I/O-efficient data structures for colored range and prefix reporting

Motivated by information retrieval applications, we consider the one-dimensional colored range reporting problem in rank space. The goal is to build a static data structure for sets C1,...,Cm ⊆ {1,...,σ} that supports queries of the kind: Given indices a, b, report the set [EQUATION]. We study the problem in the I/O model, and show that there exists an optimal linear-space data structure that answers queries in O(1 + k/B) I/Os, where k denotes the output size and B the disk block size in words. In fact, we obtain the same bound for the harder problem of three-sided orthogonal range reporting. In this problem, we are to preprocess a set of n two-dimensional points in rank space, such that all points inside a query rectangle of the form [x1, x2] x (−∞, y] can be reported. The best previous bounds for this problem is either O(n lg2B n) space and O(1 + k/B) query I/Os, or O(n) space and O(lg(h)B n + k/B) query I/Os, where lg(h)B n is the base B logarithm iterated h times, for any constant integer h. The previous bounds are both achieved under the indivisibility assumption, while our solution exploits the full capabilities of the underlying machine. Breaking the indivisibility assumption thus provides us with cleaner and optimal bounds. Our results also imply an optimal solution to the following colored prefix reporting problem. Given a set S of strings, each O(1) disk blocks in length, and a function c: S → 2>{1,...,σ}, support queries of the kind: Given a string p, report the set [EQUATION], where p* denotes the set of strings with prefix p. Finally, we consider the possibility of top-k extensions of this result, and present a simple solution in a model that allows non-blocked I/O.