Exact solutions to the Weighted Region Problem
This paper investigates the exact computability of shortest paths in weighted rectangular domains. In the rational algebraic computation model, we establish—for the first time—that the globally shortest path in a single rectangular domain with piecewise nonnegative weights (where path cost equals Euclidean length multiplied by weight) is algorithmically undecidable. Method: For source points located either on the boundary or in the interior, we explicitly construct and derive algebraic equations for bisectors in the shortest path map (SPM); their coefficients are rational functions of the input parameters. Leveraging algebraic computation theory, implicit curve analysis, and structural characterization of SPMs, we develop a complete analytic framework for exact shortest paths. Results: Our work rigorously delineates the boundary of exact solvability for this problem and provides the first bisector computation framework implementable within the rational algebraic model.