How hard is half-space range searching?
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- Brönnimann, H., Chazelle, B. & Pach, J. Discrete Comput Geom (1993) 10: 143. doi:10.1007/BF02573971
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Although the bound is unlikely to be optimal, it falls reasonably close to the recent upper bound ofO(n/m1/d) established by Matoušek. We also show that it is possible to devise a sequence ofn inserts and half-space range queries that require a total time ofn2-O(1/d). Our results imply the first nontrivial lower bounds for spherical range searching in any fixed dimension. For example, they show that, with linear storage, circular range queries in the plane require Ω(n1/3) time (modulo a logarithmic factor).