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

Abstract

We establish lower bounds on the complexity of orthogonal range reporting in the static case. Given a collection of n points in d -space and a box [ a 1 , b 1 ] X … X [ a d , b d ], report every point whose i th coordinate lies in [ a i , b i ], for each i = l, … , d . 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 O ( k + polylog( n )), where k is the number of points to be reported, can only be achieved at the expense of Ω( n (log n /log log n ) d -1 ) 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.