Towards optimal set-disjointness and set-intersection data structures

In the online set-disjointness problem the goal is to preprocess a family of sets F, so that given two sets S, S⁰ ∈ F, one can quickly establish whether the two sets are disjoint or not. If N = ^P_S∈F |S|, then let N^p be the preprocessing time and let N^q be the query time. The most efficient known combinatorial algorithm is a generalization of an algorithm by Cohen and Porat [TCS'10] which has a tradeoff curve of p + q = 2. Kopelowitz, Pettie, and Porat [SODA'16] showed that, based on the 3SUM hypothesis, there is a conditional lower bound curve of p + 2q ≥ 2. Thus, the current state-of-the-art exhibits a large gap. The online set-intersection problem is the reporting version of the online set-disjointness problem, and given a query, the goal is to report all of the elements in the intersection. When considering algorithms with N^p preprocessing time and N^q +O(op) query time, where op is the size of the output, the combinatorial algorithm for online set-disjointess can be extended to solve online set-intersection with a tradeoff curve of p + q = 2. Kopelowitz, Pettie, and Porat [SODA'16] showed that, assuming the 3SUM hypothesis, for 0 ≤ q ≤ 2/3 this curve is tight. However, for 2/3 ≤ q < 1 there is no known lower bound. In this paper we close both gaps by showing the following: For online set-disjointness we design an algorithm whose runtime, assuming ω = 2 (where ω is the exponent in the fastest matrix multiplication algorithm), matches the lower bound curve of Kopelowitz et al., for q ≤ 1/3. We then complement the new algorithm by a matching conditional lower bound for q > 1/3 which is based on a natural hypothesis on the time required to detect a triangle in an unbalanced tripartite graph. Remarkably, even if ω > 2, the algorithm matches the lower bound curve of Kopelowitz et al. for p ≥ 1.73688 and q ≤ 0.13156. For set-intersection, we prove a conditional lower bound that matches the combinatorial upper bound curve for q ≥ 1/2 which is based on a hypothesis on the time required to enumerate all triangles in an unbalanced tripartite graph. Finally, we design algorithms for detecting and enumerating triangles in unbalanced tripartite graphs which match the lower bounds of the corresponding hypotheses, assuming ω = 2.