On the difficulty of range searching
The problem of range searching is fundamental and well studied, and a large number of solutions have been suggested in the literature. The only existing non-trivial lower bound that closely matches known upper bounds with respect to time/space tradeoff is given for the pointer machine model. However, the pointer machine prohibits a number of possible and natural operations, such as the use of arrays and bit manipulation. In particular, such operations have proven useful in some special cases such as one-dimensional and rectilinear queries.
In this article, 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 (poly-) logarithmic query time can not 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, where k is the number of reported elements, for any growing function T(n). In some special cases, as for rectilinear queries, these partitions may be stored in compressed form, which has been exploited by the M-structure of Chazelle. However, so far there has been no indication that such compression would be feasible in the general case, in which case any algorithm based on our model, and supporting range searching in O(logcn+k) time requires Ω (n log n/log log n) space. (Note that it may be possible to obtain a better upper bound with an algorithm not adhering to the model of layered partitions.) Hence, we show that removing the restrictions of the pointer machine model does not help in obtaining a significantly improved time/space tradeoff — any solution based on traditional representations of point sets cannot combine linear space and polylogarithmic time.
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