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 $(nD)^{n\log^2 n}$.

翻译：由系数取自某个实数域（例如$\mathbb{R}$）的多项式定义的代数集$V$的路线图，是包含于$V$中的一条代数曲线，该曲线与$V\cap\mathbb{R}^{n}$的所有连通分支的交集是连通的。这些由Canny引入的对象可用于回答关于$V\cap \mathbb{R}^{n}$的连通性查询，前提是要求它们包含有限查询点集$\mathcal{P}\subset V$；此时，我们称该路线图与$(V, \mathcal{P})$相关联。在本文中，我们将先前证明的一个连通性结果付诸实践，设计了一个蒙特卡洛算法，该算法输入(i)定义$V$（并满足某些正则性假设）的有限多项式序列，以及(ii)$V$中有限多个查询点$\mathcal{P}$的代数表示，输出$(V, \mathcal{P})$的路线图。该算法通过去除对$V$实迹的有界性假设，推广了由最后两位作者提出的近乎最优算法。我们的算法输出规模与运行时间均为$(nD)^{n\log d}$的多项式，其中$D$为输入方程的最大度数，$d$为$V$的维数。据我们所知，此前处理此类集合的最佳已知算法，其输出规模与运行时间为$(nD)^{n\log^2 n}$的多项式。