An Optimal-Time Algorithm for Shortest Paths on Realistic Polyhedra - Schreiber, Yevgeny (Discrete & computational geometry, 2010)
We generalize our optimal-time algorithm for computing (an implicit representation of) the shortest-path map from a fixed source s on the surface of a convex polytope P to three realistic scenarios where P is a possibly nonconvex polyhedron. In the first scenario, ∂ P is a terrain whose maximum facet slope is bounded by any fixed constant. In the second scenario, P is an uncrowded polyhedron—each axis-parallel square h of side length l ( h ) whose smallest Euclidean distance to a vertex of P is at least l ( h ) is intersected by at most O (1) facets of ∂ P —an input model which, as we show, is a generalization of the well-known low-density model. In the third scenario, P is self-conforming —here, for each edge e of P , there is a connected region R ( e ) of O (1) facets whose union contains e , so that the shortest path distance from e to any edge e ′ of ∂ R ( e ) is at least c ⋅max {| e |,| e ′|}, where c is some positive constant. In particular, it includes the case where each facet of ∂ P is fat and each vertex is incident to at most O (1) facets of ∂ P . In all the above cases the algorithm runs in O ( n log n ) time and space, where n is the number of edges of P , and produces an implicit representation of the shortest-path map, so that the shortest path from s to any query point q can be determined in O (log n ) time. The constants of proportionality depend on the various parameters (maximum facet slope, crowdedness, etc.). We also note that the self-conforming model allows for a major simplification of the algorithm.