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A limit field for orthogonal range searches in two-dimensional random point search trees. (English) Zbl 1422.60020
Summary: We consider the cost of general orthogonal range queries in random quadtrees. The cost of a given query is encoded into a (random) function of four variables which characterize the coordinates of two opposite corners of the query rectangle. We prove that, when suitably shifted and rescaled, the random cost function converges uniformly in probability towards a random field that is characterized as the unique solution to a distributional fixed-point equation. We also state similar results for 2-d trees. Our results imply for instance that the worst case query satisfies the same asymptotic estimates as a typical query, and thereby resolve an open question of P. Chanzy et al. [Acta Inf. 37, No. 4–5, 355–383 (2001; Zbl 0970.68046)].
MSC:
60C05 Combinatorial probability
60F17 Functional limit theorems; invariance principles
68P20 Information storage and retrieval of data
60D05 Geometric probability and stochastic geometry
60G60 Random fields
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