### Abstract

An open question in Exact Geometric Computation is whether there are transcendental computations that can be made "geometrically exact". Perhaps the simplest such problem in computational geometry is that of computing the shortest obstacle-avoiding path between two points p, q in the plane, where the obstacles are a collection of n discs. This problem can be solved in O(n^{2} log n) time in the Real RAM model, but nothing was known about its computability in the standard (Turing) model of computation. We first give a direct proof of the Turing-computability of this problem, provided the radii of the discs are rationally related. We make the usual assumption that the numerical input data are real algebraic numbers. By appealing to effective bounds from transcendental number theory, we further show a single-exponential time upper bound when the input numbers are rational. Our result appears to be the first example of a non-algebraic combinatorial problem which is shown computable. It is also a rare example of transcendental number theory yielding positive computational results.

Original language | English (US) |
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Pages (from-to) | 567-590 |

Number of pages | 24 |

Journal | International Journal of Computational Geometry and Applications |

Volume | 16 |

Issue number | 5-6 |

DOIs | |

State | Published - Dec 2006 |

### Keywords

- Computability
- Disc obstacles
- Exact geometric computation
- Exponential complexity
- Guaranteed precision computation
- Real RAM model
- Robust numerical algorithms
- Shortest path
- Transcendental number theory

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Theory and Mathematics
- Computational Mathematics
- Applied Mathematics

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## Cite this

*International Journal of Computational Geometry and Applications*,

*16*(5-6), 567-590. https://doi.org/10.1142/s0218195906002191