Discrete & Computational Geometry

, Volume 45, Issue 1, pp 181–220 | Cite as

A Baby Steps/Giant Steps Probabilistic Algorithm for Computing Roadmaps in Smooth Bounded Real Hypersurface

  • Mohab Safey el Din
  • Éric Schost


We consider the problem of constructing roadmaps of real algebraic sets. This 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 ℚ; a deterministic version runs in time \(s^{n}\log(s)D^{O(n^{4})}\) . A subsequent 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 a semi-algebraic set (dn is the dimension of an associated object).

We give a probabilistic algorithm of complexity \((nD)^{O(n^{1.5})}\) for the problem of computing a roadmap of a closed and bounded hypersurface V of degree D in n variables, with a finite number of singular points. Even under these extra assumptions, no previous algorithm featured a cost better than \(D^{O(n^{2})}\) .


Computational real algebraic geometry Algorithms Roadmaps Complexity 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.SALSA Project, CNRS, UMR 7606, LIP6, Case 169UPMC, Univ Paris 06, INRIA, Paris-Rocquencourt CenterParisFrance
  2. 2.Computer Science Department, Room 415, Middlesex CollegeThe University of Western OntarioLondonCanada

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