  A roadmap for an algebraic set $V$ defined by polynomials with coefficients
in some real field, say $\mathbb{R}$, 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 polynomial
in $(nD)^{n\log^2 n}$.
