Can Nearest Neighbor Searching Be Simple and Always Fast?
Nearest Neighbor Searching, i.e. determining from a set S of n sites in the plane the one that is closest to a given query point q, is a classical problem in computational geometry. Fast theoretical solutions are known, e.g. point location in the Voronoi Diagram of S, or specialized structures such as so-called Delaunay hierarchies. However, practitioners tend to deem these solutions as too complicated or computationally too costly to be actually useful.
Recently in ALENEX 2010 Birn et al. proposed a simple and practical randomized solution. They reported encouraging experimental results and presented a partial performance analysis. They argued that in many cases their method achieves logarithmic expected query time but they also noted that in some cases linear expected query time is incurred. They raised the question whether some variant of their approach can achieve logarithmic expected query time in all cases.
The approach of Birn et al. derives its simplicity mostly from the fact that it applies only one simple type of geometric predicate: which one of two sites in S is closer to the query point q. In this paper we show that any method for planar nearest neighbor searching that relies just on this one type of geometric predicate can be forced to make at least n − 1 such predicate evaluations during a worst case query.
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- 2.Birn, M., Holtgrewe, M., Sanders, P., Singler, J.: Simple and Fast Nearest Neighbor Search. In: Proceedings of the Twelfth Workshop on Algorithm Engineering and Experiments (ALENEX). SIAM, Philadelphia (2010)Google Scholar
- 3.Canny, J.F., Donald, B.R., Ressler, E.K.: A Rational Rotation Method for Robust Geometric Algorithms. In: Proc. of the Eigth ACM Symposium on Computational Geometry (SOCG), pp. 251–260 (1992)Google Scholar
- 7.Millman, D., Snoeyink, J.: Computing Planar Voronoi Diagrams in Double Precision: A Further Example of Degree-driven Algorithm Design. In: Proceedings of the Annual Symposium on Computational Geometry (SoCG), pp. 386–392 (2010)Google Scholar
- 8.Mount, D.M., Arya, S.: ANN: A Library for Approximate Nearest Neighbor Searching. In: CGC 2nd Annual Fall Workshop on Computational Geometry (1997)Google Scholar
- 9.Shamos, M. I.: Geometric Complexity. In: Proceedings of Seventh Annual ACM Symposium on Theory of Computing (STOC), pp. 224–233 (1975)Google Scholar