Lower bounds for off-line range searching

This paper proves three lower bounds for variants of the following rangesearching problem: Given n weighted points inR ^d andn axis-parallel boxes, compute the sum of the weights within each box: (1) if both additions and subtractions are allowed, we prove that Ω(n log logn) is a lower bound on the number of arithmetic operations; (2) if subtractions are disallowed the lower bound becomes Ω(n(logn/loglogn)^d-1), which is nearly optimal; (3) finally, for the case where boxes are replaced by simplices, we establish a quasi-optimal lower bound of Ω(n^2-2/(d+1))/polylog(n). A preliminary version of this paper appeared inProc. 27th Ann. ACM Symp. on Theory of Computing, May 1995, pp. 733–740. This work was supported in part by NSF Grant CCR-93-01254 and the Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc.