A roadmap for an algebraic set $V$ defined by polynomials with coefficients in the field $\mathbb{ Q}$ of rational numbers is an algebraic curve contained in $V$ whose intersection with all connected components of $V\cap\mathbb{ R}^{ n}$ is connected. These objects, introduced by Canny, can be used to answer connectivity queries over $V\cap \mathbb{ R}^{ n}$ provided that they are required to contain the finite set of query points $\mathcal{ P}\subset V$; in this case, we say that the roadmap is associated to $(V, \mathcal{ P})$. In this paper, we make effective a connectivity result we previously proved, to design a Monte Carlo algorithm which, on input (i) a finite sequence of polynomials defining $V$ (and satisfying some regularity assumptions) and (ii) an algebraic representation of finitely many query points $\mathcal{ P}$ in $V$, computes a roadmap for $(V, \mathcal{ P})$. This algorithm generalizes the nearly optimal one introduced by the last two authors by dropping a boundedness assumption on the real trace of $V$. The output size and running times of our algorithm are both polynomial in $(nD)^{ n\log d}$, where $D$ is the maximal degree of the input equations and $d$ is the dimension of $V$. As far as we know, the best previously known algorithm dealing with such sets has an output size and running time respectively polynomial in $(n^{ \log{ n}}D)^{ n\log n}$ and $(n^{ \log{ n}}D)^{ n\log^2 n}$.
Categories: cs.SC math.AG