A rational rotation method for robust geometric algorithms
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
Robust Proximity Queries: An Illustration of Degree-Driven Algorithm Design
SIAM Journal on Computing
Multidimensional binary search trees used for associative searching
Communications of the ACM
STOC '75 Proceedings of seventh annual ACM symposium on Theory of computing
Proceedings of the twenty-sixth annual symposium on Computational geometry
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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.