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Computing \(L_1\) Shortest Paths Among Polygonal Obstacles in the Plane

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Abstract

Given a point s and a set of h pairwise disjoint polygonal obstacles with a total of n vertices in the plane, suppose a triangulation of the space outside the obstacles is given; we present an \(O(n+h\log h)\) time and O(n) space algorithm for building a data structure (called shortest path map) of size O(n) such that for any query point t, the length of an \(L_1\) shortest obstacle-avoiding path from s to t can be computed in \(O(\log n)\) time and the actual path can be reported in additional time proportional to the number of edges of the path. The previously best algorithm computes such a shortest path map in \(O(n\log n)\) time and O(n) space. So our algorithm is faster when h is relatively small. Further, our techniques can be extended to obtain improved results for other related problems, e.g., computing the \(L_1\) geodesic Voronoi diagram for a set of point sites among the obstacles.

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Notes

  1. For example, since the Euclidean shortest path between any two points in a simple polygon is also an \(L_1\) shortest path [17], a Euclidean SPM in a simple polygon is also an \(L_1\) one. Thus, we can use a corresponding shortest path algorithm for the Euclidean case (e.g., [16]) to compute an \(L_1\) SPM in \(bay({\overline{cd}})\) with respect to r in linear time.

  2. This was pointed out by an anonymous reviewer. Indeed, if \(m\le \sqrt{n}\), then \(m\log (n+m)=O(n)\); otherwise, \(m\log (n+m)=O(m\log m)\).

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Acknowledgements

We would like to thank an anonymous reviewer for numerous suggestions that significantly improve the presentation of the paper. D.Z. Chen was supported in part by NSF under Grants CCF-0916606, CCF-1217906, and CCF-1617735. H. Wang was supported in part by NSF under Grant CCF-1317143.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, 46556, USA

    Danny Z. Chen

  2. Department of Computer Science, Utah State University, Logan, UT, 84322, USA

    Haitao Wang

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  1. Danny Z. Chen
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  2. Haitao Wang
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Correspondence to Haitao Wang.

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Preliminary results of this paper appeared in the Proceedings of the 19th European Symposium on Algorithms (ESA 2011) and the Proceedings of the 30th Symposium on Theoretical Aspects of Computer Science (STACS 2013).

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Chen, D.Z., Wang, H. Computing \(L_1\) Shortest Paths Among Polygonal Obstacles in the Plane. Algorithmica 81, 2430–2483 (2019). https://doi.org/10.1007/s00453-018-00540-x

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