Dynamic Set Intersection

Abstract

Consider the problem of maintaining a family F of dynamic sets subject to insertions, deletions, and set-intersection reporting queries: given \(S,S'\in F\), report every member of \(S\cap S'\) 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 \(O(\frac{d}{w/\log ^2 w})\) 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 \(O(m+\frac{m\alpha }{w/\log ^2 w} +t)\) expected time, where \(m=|E|\) and \(\alpha \) is the arboricity of G. This improves a 30-year old triangle enumeration algorithm of Chiba and Nishizeki running in \(O(m \alpha )\) time.

We provide an incremental data structure on F that supports intersection witness queries, where we only need to find one \(e\in S\cap S'\). Both queries and insertions take \(O\left( {\sqrt{\frac{N}{w/\log ^2 w}}}\right) \) expected time, where \(N=\sum _{S\in F} |S|\). Finally, we provide time/space tradeoffs for the fully dynamic set intersection reporting problem. Using M words of space, each update costs \(O(\sqrt{M \log N})\) expected time, each reporting query costs \(O(\frac{N\sqrt{\log N}}{\sqrt{M}}\sqrt{op+1})\) expected time where op is the size of the output, and each witness query costs \(O(\frac{N\sqrt{\log N}}{\sqrt{M}} + \log N)\) expected time.