On the difficulty of range searching

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We consider the general problem of (2-dimensional) range reporting allowing arbitrarily convex queries. We show that using a traditional approach, even when incorporating techniques like those used in fusion trees, a polylogarithmic query time cannot be achieved unless more than linear space is used. Our arguments are based on a new non-trivial lower bound in a model of computation which, in contrast to the pointer machine model, allows for the use of arrays and bit manipulation. The crucial property of our model, Layered partitions , is that it can be used to describe all known algorithms for processing range queries, as well as many other data structures used to represent multi-dimensional data. We show that Ω( log n log T(n) ) partitions must be used to allow queries in O( T ( n ) + k ) time, for n total and k reported elements, and for any growing function T ( n ).