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Algorithmica

pp 1–54 | Cite as

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

  • Danny Z. Chen
  • Haitao Wang
Article
  • 6 Downloads

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.

Keywords

Shortest paths Polygonal domains L1 metric Voronoi diagrams Computational geometry Algorithms and data structures 

Notes

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. 1.Department of Computer Science and EngineeringUniversity of Notre DameNotre DameUSA
  2. 2.Department of Computer ScienceUtah State UniversityLoganUSA

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