Dynamic set intersection

Consider the problem of maintaining a family F of dynamic sets subject to insertions, deletions, and set-intersection reporting queries: given S, S′ ∈ F, report every member of S ∩S′ in any order. We show that in the word RAM model, where w is the word size, given a cap d on the maximum size of any set, we can support set intersection queries in O(Equation found) expected time, and updates in O(1) expected time. Using this algorithm we can list all t triangles of a graph G = (V, E) in O(Equation found) expected time, where m = |E| and α is the arboricity of G. This improves a 30-year old triangle enumeration algorithm of Chiba and Nishizeki running in O(mα) time. We provide an incremental data structure on F that supports intersection witness queries, where we only need to find one e ∈ S ∩ S′. Both queries and insertions take O (Equation found) expected time, where N = Σ_{S ∈F} |S|. Finally, we provide time/space tradeoffs for the fully dynamic set intersection reporting problem. Using M words of space, each update costs O(√M logN) expected time, each reporting query costs O(Equation found) expected time where op is the size of the output, and each witness query costs O(Equation found) expected time.