Shortest path amidst disc obstacles is computable

Ee Chien Chang, Sung Woo Choi, DoYong Kwon, Hyungju Park, Chee Yap

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 show 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 languageEnglish (US)
Title of host publicationProceedings of the 21st Annual Symposium on Computational Geometry, SCG'05
Pages116-125
Number of pages10
StatePublished - 2005
Event21st Annual Symposium on Computational Geometry, SCG'05 - Pisa, Italy
Duration: Jun 6 2005Jun 8 2005

Other

Other21st Annual Symposium on Computational Geometry, SCG'05
CountryItaly
CityPisa
Period6/6/056/8/05

Fingerprint

Shortest path
Transcendental number
Number theory
Computability
Turing
Exact Geometric Computation
Computational geometry
Algebraic number
Models of Computation
Computational Geometry
Exponential time
Transcendental
Random access storage
Combinatorial Problems
Computational Results
Radius
Upper bound
Path
Computing
Model

Keywords

  • Disc obstacles
  • Exact geometric computation
  • Exponential complexity
  • Guaranteed precision computation
  • Real RAM model
  • Robust numerical algorithms
  • Shortest path

ASJC Scopus subject areas

  • Software
  • Geometry and Topology
  • Safety, Risk, Reliability and Quality
  • Chemical Health and Safety

Cite this

Chang, E. C., Choi, S. W., Kwon, D., Park, H., & Yap, C. (2005). Shortest path amidst disc obstacles is computable. In Proceedings of the 21st Annual Symposium on Computational Geometry, SCG'05 (pp. 116-125)

Shortest path amidst disc obstacles is computable. / Chang, Ee Chien; Choi, Sung Woo; Kwon, DoYong; Park, Hyungju; Yap, Chee.

Proceedings of the 21st Annual Symposium on Computational Geometry, SCG'05. 2005. p. 116-125.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Chang, EC, Choi, SW, Kwon, D, Park, H & Yap, C 2005, Shortest path amidst disc obstacles is computable. in Proceedings of the 21st Annual Symposium on Computational Geometry, SCG'05. pp. 116-125, 21st Annual Symposium on Computational Geometry, SCG'05, Pisa, Italy, 6/6/05.
Chang EC, Choi SW, Kwon D, Park H, Yap C. Shortest path amidst disc obstacles is computable. In Proceedings of the 21st Annual Symposium on Computational Geometry, SCG'05. 2005. p. 116-125
Chang, Ee Chien ; Choi, Sung Woo ; Kwon, DoYong ; Park, Hyungju ; Yap, Chee. / Shortest path amidst disc obstacles is computable. Proceedings of the 21st Annual Symposium on Computational Geometry, SCG'05. 2005. pp. 116-125
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