Shortest paths among obstacles in the plane

We give a subquadratic <italic>(O(n<supscrpt>5/3+ε</supscrpt></italic>) time and space) algorithm for computing Euclidean shortest paths in the plane in the presence of polygonal obstacles; previous time bounds were at least quadratic in <italic>n</italic>, in the worst-case. The method avoids use of visibility graphs, relying instead on the continuous Dijkstra paradigm. The output is a shortest path map (of size <italic>O(n)</italic>) with respect to a given source point, which allows shortest path length queries to be answered in time <italic>O(log n)</italic>. The algorithm extends to the case of multiple source points, yielding a geodesic Voronoi diagram within the same time bound.

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