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We present an algorithm for determining the shortest path between a source and a destination through a planar subdivision in which each region has an associated weight. Distances are measured according to a weighted Euclidean metric: Each region of the subdivision has associated with it a weight, and the weighted distance between two points in a convex region is the product of the corresponding weight and the Euclidean distance between them. Our algorithm runs in time Ο(n7 L) and requires Ο(n3) space, where n is the number of edges of the subdivision, and L is the precision of the problem instance (including the number of bits in a user-specified tolerance ∈, which is the percentage the solution is allowed to differ from an optimal solution). The algorithm uses the fact that shortest paths obey Snell's Law of Refraction at region boundaries, a local optimality property of shortest paths that is well-known from the analogous optics model.
Mitchell et al. (Thu,) studied this question.