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

@article{Basu2014ABS,
title={A Baby Step–Giant Step Roadmap Algorithm for General Algebraic Sets},
author={Saugata Basu and Marie-Françoise Roy and Mohab Safey El Din and {\'E}ric Schost},
journal={Foundations of Computational Mathematics},
year={2014},
volume={14},
pages={1117-1172}
}
• Published 31 January 2012
• Mathematics
• Foundations of Computational Mathematics
Let $$\mathrm {R}$$R be a real closed field and $$\mathrm{D}\subset \mathrm {R}$$D⊂R an ordered domain. We present an algorithm that takes as input a polynomial $$Q \in \mathrm{D}[X_{1},\ldots ,X_{k}]$$Q∈D[X1,…,Xk] and computes a description of a roadmap of the set of zeros, $$\mathrm{Zer}(Q,\,\mathrm {R}^{k}),$$Zer(Q,Rk), of Q in $$\mathrm {R}^{k}.$$Rk. The complexity of the algorithm, measured by the number of arithmetic operations in the ordered domain $$\mathrm{D},$$D, is bounded by d^{O…
33 Citations
Divide and Conquer Roadmap for Algebraic Sets
• Mathematics
Discret. Comput. Geom.
• 2014
It is proved that for any real algebraic subset of R, Rk defined by a polynomial of degree d, any connected component of V, V contained in the unit ball, and any two points of C, there exists a semi-algebraic path connecting them in C.
Solving determinantal systems using homotopy techniques
• Mathematics, Computer Science
J. Symb. Comput.
• 2021
A Nearly Optimal Algorithm for Deciding Connectivity Queries in Smooth and Bounded Real Algebraic Sets
• Computer Science, Mathematics
J. ACM
• 2017
This article provides a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets and is the first roadmap algorithm with output size and running time polynomial in (nD)nlog (d).
Polynomial Systems Solving by Fast Linear Algebra
• Computer Science, Mathematics
ArXiv
• 2013
The complexity of solving a polynomial system is decreased from $\widetilde{O}(D^3)$ to $D(d^\omega)$ where $D$ is the number of solutions of the system and new algorithms which rely on fast linear algebra are proposed.
Critical point methods and effective real algebraic geometry: new results and trends
An overview of critical point methods and how to tune them to exploit algebraic and geometric structures in two fundamental problems, and a probabilistic algorithm that improves the long-standing $(s\, D)^{O(n^2)}$ bound obtained by Koi\-ran.
Robots, computer algebra and eight connected components
• Mathematics, Computer Science
ISSAC
• 2020
This paper shows how to combine mathematical reasoning with easy symbolic computations to study the kinematic singularities of an infinite family (depending on paramaters) modelled by the UR-series produced by the company "Universal Robots".
Numerical roadmap of smooth bounded real algebraic surface
• Mathematics, Computer Science
Comput. Aided Geom. Des.
• 2020
Efficient computation of a semi-algebraic basis of the first homology group of a semi-algebraic set
• Mathematics, Computer Science
• 2021
An algorithm for computing a semi-algebraic basis for the first homology group, H1(S,F), with coefficients in a field F, of any given semi- algebraic set S ⊂ R defined by a closed formula is given, which generalizes well known algorithms having singly exponential complexity.

References

SHOWING 1-10 OF 37 REFERENCES
A Baby Steps/Giant Steps Probabilistic Algorithm for Computing Roadmaps in Smooth Bounded Real Hypersurface
• Computer Science
Discret. Comput. Geom.
• 2011
A probabilistic algorithm of complexity $(nD)^{O(n^{1.5})}$ is given 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.
Construction of roadmaps in semi-algebraic sets
• Mathematics, Computer Science
Applicable Algebra in Engineering, Communication and Computing
• 2005
An algorithm of construction of a roadmap for a compact semi-algebraic setS ⊂ Rn is described, which is similar to the algorithm of [3], but which is simpler in the sense that it does not need the use of Whitney stratifications, and more general because it accepts as input any compact non-compact set.
COMPUTING ROADMAPS OF SEMI-ALGEBRAIC SETS ON A VARIETY
• Mathematics
• 1997
Let R be a real closed field, Z(Q) a real algebraic variety defined as the zero set of a polynomial Q ∈ R[X1, . . . , Xk] and S a semi-algebraic subset of Z(Q), defined by a Boolean formula with
Computing Roadmaps of General Semi-Algebraic Sets
• J. Canny
• Mathematics, Computer Science
Comput. J.
• 1993
The algorithm computes a one-dimensional semi-algebraic subset ℜ(S) of S (actually of an embedding of S in a space $$\hat R^n$$ for a certain real extension field R of the given field R).
Single Exponential Path Finding in Semialgebraic Sets. Part 1: The Case of a Regular Bounded Hypersurface
• Mathematics
AAECC
• 1990
An algorithm which decides in single exponential sequential time and polynomial parallel time whether x 1 and x 2 are contained in the same semialgebraically connected component of V is described, which results in the number of semial algebraically connected components of V being computable within the mentioned time bounds.
Single Exponential Path Finding in Semi-algebraic Sets, Part II: The General Case
• Mathematics
• 1994
This paper is devoted to the following result. Let S be a semi-algebraic subset of R n ; one can decide in single exponential time whether two points of S belong to the same semi-algebraically
Counting connected components of a semialgebraic set in subexponential time
• Mathematics
computational complexity
• 2005
An algorithm is exhibited which counts the number of connected components of the semialgebraic set in time (M (kd)n20)O (1) and allows us to determine whether any pair of points from the set are situated in the same connected component.
The complexity of robot motion planning
John Canny resolves long-standing problems concerning the complexity of motion planning and, for the central problem of finding a collision free path for a jointed robot in the presence of obstacles, obtains exponential speedups over existing algorithms by applying high-powered new mathematical techniques.
Algorithms in real algebraic geometry, volume 10 of Algorithms and Computation in Mathematics
• Algorithms in real algebraic geometry, volume 10 of Algorithms and Computation in Mathematics
• 2006