Some algebraic and geometric computations in PSPACE
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Motion planning in the presence of movable obstacles
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Deterministic minimal time vessel routing
Operations Research
The weighted region problem: finding shortest paths through a weighted planar subdivision
Journal of the ACM (JACM)
A single-exponential upper bound for finding shortest paths in three dimensions
Journal of the ACM (JACM)
A new algorithm for computing shortest paths in weighted planar subdivisions (extended abstract)
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Approximating weighted shortest paths on polyhedral surfaces
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Approximation algorithms for geometric shortest path problems
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
An epsilon-Approximation for Weighted Shortest Paths on Polyhedral Surfaces
SWAT '98 Proceedings of the 6th Scandinavian Workshop on Algorithm Theory
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
Complexity of the mover's problem and generalizations
SFCS '79 Proceedings of the 20th Annual Symposium on Foundations of Computer Science
Motion planning in the presence of moving obstacles
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
New lower bound techniques for robot motion planning problems
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
BUSHWHACK: An Approximation Algorithm for Minimal Paths through Pseudo-Euclidean Spaces
ISAAC '01 Proceedings of the 12th International Symposium on Algorithms and Computation
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This paper investigates the problem of time-optimum movement planning in d = 2, 3 dimensions for a point robot which has bounded control velocity through a set of n polygonal regions of given translational flow velocities. This intriguing geometric problem has immediate applications to macro-scale motion planning for ships, submarines and airplanes in the presence of significant flows of water or air. Also, it is a central motion planning problem for many of the mesoscale and micro-scale robots that recently have been constructed, that have environments with significant flows that affect their movement. In spite of these applications, there is very little literature on this problem, and prior work provided neither an upper bound on its computational complexity nor even a decision algorithm. It can easily be seen that optimum path for the d = 2 dimensional version of this problem can consist of at least an exponential number of distinct segments through flow regions. We provide the first known computational complexity hardness result for the d = 3 dimensional version of this problem; we show the problem is PSPACE hard. We give the first known decision algorithm for the d = 2 dimensional problem, but this decision algorithm has very high complexity. We also give the first known efficient approximation algorithms with bounded error.