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Shortest Paths in the Plane with Obstacle Violations

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Abstract

We study the problem of finding shortest paths in the plane among h convex obstacles, where the path is allowed to pass through (violate) up to k obstacles, for \(k \le h\). Equivalently, the problem is to find shortest paths that become obstacle-free if k obstacles are removed from the input. Given a fixed source point s, we show how to construct a map, called a shortest k-path map, so that all destinations in the same region of the map have the same combinatorial shortest path passing through at most k obstacles. We prove a tight bound of \(\varTheta (kn)\) on the size of this map, and show that it can be computed in \(O(k^2n \log n)\) time, where n is the total number of obstacle vertices.

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Notes

  1. The garage metaphor is also used in the context of finding homotopically different paths in [12], but the properties and technical details of our k-garage are quite different.

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Acknowledgements

Funding was provided by National Science Foundation (Grant No. CCF-1525817).

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Authors and Affiliations

  1. Mentor Graphics Corp., Wilsonville, OR, USA

    John Hershberger

  2. University of California, Santa Barbara, CA, USA

    Neeraj Kumar & Subhash Suri

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  1. John Hershberger
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  2. Neeraj Kumar
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Correspondence to Neeraj Kumar.

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A preliminary version of this paper [17] appeared in the Proceedings of the 25th European Symposium of Algorithms (ESA), 2017.

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Hershberger, J., Kumar, N. & Suri, S. Shortest Paths in the Plane with Obstacle Violations. Algorithmica 82, 1813–1832 (2020). https://doi.org/10.1007/s00453-020-00673-y

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