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 <italic>n</italic> 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 (Euclidean) 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 <italic>O</italic>(<italic>ES</italic>) and requires <italic>O</italic>(<italic>E</italic>) space, where <italic>E</italic> is the number of “events” in our algorithm and <italic>S</italic> is the time it takes to run a numerical search procedure. In the worst case, <italic>E</italic> is bounded above by <italic>O</italic>(<italic>n</italic><supscrpt>4</supscrpt>) (and we give an &OHgr;(<italic>n</><supscrpt>4</supscrpt>) lower bound), but it is likeky that <italic>E</italic> will be much smaller in practice. We also show that <italic>S</italic> is bounded by <italic>O</italic>(<italic>n</italic><supscrpt>4</supscrpt><italic>L</italic>), where <italic>L</italic> is the precision of the problem instance (including the number of bits in the user-specified tolerance ε). Again, the value of <italic>S</italic> 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.

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