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Consider the following special case of orthogonal range searching:

Given a set $S$ of $n$ points in $d$ dimensions, and rectangular queries with a fixed "upper-left" rectangle corner $(0,0,...0)$, report the total number of points inside the rectangle.

This is different in that all rectangles queried have a fixed corner. In this case, will we have a dynamic algorithm with a better runtime, or is the problem still as hard as general orthogonal range searching?

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The count of any rectangle $C(U, L, D, R)$ can be obtained combining the counts of 4 fixed "upper-left" retangles:

$C(U, L, D, R) = C(0, 0, D, R) - C(0, 0, U, R) - C(U, 0, D, L) + C(0, 0, U, L)$,

noting $C(up, left, down, right)$ the count in the rectangle delimited by $(up, left, down, right)$.

Thus, you cannot expect more than a factor 4 runtime improvement.

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