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Unconstrained and Curvature-Constrained Shortest-Path Distances and Their Approximation

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Abstract

We study shortest paths and their distances on a subset of a Euclidean space, and their approximation by their equivalents in a neighborhood graph defined on a sample from that subset. In particular, we recover and extend the results of Bernstein et al. (Graph approximations to geodesics on embedded manifolds, Tech. Rep., Department of Psychology, Stanford University, 2000). We do the same with curvature-constrained shortest paths and their distances, establishing what we believe are the first approximation bounds for them.

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Notes

  1. A ‘pseudo-metric’ is like a metric except that it needs not be definite.

  2. For a definition of geodesics, see [23, Chap. 4].

  3. For example, the points where a shortest path switches from the (relative) interior and the boundary can have a closure of positive measure; this is true even in the case of a domain [1].

  4. Here, \(r \nearrow s\) means that r approaches s from the left, and similarly, \(t \searrow s\) means that t approaches s from the right, on the real line.

  5. The computation of curvature-constrained shortest path distances can be done by adapting Dijkstra’s algorithm. It is implemented in [4, Alg. 1].

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Acknowledgements

The authors wish to thank Stephanie Alexander, I. David Berg, Richard Bishop, Dmitri Burago, Bruce Driver, and Bruno Pelletier for very helpful discussions. The paper was carefully read by two anonymous referees, to whom we are grateful. Some of the symbolic calculations were done with Wolfram|Alpha (http://www.wolframalpha.com). This work was partially supported by the US National Science Foundation (DMS 0915160, DMS 1513465).

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Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, CA, 92093, USA

    Ery Arias-Castro

  2. École Centrale de Marseille, 13013, Marseille, France

    Thibaut Le Gouic

  3. Higher School of Economics, National Research University, 125319, Moscow, Russian Federation

    Thibaut Le Gouic

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  1. Ery Arias-Castro
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  2. Thibaut Le Gouic
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Correspondence to Ery Arias-Castro.

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Arias-Castro, E., Le Gouic, T. Unconstrained and Curvature-Constrained Shortest-Path Distances and Their Approximation. Discrete Comput Geom 62, 1–28 (2019). https://doi.org/10.1007/s00454-019-00060-7

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