The complexity of elementary algebra and geometry
Journal of Computer and System Sciences
Collision Detection for Moving Polyhedra
IEEE Transactions on Pattern Analysis and Machine Intelligence
On approximations and incidence in cylindrical algebraic decompositions
SIAM Journal on Computing
Planning, geometry, and complexity of robot motion
Planning, geometry, and complexity of robot motion
Complexity of Quantifier Elimination in the Theory of Algebraically Closed Fields
Proceedings of the Mathematical Foundations of Computer Science 1984
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
Spatial Planning: A Configuration Space Approach
IEEE Transactions on Computers
Precise bounds for presburger arithmetic and the reals with addition: Preliminary report
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
Complexity of the mover's problem and generalizations
SFCS '79 Proceedings of the 20th Annual Symposium on Foundations of Computer Science
Constructing arrangements of lines and hyperplanes with applications
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
Algebraic cell decomposition in NC
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Proving by example and gap theorems
SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science
Generalised characteristic polynomials
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Locating a robot with angle measurements
Journal of Symbolic Computation
An Algebraic Geometry Approach to Protein Structure Determination from NMR Data
CSB '05 Proceedings of the 2005 IEEE Computational Systems Bioinformatics Conference
New lower bound techniques for robot motion planning problems
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
Computation of equilibriain noncooperative games
Computers & Mathematics with Applications
The multivariate resultant is NP-hard in any characteristic
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Rational univariate representations of bivariate systems and applications
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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We present an algorithm which solves the findpath or generalized movers' problem in single exponential sequential time. This is the first algorithm for the problem whose sequential time bound is less than double exponential. In fact, the combinatorial exponent of the algorithm is equal to the number of degrees of freedom, making it worst-case optimal, and equaling or improving the time bounds of many special purpose algorithms. The algorithm accepts a formula for a semi-algebraic set S describing the set of free configurations and produces a one-dimensional skeleton or "roadmap" of the set, which is connected within each connected component of S. Additional points may be linked to the roadmap in linear time. Our method draws from results of singularity theory, and in particular makes use of the notion of stratified sets as an efficient alternative to cell decomposition. We introduce an algebraic tool called the multivariate resultant which gives a necessary and sufficient condition for a system of homogeneous polynomials to have a solution, and show that it can be computed in polynomial parallel time. Among the consequences of this result are new methods for quantifier elimination and an improved gap theorem for the absolute value of roots of a system of polynomials.