Lower bounds for off-line range searching

This paper proves three lower bounds for variants of the following range-searching problem: given n weighted points in R^d and n 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 log n) is a lower bound on the number of arithmetic operations; (2) if subtractions are disallowed the lower bound becomes Ω(n(log n/log log n)^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).