The Weighted Region Problem: Finding Shortest Paths Through a Weighted Planar Subdivision

The problem of determining shortest paths through a weighted planar polygonal subdivision with n vertices is considered. Distances are measured according to a weighted Euclidean metric: The length of a path is defined to be the weighted sum of 1991 lengths of the subpaths within each region. An algorithm that constructs a (restricted) “shortest path map” with respect to a given source point is presented. The output is a partitioning of each edge of the subdivion into intervals of ε-optimality, allowing an ε-optimal path to be traced from the source to any query point along any edge. The algorithm runs in worst-case time O(ES) and requires O(E) space, where E is the number of “events” in our algorithm and S is the time it takes to run a numerical search procedure. In the worst case, E is bounded above by O(n⁴) (and we give an Ω(n⁴) lower bound), but it is likeky that E will be much smaller in practice. We also show that S is bounded by O(n⁴L), where L is the precision of the problem instance (including the number of bits in the user-specified tolerance ε). Again, the value of S should be smaller in practice. The algorithm applies the “continuous Dijkstra” paradigm and exploits the fact that shortest paths obey Snell's Law of Refraction at region boundaries, a local optimaly property of shortest paths that is well known from the analogous optics model. The algorithm generalizes to the multi-source case to compute Voronoi diagrams.