Lower bounds for orthogonal range searching: I. The reporting case

We establish lower bounds on the complexity of orthogonal range reporting in the static case. Given a collection of <italic>n</italic> points in <italic>d</italic>-space and a box [<italic>a</italic><subscrpt>1</subscrpt>, <italic>b</italic><subscrpt>1</subscrpt>] X … X [<italic>a<subscrpt>d</subscrpt></italic>, <italic>b<subscrpt>d</subscrpt></italic>], report every point whose <italic>i</italic>th coordinate lies in [<italic>a<subscrpt>i</subscrpt>, b<subscrpt>i</subscrpt></italic>], for each <italic>i</italic> = l, … , <italic>d</italic>. The collection of points is fixed once and for all and can be preprocessed. The box, on the other hand, constitutes a query that must be answered online. It is shown that on a pointer machine a query time of <italic>O</italic>(<italic>k</italic> + polylog(<italic>n</italic>)), where <italic>k</italic> is the number of points to be reported, can only be achieved at the expense of &OHgr;(<italic>n</italic>(log <italic>n</italic>/log log <italic>n</italic>)<supscrpt><italic>d</italic>-1</supscrpt>) storage. Interestingly, these bounds are optimal in the pointer machine model, but they can be improved (ever so slightly) on a random access machine. In a companion paper, we address the related problem of adding up weights assigned to the points in the query box.