A baby steps/giant steps Monte Carlo algorithm for computing roadmaps in smooth compact real hypersurfaces
| dc.creator | Din, Mohab Safey El | |
| dc.creator | Schost, Éric | |
| dc.date | 2009-02-10 | |
| dc.date.accessioned | 2026-07-07T13:05:31Z | |
| dc.date.available | 2026-07-07T13:05:31Z | |
| dc.description | We consider the problem of constructing roadmaps of real algebraic sets. The problem was introduced by Canny to answer connectivity questions and solve motion planning problems. Given $s$ polynomial equations with rational coefficients, of degree $D$ in $n$ variables, Canny's algorithm has a Monte Carlo cost of $s^n\log(s) D^{O(n^2)}$ operations in $\mathbb{Q}$; a deterministic version runs in time $s^n \log(s) D^{O(n^4)}$. The next improvement was due to Basu, Pollack and Roy, with an algorithm of deterministic cost $s^{d+1} D^{O(n^2)}$ for the more general problem of computing roadmaps of semi-algebraic sets ($d \le n$ is the dimension of an associated object). We give a Monte Carlo algorithm of complexity $(nD)^{O(n^{1.5})}$ for the problem of computing a roadmap of a compact hypersurface $V$ of degree $D$ in $n$ variables; we also have to assume that $V$ has a finite number of singular points. Even under these extra assumptions, no previous algorithm featured a cost better than $D^{O(n^2)}$. | |
| dc.identifier | https://arxiv.org/abs/0902.1612 | |
| dc.identifier | http://arxiv.org/abs/0902.1612 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/227513 | |
| dc.subject | Symbolic Computation | |
| dc.title | A baby steps/giant steps Monte Carlo algorithm for computing roadmaps in smooth compact real hypersurfaces | |
| dc.type | text |