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# A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon

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LIPIcs.SoCG.2018.58.pdf
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## Cite As

Chih-Hung Liu. A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SoCG.2018.58

## Abstract

The geodesic Voronoi diagram of m point sites inside a simple polygon of n vertices is a subdivision of the polygon into m cells, one to each site, such that all points in a cell share the same nearest site under the geodesic distance. The best known lower bound for the construction time is Omega(n+m log m), and a matching upper bound is a long-standing open question. The state-of-the-art construction algorithms achieve O((n+m)log (n+m)) and O(n+m log m log^2n) time, which are optimal for m=Omega(n) and m=O(n/(log^3n)), respectively. In this paper, we give a construction algorithm with O(n+m(log m+log^2 n)) time, and it is nearly optimal in the sense that if a single Voronoi vertex can be computed in O(log n) time, then the construction time will become the optimal O(n+m log m). In other words, we reduce the problem of constructing the diagram in the optimal time to the problem of computing a single Voronoi vertex in O(log n) time.
##### Keywords
• Simple polygons
• Voronoi diagrams
• Geodesic distance

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## References

1. Boris Aronov. On the geodesic Voronoi diagram of point sites in a simple polygon. Algorithmica, 4(1-4):109-140, 1989.
2. Bernard Chazelle. Triangulating a simple polygon in linear time. Discrete & Computational Geometry, 6:485-524, 1991.
3. Herbert Edelsbrunner, Leonidas J. Guibas, and Jorge Stolfi. Optimal point location in a monotone subdivision. SIAM Journal on Computing, 15(2):317-340, 1986.
4. Leonidas J. Guibas and John Hershberger. Optimal shortest path queries in a simple polygon. Journal of Computer and System Sciences, 39(2):126-152, 1989.
5. Leonidas J. Guibas, John Hershberger, Daniel Leven, Micha Sharir, and Robert E. Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2(1-4):209-233, 1987.
6. John Hershberger. A new data structure for shortest path queries in a simple polygon. Information Processing Letters, 38(5):231-235, 1991.
7. Kurt Mehlhorn and Peter Sanders. Sorted sequences. In Algorithms and Data Structures: The Basic Toolbox. Springer-Verlag Berlin Heidelberg, 2008.
8. Joseph S. B. Mitchell. Geometric shortest paths and network optimization. In Handbook of Computational Geometry, pages 633-701. Elsevier, 2000.
9. Eunjin Oh and Hee-Kap Ahn. Voronoi diagrams for a moderate-sized point-set in a simple polygon. In 33rd International Symposium on Computational Geometry, SoCG 2017, July 4-7, 2017, Brisbane, Australia, pages 52:1-52:15, 2017.
10. Evanthia Papadopoulou and D. T. Lee. A new approach for the geodesic Voronoi diagram of points in a simple polygon and other restricted polygonal domains. Algorithmica, 20(4):319-352, 1998.
11. Robert Endre Tarjan. Updating a balanced search tree in O(1) rotations. Information Processing Letters, 16(5):253-257, 1983.
12. Robert Endre Tarjan. Efficient Top-Down Updating of Red-Black Trees. Technical report, Technical Report TR-006-85. Dapartment of Computer Science, Princeton University, 1985.
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