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Shortest Path Problems on a Polyhedral Surface

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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

  1. 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].

  2. The core has also been referred to as the kernel or the antarctic in [1, 13]. Note that neither the star unfolding nor its core are necessarily star-shaped [1].

  3. In [1], this dual graph is referred to as the pasting tree.

  4. 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.

  5. 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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Authors and Affiliations

  1. Department of Computer Science, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX, 78249-0667, USA

    Atlas F. Cook IV

  2. Department of Computer Science, Tulane University, 6823 St. Charles Ave., New Orleans, LA, 70118, USA

    Carola Wenk

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  1. Atlas F. Cook IV
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  2. Carola Wenk
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Correspondence to Atlas F. Cook IV.

Additional information

This work has been supported by the National Science Foundation grant NSF CAREER CCF-0643597.

Previous versions of this work have appeared in [18–21].

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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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