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An Optimal Algorithm for Minimum-Link Rectilinear Paths in Triangulated Rectilinear Domains

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Abstract

We present a new algorithm for finding minimum-link rectilinear paths among rectilinear obstacles in the plane. Given a triangulated rectilinear domain of h pairwise-disjoint rectilinear obstacles with a total of n vertices, our algorithm can find a minimum-link rectilinear path between any two points in \(O(n+ h \log h)\) time. Further, within the same time our algorithm can build an O(n)-size data structure for any source point s, such that given any query point t, the number of edges of a minimum-link rectilinear path from s to t can be computed in \(O(\log n)\) time and the actual path can be output in additional time linear in the number of the edges of the path. The previously best algorithms for the problems run in \(O(n \log n)\) time.

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Notes

  1. Alternatively, one may replace \(dis(e_r)\ne 2i-1\) by \(dis(e_r)>2i-1\) for the definition, but we choose to use \(dis(e_r)\ne 2i-1\) only for making our discussion later easier.

  2. A possible improvement might be to implement the split operations using finger search trees as in [19], although this would not affect the running time of the overall algorithm asymptotically.

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Acknowledgements

The authors wish to thank the anonymous reviewers for their suggestions that help improve the presentation of the paper. J. Mitchell acknowledges support from the US-Israel Binational Science Foundation (Grant 2010074) and the National Science Foundation (CCF-1018388, CCF-1526406). H. Wang was supported in part by NSF under Grant CCF-1317143.

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Authors and Affiliations

  1. Stony Brook University, Stony Brook, NY, 11794, USA

    Joseph S. B. Mitchell

  2. Linköping University, Linköping, Sweden

    Valentin Polishchuk

  3. Google, Zurich, Switzerland

    Mikko Sysikaski

  4. Utah State University, Logan, UT, 84322, USA

    Haitao Wang

Authors
  1. Joseph S. B. Mitchell
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  2. Valentin Polishchuk
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  3. Mikko Sysikaski
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  4. Haitao Wang
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Corresponding author

Correspondence to Haitao Wang.

Additional information

A preliminary version of this paper appeared in the Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming (ICALP 2015).

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Mitchell, J.S.B., Polishchuk, V., Sysikaski, M. et al. An Optimal Algorithm for Minimum-Link Rectilinear Paths in Triangulated Rectilinear Domains. Algorithmica 81, 289–316 (2019). https://doi.org/10.1007/s00453-018-0446-1

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