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- W2090626328 abstract "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(nlog 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." @default.
- W2090626328 created "2016-06-24" @default.
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- W2090626328 date "2009-01-30" @default.
- W2090626328 modified "2023-09-23" @default.
- W2090626328 title "An Optimal-Time Algorithm for Shortest Paths on Realistic Polyhedra" @default.
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- W2090626328 doi "https://doi.org/10.1007/s00454-009-9136-8" @default.
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