Let S be a subdivision of the plane into polygonal regions, where each region has an associated positive weight. The weighted region shortest path problem is to determine a shortest path in S between two points s,tεR2, where the distances are measured according to the weighted Euclidean metric - the length of a path is defined to be the weighted sum of (Euclidean) lengths of the sub-paths within each region. We show that this problem cannot be solved in the Algebraic Computation Model over the Rational Numbers (ACMQ). In the ACMQ, one can compute exactly any number that can be obtained from the rationals Q by applying a finite number of operations from +, -, ×, ÷, √k, for any integer k≥2. Our proof uses Galois theory and is based on Bajaj's technique.

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Keywords Computational geometry, Galois theory, Unsolvability, Weighted region shortest paths
Persistent URL dx.doi.org/10.1016/j.comgeo.2014.02.004
Journal Computational Geometry
De Carufel, J.-L. (Jean-Lou), Grimm, C. (Carsten), Maheshwari, A, Owen, M. (Megan), & Smid, M. (2014). A note on the unsolvability of the weighted region shortest path problem. Computational Geometry, 47(7), 724–727. doi:10.1016/j.comgeo.2014.02.004