Shortest paths on realistic polyhedra
Yevgeny Schreiber · 2007
We generalize our optimal-time algorithm for computing (an implicitrepresentation of) the shortest-path map from a fixed source s onthe surface of a convex polytope P into three scenarios where P is a possibly nonconvex polyhedron. In the first scenario, ∂ P is a terrain whose maximal facet slope is bounded by a constant. In the second scenario P is an uncrowded polyhedron -- each axis-parallel square h of 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 that, as we show, is a generalization of the well-known low density model. In the third scenario P is self-conforming -- that is, for each surface edge e of P, there is only a constant number of facets of ∂ P within shortest path distance O(|e|); in particular, it includes the case that 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. We also show that the self-conforming model allows for a major simplification of the algorithm.