2013/05/22 by Jean-Lou De Carufel, Carsten Grimm, De Carufel, Jean-Lou +7 · 1 citation
Computer Science · #Complexity and Algorithms in Graphs #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Computer and information sciences
paper · pdf · doi:10.48550/arxiv.1305.5209
openalex publication_date 2013/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, t in R2, 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.