Towards Optimal Dynamic Indexes for Approximate (and Exact) Triangle Counting

In ICDT’19, Kara, Ngo, Nikolic, Olteanu, and Zhang gave a structure which maintains the number T of triangles in an undirected graph G = (V, E) along with the edge insertions/deletions in G. Using O(m) space (m = |E|), their structure supports an update in O( √ m log m) amortized time which is optimal (up to polylog factors) subject to the OMv-conjecture (Henzinger, Krinninger, Nanongkai, and Saranurak, STOC’15). Aiming to improve the update efficiency, we study: the optimal tradeoff between update time and approximation quality. We require a structure to provide the (ε, Γ)-guarantee: when queried, it should return an estimate t of T that has relative error at most ε if T ≥ Γ, or an absolute error at most ε · Γ, otherwise. We prove that, under any ε ≤ 0.49 and subject to the OMv-conjecture, no structure can guarantee O(m0.5−δ/Γ) expected amortized update time and O(m2/3−δ) query time simultaneously for any constant δ > 0; this is true for Γ = m of any constant c in [0, 1/2). We match the lower bound with a structure that ensures Õ((1/ε)3 · √ m/Γ) amortized update time with high probability, and O(1) query time. (for exact counting) how to achieve arboricity-sensitive update time. For any 1 ≤ Γ ≤ √ m, we describe a structure of O(min{αm + m log m, (m/Γ)2}) space that maintains T precisely, and supports an update in Õ(min{α + Γ, √ m}) amortized time, where α is the largest arboricity of G in history (and does not need to be known). Our structure reconstructs the aforementioned ICDT’19 result up to polylog factors by setting Γ = √ m, but achieves Õ(m0.5−δ) update time as long as α = O(m0.5−δ). 2012 ACM Subject Classification Theory of computation → Database query processing and optimization (theory)