Abstract
We describe algorithms to compute edge sequences, a shortest path map, and the Fréchet distance for a convex polyhedral surface. Distances on the surface are measured by the length of a Euclidean shortest path. We describe how the star unfolding changes as a source point slides continuously along an edge of the convex polyhedral surface. We describe alternative algorithms to the edge sequence algorithm of Agarwal et al. (SIAM J. Comput. 26(6):1689–1713, 1997) for a convex polyhedral surface. Our approach uses persistent trees, star unfoldings, and kinetic Voronoi diagrams. We also show that the core of the star unfolding can overlap itself when the polyhedral surface is non-convex.
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Notes
If multiple shortest paths exist from s to v j , then any of these shortest paths can be used to represent π(s,v j ) as in [13].
In [1], this dual graph is referred to as the pasting tree.
This construction is based on a near-linear function λ s+2(k) that is defined by the length of a Davenport-Schinzel sequence. Here, s is a constant that represents the maximum number of times that a pair of shapes can intersect, and n is the total number of shapes.
Queries in O(logn) time are also possible by [23] but at the cost of essentially squaring both the time and space preprocessing bounds.
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Cook, A.F., Wenk, C. Shortest Path Problems on a Polyhedral Surface. Algorithmica 69, 58–77 (2014). https://doi.org/10.1007/s00453-012-9723-6
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DOI: https://doi.org/10.1007/s00453-012-9723-6