New results on shortest paths in three dimensions

We revisit the problem of computing shortest obstacle-avoid-ing paths among obstacles in three dimensions. We prove new hardness results, showing, e.g., that computing Eu-clidean shortest paths among sets of “stacked ” axis-aligned rectangles is NP-complete, and that computing L1-shortest paths among disjoint balls is NP-complete. On the positive side, we present an efficient algorithm for computing an L1-shortest path between two given points that lies on or above a given polyhedral terrain. We also give polynomial-time algorithms for some versions of stacked polygonal obstacles that are “terrain-like ” and analyze the complexity of short-est path maps in the presence of parallel halfplane “walls.