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Title: Shortest path amidst disc obstacles is computable
Authors: Chang, E.-C. 
Choi, S.W.
Kwon, D.
Park, H.
Yap, C.K.
Keywords: Disc obstacles
Exact geometric computation
Exponential complexity
Guaranteed precision computation
Real RAM model
Robust numerical algorithms
Shortest path
Issue Date: 2005
Citation: Chang, E.-C.,Choi, S.W.,Kwon, D.,Park, H.,Yap, C.K. (2005). Shortest path amidst disc obstacles is computable. Proceedings of the Annual Symposium on Computational Geometry : 116-125. ScholarBank@NUS Repository.
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. Copyright 2005 ACM.
Source Title: Proceedings of the Annual Symposium on Computational Geometry
Appears in Collections:Staff Publications

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