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

Abstract

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 (d ≤ n 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 ) .

Connectivity in Semi-algebraic Sets

“…Due to its importance, there has been intense research effort on the problem [7], [12], [1], [2], [3], [4], [6], [11].…”

A Baby Step–Giant Step Roadmap Algorithm for General Algebraic Sets

“…A new construction for computing roadmaps, with an improved recursive scheme of baby step -giant step type, has been proposed, and applied successfully in the case of smooth bounded real algebraic hypersurfaces in [8]. In this new recursive scheme, the dimension drops by √ k in each recursive call.…”

“…As a result, the depth of the recursive calls in this new algorithm is at most √ k, and consequently the algorithm has a complexity of d O(k √ k) . The proof of correctness of the algorithm in [8] depends on certain results from commutative algebra and complex algebraic geometry, in order to prove smoothness of polar varieties corresponding to generic projections of a non-singular hypersurface. Choosing generic coordinates in the algorithm is necessary since the non-singularity of polar varieties does not hold for all projections, but only for a Zariski-dense set of projections.…”

“…The goal of this paper is to obtain a deterministic algorithm for computing the roadmap of a general algebraic set, combining a baby step -giant step recursive scheme similar to that used in [8] and extending techniques coming from [2].…”

Critical point methods and effective real algebraic geometry

“…Once these properties are understood, they can be exploited to design geometric procedures for solving. For instance, the first improvement of the long-standing O(n 2 ) exponent in the complexity of Canny's probabilistic algorithm [12] to O(n 3/2 ) is based on a new geometric connectivity result obtained by investigating properties of polar varieties in [25] (see also [10] for a further generalization to general algebraic sets).…”