Abstract
We prove that the red—blue discrepancy of the set system formed by n points and n axis-parallel boxes in <bo>R</bo>d can be as high as n Ω(1) in any dimension d= Ω(log n) . This contrasts with the fixed-dimensional case d=O(1) , where the discrepancy is always polylogarithmic. More generally we show that in any dimension 1<d= O(log n) the maximum discrepancy is 2 Ω(d) . Our result also leads to a new lower bound on the complexity of off-line orthogonal range searching. This is the problem of summing up weights in boxes, given n weighted points and n boxes in <bo>R</bo>d . We prove that the number of arithmetic operations is at least Ω(nd+ nlog log n) in any dimension d=O(log n) .
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Received June 30, 2000, and in revised form November 9, 2000. Online publication April 6, 2001.
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Chazelle, B., Lvov, A. The Discrepancy of Boxes in Higher Dimension. Discrete Comput Geom 25, 519–524 (2001). https://doi.org/10.1007/s00454-001-0014-2
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DOI: https://doi.org/10.1007/s00454-001-0014-2