LOWER BOUNDS FOR SET INTERSECTION QUERIES

We consider the following set intersection reporting problem. We have a collection of initially empty sets and would like to process an intermixed sequence of n updates (insertions into and deletions from individual sets) and q queries (reporting the intersection of two sets). We cast this problem in the arithmetic model of computation of Fredman [F1] and Yao [Ya2] and show that any algorithm that fits in this model must take time Omega(q+n root q) to process a sequence of n updates and q queries, ignoring factors that are polynomial in log n. We also show that this bound is tight in this model of computation, again to within a polynomial in log n factor, improving upon a result of Yellin [Ye]. Furthermore, we consider the case q=O(n) with an additional space restriction. We only allow the use of m memory locations, where m less than or equal to n(3/2). We show a tight bound of Theta(n(2)/m(1/3)) for a sequence of n operations, again ignoring the polynomial in log n factors.