Abstract
In the online setdisjointness problem the goal is to preprocess a family of sets ℱ, so that given two sets S,S' ∈ ℱ, one can quickly establish whether the two sets are disjoint or not. If N = ∑_{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 stateoftheart exhibits a large gap.
The online setintersection problem is the reporting version of the online setdisjointness 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 setdisjointess can be extended to solve online setintersection 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 setdisjointness 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 setintersection, 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.
BibTeX  Entry
@InProceedings{kopelowitz_et_al:LIPIcs:2020:12481,
author = {Tsvi Kopelowitz and Virginia Vassilevska Williams},
title = {{Towards Optimal SetDisjointness and SetIntersection Data Structures}},
booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
pages = {74:174:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771382},
ISSN = {18688969},
year = {2020},
volume = {168},
editor = {Artur Czumaj and Anuj Dawar and Emanuela Merelli},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12481},
URN = {urn:nbn:de:0030drops124813},
doi = {10.4230/LIPIcs.ICALP.2020.74},
annote = {Keywords: Setdisjointness data structures, Triangle detection, Triangle enumeration, Finegrained complexity, Fast matrix multiplication}
}
Keywords: 

Setdisjointness data structures, Triangle detection, Triangle enumeration, Finegrained complexity, Fast matrix multiplication 
Collection: 

47th International Colloquium on Automata, Languages, and Programming (ICALP 2020) 
Issue Date: 

2020 
Date of publication: 

29.06.2020 