Discrete & Computational Geometry

, Volume 10, Issue 2, pp 143-155

First online:

How hard is half-space range searching?

  • Hervé BrönnimannAffiliated withDepartment of Computer Science, Princeton University
  • , Bernard ChazelleAffiliated withDepartment of Computer Science, Princeton University
  • , János PachAffiliated withDepartment of Computer Science, City College, C.U.N.Y.Courant Institute, N.Y.U.

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We investigate the complexity ofhalf-space range searching: givenn points ind-space, build a data structure that allows us to determine efficiently how many points lie in a query half-space. We establish a tradeoff between the storagem and the worst-case query timet in the Fredman/Yao arithmetic model of computation. We show thatt must be at least on the order of
$$\frac{{(n/\log n)^{1 - (d - 1)/d(d + 1)} }}{{m^{1/d} }}$$

Although the bound is unlikely to be optimal, it falls reasonably close to the recent upper bound ofO(n/m 1/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 ofn 2-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(n 1/3) time (modulo a logarithmic factor).