# Optimal Algorithm for Geodesic Nearest-point Voronoi Diagrams in Simple Polygons

@inproceedings{Oh2019OptimalAF, title={Optimal Algorithm for Geodesic Nearest-point Voronoi Diagrams in Simple Polygons}, author={Eunjin Oh}, booktitle={ACM-SIAM Symposium on Discrete Algorithms}, year={2019} }

Given a set of m point sites in a simple polygon, the geodesic nearest-point Voronoi diagram of the sites partitions the polygon into m Voronoi cells, one cell per site, such that every point in a cell has the same nearest site under the geodesic metric. In this paper, we present an O(n + m log m)-time algorithm for computing the geodesic nearest-point Voronoi diagram of m points in a simple n-gon. This matches the best known lower bound of Ω(n + m log m) as well as improving the previously…

## 7 Citations

### An Optimal Deterministic Algorithm for Geodesic Farthest-Point Voronoi Diagrams in Simple Polygons

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- 2021

A deterministic algorithm of O(n + m log m) time is presented, which is optimal and answers an open question posed by Mitchell in the Handbook of Computational Geometry two decades ago.

### Optimal Algorithm for Geodesic Farthest-Point Voronoi Diagrams

- MathematicsSoCG
- 2019

The first randomized algorithm to compute the geodesic farthest-point Voronoi diagram of S in P running in expected O(n + m) time is presented, which is a partition of P into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to theGeodesic distance.

### Kinetic Geodesic Voronoi Diagrams in a Simple Polygon

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A kinetic data structure is developed that maintains the geodesic Voronoi diagram as the sites move, and it is proved that the number of such changes is at most $O(m^3)$, and that this is tight in the worst case.

### Farthest-Point Voronoi Diagrams in the Presence of Rectangular Obstacles

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- 2022

This work presents an algorithm to compute the geodesic L 1, the farthest-point Voronoi diagram of m point sites in the presence of n rectangular obstacles in the plane that takes O ( nm) space and can construct a data structure in the same construction time and space that answers a farthest neighbours query.

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This paper builds a shortest path map for a source point s, so that given any query point t, the shortest path length from s to t can be computed in O(logn) time and a shortest s-t path can be produced in additional time linear in the number of edges of the path.

### Towards Space Efficient Two-Point Shortest Path Queries in a Polygonal Domain

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The main result can be extended to include a space-time trade-off and the first improvement upon a conference paper by Chiang and Mitchell from 1999 is improved.

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The problem of finding an obstacle-avoiding Euclidean shortest path between two points is solved by solving the problem in O(n\log n) time and O( n) space, which is optimal in both time and space.