm with output size and running time polynomial in $(nD)^{n\log(n)}$.
Domaines
Calcul formel [cs.SC]
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